example of descriptive quantitative research paper

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• What is descriptive research?

Last updated

5 February 2023

Reviewed by

Cathy Heath

Descriptive research is a common investigatory model used by researchers in various fields, including social sciences, linguistics, and academia.

Read on to understand the characteristics of descriptive research and explore its underlying techniques, processes, and procedures.

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Descriptive research is an exploratory research method. It enables researchers to precisely and methodically describe a population, circumstance, or phenomenon.

As the name suggests, descriptive research describes the characteristics of the group, situation, or phenomenon being studied without manipulating variables or testing hypotheses . This can be reported using surveys , observational studies, and case studies. You can use both quantitative and qualitative methods to compile the data.

Besides making observations and then comparing and analyzing them, descriptive studies often develop knowledge concepts and provide solutions to critical issues. It always aims to answer how the event occurred, when it occurred, where it occurred, and what the problem or phenomenon is.

• Characteristics of descriptive research

The following are some of the characteristics of descriptive research:

Quantitativeness

Descriptive research can be quantitative as it gathers quantifiable data to statistically analyze a population sample. These numbers can show patterns, connections, and trends over time and can be discovered using surveys, polls, and experiments.

Qualitativeness

Descriptive research can also be qualitative. It gives meaning and context to the numbers supplied by quantitative descriptive research .

Researchers can use tools like interviews, focus groups, and ethnographic studies to illustrate why things are what they are and help characterize the research problem. This is because it’s more explanatory than exploratory or experimental research.

Uncontrolled variables

Descriptive research differs from experimental research in that researchers cannot manipulate the variables. They are recognized, scrutinized, and quantified instead. This is one of its most prominent features.

Cross-sectional studies

Descriptive research is a cross-sectional study because it examines several areas of the same group. It involves obtaining data on multiple variables at the personal level during a certain period. It’s helpful when trying to understand a larger community’s habits or preferences.

Carried out in a natural environment

Descriptive studies are usually carried out in the participants’ everyday environment, which allows researchers to avoid influencing responders by collecting data in a natural setting. You can use online surveys or survey questions to collect data or observe.

Basis for further research

You can further dissect descriptive research’s outcomes and use them for different types of investigation. The outcomes also serve as a foundation for subsequent investigations and can guide future studies. For example, you can use the data obtained in descriptive research to help determine future research designs.

• Descriptive research methods

There are three basic approaches for gathering data in descriptive research: observational, case study, and survey.

You can use surveys to gather data in descriptive research. This involves gathering information from many people using a questionnaire and interview .

Surveys remain the dominant research tool for descriptive research design. Researchers can conduct various investigations and collect multiple types of data (quantitative and qualitative) using surveys with diverse designs.

You can conduct surveys over the phone, online, or in person. Your survey might be a brief interview or conversation with a set of prepared questions intended to obtain quick information from the primary source.

Observation

This descriptive research method involves observing and gathering data on a population or phenomena without manipulating variables. It is employed in psychology, market research , and other social science studies to track and understand human behavior.

Observation is an essential component of descriptive research. It entails gathering data and analyzing it to see whether there is a relationship between the two variables in the study. This strategy usually allows for both qualitative and quantitative data analysis.

Case studies

A case study can outline a specific topic’s traits. The topic might be a person, group, event, or organization.

It involves using a subset of a larger group as a sample to characterize the features of that larger group.

You can generalize knowledge gained from studying a case study to benefit a broader audience.

This approach entails carefully examining a particular group, person, or event over time. You can learn something new about the study topic by using a small group to better understand the dynamics of the entire group.

• Types of descriptive research

There are several types of descriptive study. The most well-known include cross-sectional studies, census surveys, sample surveys, case reports, and comparison studies.

Case reports and case series

In the healthcare and medical fields, a case report is used to explain a patient’s circumstances when suffering from an uncommon illness or displaying certain symptoms. Case reports and case series are both collections of related cases. They have aided the advancement of medical knowledge on countless occasions.

The normative component is an addition to the descriptive survey. In the descriptive–normative survey, you compare the study’s results to the norm.

Descriptive survey

This descriptive type of research employs surveys to collect information on various topics. This data aims to determine the degree to which certain conditions may be attained.

You can extrapolate or generalize the information you obtain from sample surveys to the larger group being researched.

Correlative survey

Correlative surveys help establish if there is a positive, negative, or neutral connection between two variables.

Performing census surveys involves gathering relevant data on several aspects of a given population. These units include individuals, families, organizations, objects, characteristics, and properties.

During descriptive research, you gather different degrees of interest over time from a specific population. Cross-sectional studies provide a glimpse of a phenomenon’s prevalence and features in a population. There are no ethical challenges with them and they are quite simple and inexpensive to carry out.

Comparative studies

These surveys compare the two subjects’ conditions or characteristics. The subjects may include research variables, organizations, plans, and people.

Comparison points, assumption of similarities, and criteria of comparison are three important variables that affect how well and accurately comparative studies are conducted.

For instance, descriptive research can help determine how many CEOs hold a bachelor’s degree and what proportion of low-income households receive government help.

• Pros and cons

The primary advantage of descriptive research designs is that researchers can create a reliable and beneficial database for additional study. To conduct any inquiry, you need access to reliable information sources that can give you a firm understanding of a situation.

Quantitative studies are time- and resource-intensive, so knowing the hypotheses viable for testing is crucial. The basic overview of descriptive research provides helpful hints as to which variables are worth quantitatively examining. This is why it’s employed as a precursor to quantitative research designs.

Some experts view this research as untrustworthy and unscientific. However, there is no way to assess the findings because you don’t manipulate any variables statistically.

Cause-and-effect correlations also can’t be established through descriptive investigations. Additionally, observational study findings cannot be replicated, which prevents a review of the findings and their replication.

The absence of statistical and in-depth analysis and the rather superficial character of the investigative procedure are drawbacks of this research approach.

• Descriptive research examples and applications

Several descriptive research examples are emphasized based on their types, purposes, and applications. Research questions often begin with “What is …” These studies help find solutions to practical issues in social science, physical science, and education.

Here are some examples and applications of descriptive research:

Determining consumer perception and behavior

Organizations use descriptive research designs to determine how various demographic groups react to a certain product or service.

For example, a business looking to sell to its target market should research the market’s behavior first. When researching human behavior in response to a cause or event, the researcher pays attention to the traits, actions, and responses before drawing a conclusion.

Scientific classification

Scientific descriptive research enables the classification of organisms and their traits and constituents.

Measuring data trends

A descriptive study design’s statistical capabilities allow researchers to track data trends over time. It’s frequently used to determine the study target’s current circumstances and underlying patterns.

Conduct comparison

Organizations can use a descriptive research approach to learn how various demographics react to a certain product or service. For example, you can study how the target market responds to a competitor’s product and use that information to infer their behavior.

• Bottom line

A descriptive research design is suitable for exploring certain topics and serving as a prelude to larger quantitative investigations. It provides a comprehensive understanding of the “what” of the group or thing you’re investigating.

This research type acts as the cornerstone of other research methodologies . It is distinctive because it can use quantitative and qualitative research approaches at the same time.

What is descriptive research design?

Descriptive research design aims to systematically obtain information to describe a phenomenon, situation, or population. More specifically, it helps answer the what, when, where, and how questions regarding the research problem rather than the why.

How does descriptive research compare to qualitative research?

Despite certain parallels, descriptive research concentrates on describing phenomena, while qualitative research aims to understand people better.

How do you analyze descriptive research data?

Data analysis involves using various methodologies, enabling the researcher to evaluate and provide results regarding validity and reliability.

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• Descriptive Research Designs: Types, Examples & Methods

One of the components of research is getting enough information about the research problem—the what, how, when and where answers, which is why descriptive research is an important type of research. It is very useful when conducting research whose aim is to identify characteristics, frequencies, trends, correlations, and categories.

This research method takes a problem with little to no relevant information and gives it a befitting description using qualitative and quantitative research method s. Descriptive research aims to accurately describe a research problem.

In the subsequent sections, we will be explaining what descriptive research means, its types, examples, and data collection methods.

What is Descriptive Research?

Descriptive research is a type of research that describes a population, situation, or phenomenon that is being studied. It focuses on answering the how, what, when, and where questions If a research problem, rather than the why.

This is mainly because it is important to have a proper understanding of what a research problem is about before investigating why it exists in the first place. 

For example, an investor considering an investment in the ever-changing Amsterdam housing market needs to understand what the current state of the market is, how it changes (increasing or decreasing), and when it changes (time of the year) before asking for the why. This is where descriptive research comes in.

What Are The Types of Descriptive Research?

Descriptive research is classified into different types according to the kind of approach that is used in conducting descriptive research. The different types of descriptive research are highlighted below:

• Descriptive-survey

Descriptive survey research uses surveys to gather data about varying subjects. This data aims to know the extent to which different conditions can be obtained among these subjects.

For example, a researcher wants to determine the qualification of employed professionals in Maryland. He uses a survey as his research instrument , and each item on the survey related to qualifications is subjected to a Yes/No answer. 

This way, the researcher can describe the qualifications possessed by the employed demographics of this community. 

• Descriptive-normative survey

This is an extension of the descriptive survey, with the addition being the normative element. In the descriptive-normative survey, the results of the study should be compared with the norm.

For example, an organization that wishes to test the skills of its employees by a team may have them take a skills test. The skills tests are the evaluation tool in this case, and the result of this test is compared with the norm of each role.

If the score of the team is one standard deviation above the mean, it is very satisfactory, if within the mean, satisfactory, and one standard deviation below the mean is unsatisfactory.

• Descriptive-status

This is a quantitative description technique that seeks to answer questions about real-life situations. For example, a researcher researching the income of the employees in a company, and the relationship with their performance.

A survey will be carried out to gather enough data about the income of the employees, then their performance will be evaluated and compared to their income. This will help determine whether a higher income means better performance and low income means lower performance or vice versa.

• Descriptive-analysis

The descriptive-analysis method of research describes a subject by further analyzing it, which in this case involves dividing it into 2 parts. For example, the HR personnel of a company that wishes to analyze the job role of each employee of the company may divide the employees into the people that work at the Headquarters in the US and those that work from Oslo, Norway office.

A questionnaire is devised to analyze the job role of employees with similar salaries and who work in similar positions.

• Descriptive classification

This method is employed in biological sciences for the classification of plants and animals. A researcher who wishes to classify the sea animals into different species will collect samples from various search stations, then classify them accordingly.

• Descriptive-comparative

In descriptive-comparative research, the researcher considers 2 variables that are not manipulated, and establish a formal procedure to conclude that one is better than the other. For example, an examination body wants to determine the better method of conducting tests between paper-based and computer-based tests.

A random sample of potential participants of the test may be asked to use the 2 different methods, and factors like failure rates, time factors, and others will be evaluated to arrive at the best method.

• Correlative Survey

Correlative surveys are used to determine whether the relationship between 2 variables is positive, negative, or neutral. That is, if 2 variables say X and Y are directly proportional, inversely proportional or are not related to each other.

Examples of Descriptive Research

There are different examples of descriptive research, that may be highlighted from its types, uses, and applications. However, we will be restricting ourselves to only 3 distinct examples in this article.

• Comparing Student Performance:

An academic institution may wish 2 compare the performance of its junior high school students in English language and Mathematics. This may be used to classify students based on 2 major groups, with one group going ahead to study while courses, while the other study courses in the Arts & Humanities field.

Students who are more proficient in mathematics will be encouraged to go into STEM and vice versa. Institutions may also use this data to identify students’ weak points and work on ways to assist them.

• Scientific Classification

During the major scientific classification of plants, animals, and periodic table elements, the characteristics and components of each subject are evaluated and used to determine how they are classified.

For example, living things may be classified into kingdom Plantae or kingdom animal is depending on their nature. Further classification may group animals into mammals, pieces, vertebrae, invertebrae, etc. 

All these classifications are made a result of descriptive research which describes what they are.

• Human Behavior

When studying human behaviour based on a factor or event, the researcher observes the characteristics, behaviour, and reaction, then use it to conclude. A company willing to sell to its target market needs to first study the behaviour of the market.

This may be done by observing how its target reacts to a competitor’s product, then use it to determine their behaviour.

What are the Characteristics of Descriptive Research?  

The characteristics of descriptive research can be highlighted from its definition, applications, data collection methods, and examples. Some characteristics of descriptive research are:

• Quantitativeness

Descriptive research uses a quantitative research method by collecting quantifiable information to be used for statistical analysis of the population sample. This is very common when dealing with research in the physical sciences.

• Qualitativeness

It can also be carried out using the qualitative research method, to properly describe the research problem. This is because descriptive research is more explanatory than exploratory or experimental.

• Uncontrolled variables

In descriptive research, researchers cannot control the variables like they do in experimental research.

• The basis for further research

The results of descriptive research can be further analyzed and used in other research methods. It can also inform the next line of research, including the research method that should be used.

This is because it provides basic information about the research problem, which may give birth to other questions like why a particular thing is the way it is.

Why Use Descriptive Research Design?  

Descriptive research can be used to investigate the background of a research problem and get the required information needed to carry out further research. It is used in multiple ways by different organizations, and especially when getting the required information about their target audience.

• Define subject characteristics :

It is used to determine the characteristics of the subjects, including their traits, behaviour, opinion, etc. This information may be gathered with the use of surveys, which are shared with the respondents who in this case, are the research subjects.

For example, a survey evaluating the number of hours millennials in a community spends on the internet weekly, will help a service provider make informed business decisions regarding the market potential of the community.

• Measure Data Trends

It helps to measure the changes in data over some time through statistical methods. Consider the case of individuals who want to invest in stock markets, so they evaluate the changes in prices of the available stocks to make a decision investment decision.

Brokerage companies are however the ones who carry out the descriptive research process, while individuals can view the data trends and make decisions.

Descriptive research is also used to compare how different demographics respond to certain variables. For example, an organization may study how people with different income levels react to the launch of a new Apple phone.

This kind of research may take a survey that will help determine which group of individuals are purchasing the new Apple phone. Do the low-income earners also purchase the phone, or only the high-income earners do?

Further research using another technique will explain why low-income earners are purchasing the phone even though they can barely afford it. This will help inform strategies that will lure other low-income earners and increase company sales.

• Validate existing conditions

When you are not sure about the validity of an existing condition, you can use descriptive research to ascertain the underlying patterns of the research object. This is because descriptive research methods make an in-depth analysis of each variable before making conclusions.

• Conducted Overtime

Descriptive research is conducted over some time to ascertain the changes observed at each point in time. The higher the number of times it is conducted, the more authentic the conclusion will be.

What are the Disadvantages of Descriptive Research?  

• Response and Non-response Bias

Respondents may either decide not to respond to questions or give incorrect responses if they feel the questions are too confidential. When researchers use observational methods, respondents may also decide to behave in a particular manner because they feel they are being watched.

• The researcher may decide to influence the result of the research due to personal opinion or bias towards a particular subject. For example, a stockbroker who also has a business of his own may try to lure investors into investing in his own company by manipulating results.
• A case-study or sample taken from a large population is not representative of the whole population.
• Limited scope:The scope of descriptive research is limited to the what of research, with no information on why thereby limiting the scope of the research.

What are the Data Collection Methods in Descriptive Research?  

There are 3 main data collection methods in descriptive research, namely; observational method, case study method, and survey research.

1. Observational Method

The observational method allows researchers to collect data based on their view of the behaviour and characteristics of the respondent, with the respondents themselves not directly having an input. It is often used in market research, psychology, and some other social science research to understand human behaviour.

It is also an important aspect of physical scientific research, with it being one of the most effective methods of conducting descriptive research . This process can be said to be either quantitative or qualitative.

Quantitative observation involved the objective collection of numerical data , whose results can be analyzed using numerical and statistical methods. 

Qualitative observation, on the other hand, involves the monitoring of characteristics and not the measurement of numbers. The researcher makes his observation from a distance, records it, and is used to inform conclusions.

2. Case Study Method

A case study is a sample group (an individual, a group of people, organizations, events, etc.) whose characteristics are used to describe the characteristics of a larger group in which the case study is a subgroup. The information gathered from investigating a case study may be generalized to serve the larger group.

This generalization, may, however, be risky because case studies are not sufficient to make accurate predictions about larger groups. Case studies are a poor case of generalization.

3. Survey Research

This is a very popular data collection method in research designs. In survey research, researchers create a survey or questionnaire and distribute it to respondents who give answers.

Generally, it is used to obtain quick information directly from the primary source and also conducting rigorous quantitative and qualitative research. In some cases, survey research uses a blend of both qualitative and quantitative strategies.

Survey research can be carried out both online and offline using the following methods

• Online Surveys: This is a cheap method of carrying out surveys and getting enough responses. It can be carried out using Formplus, an online survey builder. Formplus has amazing tools and features that will help increase response rates.
• Offline Surveys: This includes paper forms, mobile offline forms , and SMS-based forms.

What Are The Differences Between Descriptive and Correlational Research?  

Before going into the differences between descriptive and correlation research, we need to have a proper understanding of what correlation research is about. Therefore, we will be giving a summary of the correlation research below.

Correlational research is a type of descriptive research, which is used to measure the relationship between 2 variables, with the researcher having no control over them. It aims to find whether there is; positive correlation (both variables change in the same direction), negative correlation (the variables change in the opposite direction), or zero correlation (there is no relationship between the variables).

Correlational research may be used in 2 situations;

(i) when trying to find out if there is a relationship between two variables, and

(ii) when a causal relationship is suspected between two variables, but it is impractical or unethical to conduct experimental research that manipulates one of the variables. 

Below are some of the differences between correlational and descriptive research:

• Definitions :

Descriptive research aims is a type of research that provides an in-depth understanding of the study population, while correlational research is the type of research that measures the relationship between 2 variables. 

• Characteristics :

Descriptive research provides descriptive data explaining what the research subject is about, while correlation research explores the relationship between data and not their description.

• Predictions :

 Predictions cannot be made in descriptive research while correlation research accommodates the possibility of making predictions.

Descriptive Research vs. Causal Research

Descriptive research and causal research are both research methodologies, however, one focuses on a subject’s behaviors while the latter focuses on a relationship’s cause-and-effect. To buttress the above point, descriptive research aims to describe and document the characteristics, behaviors, or phenomena of a particular or specific population or situation. 

It focuses on providing an accurate and detailed account of an already existing state of affairs between variables. Descriptive research answers the questions of “what,” “where,” “when,” and “how” without attempting to establish any causal relationships or explain any underlying factors that might have caused the behavior.

Causal research, on the other hand, seeks to determine cause-and-effect relationships between variables. It aims to point out the factors that influence or cause a particular result or behavior. Causal research involves manipulating variables, controlling conditions or a subgroup, and observing the resulting effects. The primary objective of causal research is to establish a cause-effect relationship and provide insights into why certain phenomena happen the way they do.

Descriptive Research vs. Analytical Research

Descriptive research provides a detailed and comprehensive account of a specific situation or phenomenon. It focuses on describing and summarizing data without making inferences or attempting to explain underlying factors or the cause of the factor. 

It is primarily concerned with providing an accurate and objective representation of the subject of research. While analytical research goes beyond the description of the phenomena and seeks to analyze and interpret data to discover if there are patterns, relationships, or any underlying factors. 

It examines the data critically, applies statistical techniques or other analytical methods, and draws conclusions based on the discovery. Analytical research also aims to explore the relationships between variables and understand the underlying mechanisms or processes involved.

Descriptive Research vs. Exploratory Research

Descriptive research is a research method that focuses on providing a detailed and accurate account of a specific situation, group, or phenomenon. This type of research describes the characteristics, behaviors, or relationships within the given context without looking for an underlying cause. 

Descriptive research typically involves collecting and analyzing quantitative or qualitative data to generate descriptive statistics or narratives. Exploratory research differs from descriptive research because it aims to explore and gain firsthand insights or knowledge into a relatively unexplored or poorly understood topic. 

It focuses on generating ideas, hypotheses, or theories rather than providing definitive answers. Exploratory research is often conducted at the early stages of a research project to gather preliminary information and identify key variables or factors for further investigation. It involves open-ended interviews, observations, or small-scale surveys to gather qualitative data.

Read More – Exploratory Research: What are its Method & Examples?

Descriptive Research vs. Experimental Research

Descriptive research aims to describe and document the characteristics, behaviors, or phenomena of a particular population or situation. It focuses on providing an accurate and detailed account of the existing state of affairs. 

Descriptive research typically involves collecting data through surveys, observations, or existing records and analyzing the data to generate descriptive statistics or narratives. It does not involve manipulating variables or establishing cause-and-effect relationships.

Experimental research, on the other hand, involves manipulating variables and controlling conditions to investigate cause-and-effect relationships. It aims to establish causal relationships by introducing an intervention or treatment and observing the resulting effects. 

Experimental research typically involves randomly assigning participants to different groups, such as control and experimental groups, and measuring the outcomes. It allows researchers to control for confounding variables and draw causal conclusions.

Related – Experimental vs Non-Experimental Research: 15 Key Differences

Descriptive Research vs. Explanatory Research

Descriptive research focuses on providing a detailed and accurate account of a specific situation, group, or phenomenon. It aims to describe the characteristics, behaviors, or relationships within the given context. 

Descriptive research is primarily concerned with providing an objective representation of the subject of study without explaining underlying causes or mechanisms. Explanatory research seeks to explain the relationships between variables and uncover the underlying causes or mechanisms. 

It goes beyond description and aims to understand the reasons or factors that influence a particular outcome or behavior. Explanatory research involves analyzing data, conducting statistical analyses, and developing theories or models to explain the observed relationships.

Descriptive Research vs. Inferential Research

Descriptive research focuses on describing and summarizing data without making inferences or generalizations beyond the specific sample or population being studied. It aims to provide an accurate and objective representation of the subject of study. 

Descriptive research typically involves analyzing data to generate descriptive statistics, such as means, frequencies, or percentages, to describe the characteristics or behaviors observed.

Inferential research, however, involves making inferences or generalizations about a larger population based on a smaller sample. 

It aims to draw conclusions about the population characteristics or relationships by analyzing the sample data. Inferential research uses statistical techniques to estimate population parameters, test hypotheses, and determine the level of confidence or significance in the findings.

Related – Inferential Statistics: Definition, Types + Examples

Conclusion  

The uniqueness of descriptive research partly lies in its ability to explore both quantitative and qualitative research methods. Therefore, when conducting descriptive research, researchers have the opportunity to use a wide variety of techniques that aids the research process.

Descriptive research explores research problems in-depth, beyond the surface level thereby giving a detailed description of the research subject. That way, it can aid further research in the field, including other research methods .

It is also very useful in solving real-life problems in various fields of social science, physical science, and education.

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• USC Libraries
• Research Guides

Organizing Your Social Sciences Research Paper

• Quantitative Methods
• Purpose of Guide
• Design Flaws to Avoid
• Independent and Dependent Variables
• Glossary of Research Terms
• Reading Research Effectively
• Narrowing a Topic Idea
• Broadening a Topic Idea
• Extending the Timeliness of a Topic Idea
• Academic Writing Style
• Choosing a Title
• Making an Outline
• Paragraph Development
• Research Process Video Series
• Executive Summary
• The C.A.R.S. Model
• Background Information
• The Research Problem/Question
• Theoretical Framework
• Citation Tracking
• Content Alert Services
• Evaluating Sources
• Primary Sources
• Secondary Sources
• Tiertiary Sources
• Scholarly vs. Popular Publications
• Qualitative Methods
• Insiderness
• Using Non-Textual Elements
• Limitations of the Study
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• Bibliography

Quantitative methods emphasize objective measurements and the statistical, mathematical, or numerical analysis of data collected through polls, questionnaires, and surveys, or by manipulating pre-existing statistical data using computational techniques . Quantitative research focuses on gathering numerical data and generalizing it across groups of people or to explain a particular phenomenon.

Babbie, Earl R. The Practice of Social Research . 12th ed. Belmont, CA: Wadsworth Cengage, 2010; Muijs, Daniel. Doing Quantitative Research in Education with SPSS . 2nd edition. London: SAGE Publications, 2010.

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Statistics & Data Research Guide

Characteristics of Quantitative Research

Your goal in conducting quantitative research study is to determine the relationship between one thing [an independent variable] and another [a dependent or outcome variable] within a population. Quantitative research designs are either descriptive [subjects usually measured once] or experimental [subjects measured before and after a treatment]. A descriptive study establishes only associations between variables; an experimental study establishes causality.

Quantitative research deals in numbers, logic, and an objective stance. Quantitative research focuses on numeric and unchanging data and detailed, convergent reasoning rather than divergent reasoning [i.e., the generation of a variety of ideas about a research problem in a spontaneous, free-flowing manner].

Its main characteristics are :

• The data is usually gathered using structured research instruments.
• The results are based on larger sample sizes that are representative of the population.
• The research study can usually be replicated or repeated, given its high reliability.
• Researcher has a clearly defined research question to which objective answers are sought.
• All aspects of the study are carefully designed before data is collected.
• Data are in the form of numbers and statistics, often arranged in tables, charts, figures, or other non-textual forms.
• Project can be used to generalize concepts more widely, predict future results, or investigate causal relationships.
• Researcher uses tools, such as questionnaires or computer software, to collect numerical data.

The overarching aim of a quantitative research study is to classify features, count them, and construct statistical models in an attempt to explain what is observed.

  Things to keep in mind when reporting the results of a study using quantitative methods :

• Explain the data collected and their statistical treatment as well as all relevant results in relation to the research problem you are investigating. Interpretation of results is not appropriate in this section.
• Report unanticipated events that occurred during your data collection. Explain how the actual analysis differs from the planned analysis. Explain your handling of missing data and why any missing data does not undermine the validity of your analysis.
• Explain the techniques you used to "clean" your data set.
• Choose a minimally sufficient statistical procedure ; provide a rationale for its use and a reference for it. Specify any computer programs used.
• Describe the assumptions for each procedure and the steps you took to ensure that they were not violated.
• When using inferential statistics , provide the descriptive statistics, confidence intervals, and sample sizes for each variable as well as the value of the test statistic, its direction, the degrees of freedom, and the significance level [report the actual p value].
• Avoid inferring causality , particularly in nonrandomized designs or without further experimentation.
• Use tables to provide exact values ; use figures to convey global effects. Keep figures small in size; include graphic representations of confidence intervals whenever possible.
• Always tell the reader what to look for in tables and figures .

NOTE:   When using pre-existing statistical data gathered and made available by anyone other than yourself [e.g., government agency], you still must report on the methods that were used to gather the data and describe any missing data that exists and, if there is any, provide a clear explanation why the missing data does not undermine the validity of your final analysis.

Babbie, Earl R. The Practice of Social Research . 12th ed. Belmont, CA: Wadsworth Cengage, 2010; Brians, Craig Leonard et al. Empirical Political Analysis: Quantitative and Qualitative Research Methods . 8th ed. Boston, MA: Longman, 2011; McNabb, David E. Research Methods in Public Administration and Nonprofit Management: Quantitative and Qualitative Approaches . 2nd ed. Armonk, NY: M.E. Sharpe, 2008; Quantitative Research Methods. Writing@CSU. Colorado State University; Singh, Kultar. Quantitative Social Research Methods . Los Angeles, CA: Sage, 2007.

Basic Research Design for Quantitative Studies

Before designing a quantitative research study, you must decide whether it will be descriptive or experimental because this will dictate how you gather, analyze, and interpret the results. A descriptive study is governed by the following rules: subjects are generally measured once; the intention is to only establish associations between variables; and, the study may include a sample population of hundreds or thousands of subjects to ensure that a valid estimate of a generalized relationship between variables has been obtained. An experimental design includes subjects measured before and after a particular treatment, the sample population may be very small and purposefully chosen, and it is intended to establish causality between variables. Introduction The introduction to a quantitative study is usually written in the present tense and from the third person point of view. It covers the following information:

• Identifies the research problem -- as with any academic study, you must state clearly and concisely the research problem being investigated.
• Reviews the literature -- review scholarship on the topic, synthesizing key themes and, if necessary, noting studies that have used similar methods of inquiry and analysis. Note where key gaps exist and how your study helps to fill these gaps or clarifies existing knowledge.
• Describes the theoretical framework -- provide an outline of the theory or hypothesis underpinning your study. If necessary, define unfamiliar or complex terms, concepts, or ideas and provide the appropriate background information to place the research problem in proper context [e.g., historical, cultural, economic, etc.].

Methodology The methods section of a quantitative study should describe how each objective of your study will be achieved. Be sure to provide enough detail to enable the reader can make an informed assessment of the methods being used to obtain results associated with the research problem. The methods section should be presented in the past tense.

• Study population and sampling -- where did the data come from; how robust is it; note where gaps exist or what was excluded. Note the procedures used for their selection;
• Data collection – describe the tools and methods used to collect information and identify the variables being measured; describe the methods used to obtain the data; and, note if the data was pre-existing [i.e., government data] or you gathered it yourself. If you gathered it yourself, describe what type of instrument you used and why. Note that no data set is perfect--describe any limitations in methods of gathering data.
• Data analysis -- describe the procedures for processing and analyzing the data. If appropriate, describe the specific instruments of analysis used to study each research objective, including mathematical techniques and the type of computer software used to manipulate the data.

Results The finding of your study should be written objectively and in a succinct and precise format. In quantitative studies, it is common to use graphs, tables, charts, and other non-textual elements to help the reader understand the data. Make sure that non-textual elements do not stand in isolation from the text but are being used to supplement the overall description of the results and to help clarify key points being made. Further information about how to effectively present data using charts and graphs can be found here .

• Statistical analysis -- how did you analyze the data? What were the key findings from the data? The findings should be present in a logical, sequential order. Describe but do not interpret these trends or negative results; save that for the discussion section. The results should be presented in the past tense.

Discussion Discussions should be analytic, logical, and comprehensive. The discussion should meld together your findings in relation to those identified in the literature review, and placed within the context of the theoretical framework underpinning the study. The discussion should be presented in the present tense.

• Interpretation of results -- reiterate the research problem being investigated and compare and contrast the findings with the research questions underlying the study. Did they affirm predicted outcomes or did the data refute it?
• Description of trends, comparison of groups, or relationships among variables -- describe any trends that emerged from your analysis and explain all unanticipated and statistical insignificant findings.
• Discussion of implications – what is the meaning of your results? Highlight key findings based on the overall results and note findings that you believe are important. How have the results helped fill gaps in understanding the research problem?
• Limitations -- describe any limitations or unavoidable bias in your study and, if necessary, note why these limitations did not inhibit effective interpretation of the results.

Conclusion End your study by to summarizing the topic and provide a final comment and assessment of the study.

• Summary of findings – synthesize the answers to your research questions. Do not report any statistical data here; just provide a narrative summary of the key findings and describe what was learned that you did not know before conducting the study.
• Recommendations – if appropriate to the aim of the assignment, tie key findings with policy recommendations or actions to be taken in practice.
• Future research – note the need for future research linked to your study’s limitations or to any remaining gaps in the literature that were not addressed in your study.

Black, Thomas R. Doing Quantitative Research in the Social Sciences: An Integrated Approach to Research Design, Measurement and Statistics . London: Sage, 1999; Gay,L. R. and Peter Airasain. Educational Research: Competencies for Analysis and Applications . 7th edition. Upper Saddle River, NJ: Merril Prentice Hall, 2003; Hector, Anestine. An Overview of Quantitative Research in Composition and TESOL . Department of English, Indiana University of Pennsylvania; Hopkins, Will G. “Quantitative Research Design.” Sportscience 4, 1 (2000); "A Strategy for Writing Up Research Results. The Structure, Format, Content, and Style of a Journal-Style Scientific Paper." Department of Biology. Bates College; Nenty, H. Johnson. "Writing a Quantitative Research Thesis." International Journal of Educational Science 1 (2009): 19-32; Ouyang, Ronghua (John). Basic Inquiry of Quantitative Research . Kennesaw State University.

Strengths of Using Quantitative Methods

Quantitative researchers try to recognize and isolate specific variables contained within the study framework, seek correlation, relationships and causality, and attempt to control the environment in which the data is collected to avoid the risk of variables, other than the one being studied, accounting for the relationships identified.

Among the specific strengths of using quantitative methods to study social science research problems:

• Allows for a broader study, involving a greater number of subjects, and enhancing the generalization of the results;
• Allows for greater objectivity and accuracy of results. Generally, quantitative methods are designed to provide summaries of data that support generalizations about the phenomenon under study. In order to accomplish this, quantitative research usually involves few variables and many cases, and employs prescribed procedures to ensure validity and reliability;
• Applying well established standards means that the research can be replicated, and then analyzed and compared with similar studies;
• You can summarize vast sources of information and make comparisons across categories and over time; and,
• Personal bias can be avoided by keeping a 'distance' from participating subjects and using accepted computational techniques .

Babbie, Earl R. The Practice of Social Research . 12th ed. Belmont, CA: Wadsworth Cengage, 2010; Brians, Craig Leonard et al. Empirical Political Analysis: Quantitative and Qualitative Research Methods . 8th ed. Boston, MA: Longman, 2011; McNabb, David E. Research Methods in Public Administration and Nonprofit Management: Quantitative and Qualitative Approaches . 2nd ed. Armonk, NY: M.E. Sharpe, 2008; Singh, Kultar. Quantitative Social Research Methods . Los Angeles, CA: Sage, 2007.

Limitations of Using Quantitative Methods

Quantitative methods presume to have an objective approach to studying research problems, where data is controlled and measured, to address the accumulation of facts, and to determine the causes of behavior. As a consequence, the results of quantitative research may be statistically significant but are often humanly insignificant.

Some specific limitations associated with using quantitative methods to study research problems in the social sciences include:

• Quantitative data is more efficient and able to test hypotheses, but may miss contextual detail;
• Uses a static and rigid approach and so employs an inflexible process of discovery;
• The development of standard questions by researchers can lead to "structural bias" and false representation, where the data actually reflects the view of the researcher instead of the participating subject;
• Results provide less detail on behavior, attitudes, and motivation;
• Researcher may collect a much narrower and sometimes superficial dataset;
• Results are limited as they provide numerical descriptions rather than detailed narrative and generally provide less elaborate accounts of human perception;
• The research is often carried out in an unnatural, artificial environment so that a level of control can be applied to the exercise. This level of control might not normally be in place in the real world thus yielding "laboratory results" as opposed to "real world results"; and,
• Preset answers will not necessarily reflect how people really feel about a subject and, in some cases, might just be the closest match to the preconceived hypothesis.

Research Tip

Finding Examples of How to Apply Different Types of Research Methods

SAGE publications is a major publisher of studies about how to design and conduct research in the social and behavioral sciences. Their SAGE Research Methods Online and Cases database includes contents from books, articles, encyclopedias, handbooks, and videos covering social science research design and methods including the complete Little Green Book Series of Quantitative Applications in the Social Sciences and the Little Blue Book Series of Qualitative Research techniques. The database also includes case studies outlining the research methods used in real research projects. This is an excellent source for finding definitions of key terms and descriptions of research design and practice, techniques of data gathering, analysis, and reporting, and information about theories of research [e.g., grounded theory]. The database covers both qualitative and quantitative research methods as well as mixed methods approaches to conducting research.

SAGE Research Methods Online and Cases

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• What Is Quantitative Research? | Definition & Methods

What Is Quantitative Research? | Definition & Methods

Published on 4 April 2022 by Pritha Bhandari . Revised on 10 October 2022.

Quantitative research is the process of collecting and analysing numerical data. It can be used to find patterns and averages, make predictions, test causal relationships, and generalise results to wider populations.

Quantitative research is the opposite of qualitative research , which involves collecting and analysing non-numerical data (e.g. text, video, or audio).

Quantitative research is widely used in the natural and social sciences: biology, chemistry, psychology, economics, sociology, marketing, etc.

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Table of contents

Quantitative research methods, quantitative data analysis, advantages of quantitative research, disadvantages of quantitative research, frequently asked questions about quantitative research.

You can use quantitative research methods for descriptive, correlational or experimental research.

• In descriptive research , you simply seek an overall summary of your study variables.
• In correlational research , you investigate relationships between your study variables.
• In experimental research , you systematically examine whether there is a cause-and-effect relationship between variables.

Correlational and experimental research can both be used to formally test hypotheses , or predictions, using statistics. The results may be generalised to broader populations based on the sampling method used.

To collect quantitative data, you will often need to use operational definitions that translate abstract concepts (e.g., mood) into observable and quantifiable measures (e.g., self-ratings of feelings and energy levels).

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Once data is collected, you may need to process it before it can be analysed. For example, survey and test data may need to be transformed from words to numbers. Then, you can use statistical analysis to answer your research questions .

Descriptive statistics will give you a summary of your data and include measures of averages and variability. You can also use graphs, scatter plots and frequency tables to visualise your data and check for any trends or outliers.

Using inferential statistics , you can make predictions or generalisations based on your data. You can test your hypothesis or use your sample data to estimate the population parameter .

You can also assess the reliability and validity of your data collection methods to indicate how consistently and accurately your methods actually measured what you wanted them to.

Quantitative research is often used to standardise data collection and generalise findings . Strengths of this approach include:

• Replication

Repeating the study is possible because of standardised data collection protocols and tangible definitions of abstract concepts.

• Direct comparisons of results

The study can be reproduced in other cultural settings, times or with different groups of participants. Results can be compared statistically.

• Large samples

Data from large samples can be processed and analysed using reliable and consistent procedures through quantitative data analysis.

• Hypothesis testing

Using formalised and established hypothesis testing procedures means that you have to carefully consider and report your research variables, predictions, data collection and testing methods before coming to a conclusion.

Despite the benefits of quantitative research, it is sometimes inadequate in explaining complex research topics. Its limitations include:

• Superficiality

Using precise and restrictive operational definitions may inadequately represent complex concepts. For example, the concept of mood may be represented with just a number in quantitative research, but explained with elaboration in qualitative research.

• Narrow focus

Predetermined variables and measurement procedures can mean that you ignore other relevant observations.

• Structural bias

Despite standardised procedures, structural biases can still affect quantitative research. Missing data , imprecise measurements or inappropriate sampling methods are biases that can lead to the wrong conclusions.

• Lack of context

Quantitative research often uses unnatural settings like laboratories or fails to consider historical and cultural contexts that may affect data collection and results.

Quantitative research deals with numbers and statistics, while qualitative research deals with words and meanings.

Quantitative methods allow you to test a hypothesis by systematically collecting and analysing data, while qualitative methods allow you to explore ideas and experiences in depth.

In mixed methods research , you use both qualitative and quantitative data collection and analysis methods to answer your research question .

Data collection is the systematic process by which observations or measurements are gathered in research. It is used in many different contexts by academics, governments, businesses, and other organisations.

Operationalisation means turning abstract conceptual ideas into measurable observations.

For example, the concept of social anxiety isn’t directly observable, but it can be operationally defined in terms of self-rating scores, behavioural avoidance of crowded places, or physical anxiety symptoms in social situations.

Before collecting data , it’s important to consider how you will operationalise the variables that you want to measure.

Reliability and validity are both about how well a method measures something:

• Reliability refers to the  consistency of a measure (whether the results can be reproduced under the same conditions).
• Validity   refers to the  accuracy of a measure (whether the results really do represent what they are supposed to measure).

If you are doing experimental research , you also have to consider the internal and external validity of your experiment.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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12 Quantitative Descriptive and Correlational Research

• Published: February 2023
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This chapter presents research designs for descriptive and correlational quantitative research. Descriptive research designs are used to address the question “What is x?” Correlational research designs are used to address the question “How are things related?” In contrast to some experimental research designs, in these design types the primary area of interest under investigation is not manipulated by the researcher. Researchers investigating descriptive or correlational research questions commonly use surveys or observational methods to gather data. Surveys are an efficient method for gathering large amounts of information about such things as individuals’ experiences, beliefs, and attitudes. When designing a survey, researchers must consider many things, such as how long it will be and what it will cover. Observation is an important means of gathering data, as when researchers observe video recordings of teachers or students in various situations. Another approach to observational research is the experience sampling method (ESM). In ESM, participants are interrupted at random times throughout the day and asked to respond to questions concerning their experiences in real time. In other words, researchers ask participants what they are doing at the moment they are contacted.

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Child Care and Early Education Research Connections

Descriptive research studies.

Descriptive research is a type of research that is used to describe the characteristics of a population. It collects data that are used to answer a wide range of what, when, and how questions pertaining to a particular population or group. For example, descriptive studies might be used to answer questions such as: What percentage of Head Start teachers have a bachelor's degree or higher? What is the average reading ability of 5-year-olds when they first enter kindergarten? What kinds of math activities are used in early childhood programs? When do children first receive regular child care from someone other than their parents? When are children with developmental disabilities first diagnosed and when do they first receive services? What factors do programs consider when making decisions about the type of assessments that will be used to assess the skills of the children in their programs? How do the types of services children receive from their early childhood program change as children age?

Descriptive research does not answer questions about why a certain phenomenon occurs or what the causes are. Answers to such questions are best obtained from  randomized and quasi-experimental studies . However, data from descriptive studies can be used to examine the relationships (correlations) among variables. While the findings from correlational analyses are not evidence of causality, they can help to distinguish variables that may be important in explaining a phenomenon from those that are not. Thus, descriptive research is often used to generate hypotheses that should be tested using more rigorous designs.

A variety of data collection methods may be used alone or in combination to answer the types of questions guiding descriptive research. Some of the more common methods include surveys, interviews, observations, case studies, and portfolios. The data collected through these methods can be either quantitative or qualitative. Quantitative data are typically analyzed and presenting using  descriptive statistics . Using quantitative data, researchers may describe the characteristics of a sample or population in terms of percentages (e.g., percentage of population that belong to different racial/ethnic groups, percentage of low-income families that receive different government services) or averages (e.g., average household income, average scores of reading, mathematics and language assessments). Quantitative data, such as narrative data collected as part of a case study, may be used to organize, classify, and used to identify patterns of behaviors, attitudes, and other characteristics of groups.

Descriptive studies have an important role in early care and education research. Studies such as the  National Survey of Early Care and Education  and the  National Household Education Surveys Program  have greatly increased our knowledge of the supply of and demand for child care in the U.S. The  Head Start Family and Child Experiences Survey  and the  Early Childhood Longitudinal Study Program  have provided researchers, policy makers and practitioners with rich information about school readiness skills of children in the U.S.

Each of the methods used to collect descriptive data have their own strengths and limitations. The following are some of the strengths and limitations of descriptive research studies in general.

Study participants are questioned or observed in a natural setting (e.g., their homes, child care or educational settings).

Study data can be used to identify the prevalence of particular problems and the need for new or additional services to address these problems.

Descriptive research may identify areas in need of additional research and relationships between variables that require future study. Descriptive research is often referred to as "hypothesis generating research."

Depending on the data collection method used, descriptive studies can generate rich datasets on large and diverse samples.

Limitations:

Descriptive studies cannot be used to establish cause and effect relationships.

Respondents may not be truthful when answering survey questions or may give socially desirable responses.

The choice and wording of questions on a questionnaire may influence the descriptive findings.

Depending on the type and size of sample, the findings may not be generalizable or produce an accurate description of the population of interest.

Writing Quantitative Research Studies

• Reference work entry
• First Online: 13 January 2019
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• Ankur Singh 2 ,
• Adyya Gupta 3 &
• Karen G. Peres 4  

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Summarizing quantitative data and its effective presentation and discussion can be challenging for students and researchers. This chapter provides a framework for adequately reporting findings from quantitative analysis in a research study for those contemplating to write a research paper. The rationale underpinning the reporting methods to maintain the credibility and integrity of quantitative studies is outlined. Commonly used terminologies in empirical studies are defined and discussed with suitable examples. Key elements that build consistency between different sections (background, methods, results, and the discussion) of a research study using quantitative methods in a journal article are explicated. Specifically, recommended standard guidelines for randomized controlled trials and observational studies for reporting and discussion of findings from quantitative studies are elaborated. Key aspects of methodology that include describing the study population, sampling strategy, data collection methods, measurements/variables, and statistical analysis which informs the quality of a study from the reviewer’s perspective are described. Effective use of references in the methods section to strengthen the rationale behind specific statistical techniques and choice of measures has been highlighted with examples. Identifying ways in which data can be most succinctly and effectively summarized in tables and graphs according to their suitability and purpose of information is also detailed in this chapter. Strategies to present and discuss the quantitative findings in a structured discussion section are also provided. Overall, the chapter provides the readers with a comprehensive set of tools to identify key strategies to be considered when reporting quantitative research.

• Quantitative analysis
• Research methodology
• Writing strategies
• Empirical studies

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Centre for Health Equity, Melbourne School of Population and Global Health, The University of Melbourne, Melbourne, VIC, Australia

Ankur Singh

School of Public Health, The University of Adelaide, Adelaide, SA, Australia

Adyya Gupta

Australian Research Centre for Population Oral Health (ARCPOH), Adelaide Dental School, The University of Adelaide, Adelaide, SA, Australia

Karen G. Peres

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Singh, A., Gupta, A., Peres, K.G. (2019). Writing Quantitative Research Studies. In: Liamputtong, P. (eds) Handbook of Research Methods in Health Social Sciences. Springer, Singapore. https://doi.org/10.1007/978-981-10-5251-4_117

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Descriptive Statistics | Definitions, Types, Examples

Published on July 9, 2020 by Pritha Bhandari . Revised on June 21, 2023.

Descriptive statistics summarize and organize characteristics of a data set. A data set is a collection of responses or observations from a sample or entire population.

In quantitative research , after collecting data, the first step of statistical analysis is to describe characteristics of the responses, such as the average of one variable (e.g., age), or the relation between two variables (e.g., age and creativity).

The next step is inferential statistics , which help you decide whether your data confirms or refutes your hypothesis and whether it is generalizable to a larger population.

Table of contents

Types of descriptive statistics, frequency distribution, measures of central tendency, measures of variability, univariate descriptive statistics, bivariate descriptive statistics, other interesting articles, frequently asked questions about descriptive statistics.

There are 3 main types of descriptive statistics:

• The distribution concerns the frequency of each value.
• The central tendency concerns the averages of the values.
• The variability or dispersion concerns how spread out the values are.

You can apply these to assess only one variable at a time, in univariate analysis, or to compare two or more, in bivariate and multivariate analysis.

• Go to a library
• Watch a movie at a theater
• Visit a national park

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A data set is made up of a distribution of values, or scores. In tables or graphs, you can summarize the frequency of every possible value of a variable in numbers or percentages. This is called a frequency distribution .

• Simple frequency distribution table
• Grouped frequency distribution table

From this table, you can see that more women than men or people with another gender identity took part in the study. In a grouped frequency distribution, you can group numerical response values and add up the number of responses for each group. You can also convert each of these numbers to percentages.

Measures of central tendency estimate the center, or average, of a data set. The mean, median and mode are 3 ways of finding the average.

Here we will demonstrate how to calculate the mean, median, and mode using the first 6 responses of our survey.

The mean , or M , is the most commonly used method for finding the average.

To find the mean, simply add up all response values and divide the sum by the total number of responses. The total number of responses or observations is called N .

The median is the value that’s exactly in the middle of a data set.

To find the median, order each response value from the smallest to the biggest. Then , the median is the number in the middle. If there are two numbers in the middle, find their mean.

The mode is the simply the most popular or most frequent response value. A data set can have no mode, one mode, or more than one mode.

To find the mode, order your data set from lowest to highest and find the response that occurs most frequently.

Measures of variability give you a sense of how spread out the response values are. The range, standard deviation and variance each reflect different aspects of spread.

The range gives you an idea of how far apart the most extreme response scores are. To find the range , simply subtract the lowest value from the highest value.

Standard deviation

The standard deviation ( s or SD ) is the average amount of variability in your dataset. It tells you, on average, how far each score lies from the mean. The larger the standard deviation, the more variable the data set is.

There are six steps for finding the standard deviation:

• List each score and find their mean.
• Subtract the mean from each score to get the deviation from the mean.
• Square each of these deviations.
• Add up all of the squared deviations.
• Divide the sum of the squared deviations by N – 1.
• Find the square root of the number you found.

Step 5: 421.5/5 = 84.3

Step 6: √84.3 = 9.18

The variance is the average of squared deviations from the mean. Variance reflects the degree of spread in the data set. The more spread the data, the larger the variance is in relation to the mean.

To find the variance, simply square the standard deviation. The symbol for variance is s 2 .

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Univariate descriptive statistics focus on only one variable at a time. It’s important to examine data from each variable separately using multiple measures of distribution, central tendency and spread. Programs like SPSS and Excel can be used to easily calculate these.

If you were to only consider the mean as a measure of central tendency, your impression of the “middle” of the data set can be skewed by outliers, unlike the median or mode.

Likewise, while the range is sensitive to outliers , you should also consider the standard deviation and variance to get easily comparable measures of spread.

If you’ve collected data on more than one variable, you can use bivariate or multivariate descriptive statistics to explore whether there are relationships between them.

In bivariate analysis, you simultaneously study the frequency and variability of two variables to see if they vary together. You can also compare the central tendency of the two variables before performing further statistical tests .

Multivariate analysis is the same as bivariate analysis but with more than two variables.

Contingency table

In a contingency table, each cell represents the intersection of two variables. Usually, an independent variable (e.g., gender) appears along the vertical axis and a dependent one appears along the horizontal axis (e.g., activities). You read “across” the table to see how the independent and dependent variables relate to each other.

Interpreting a contingency table is easier when the raw data is converted to percentages. Percentages make each row comparable to the other by making it seem as if each group had only 100 observations or participants. When creating a percentage-based contingency table, you add the N for each independent variable on the end.

From this table, it is more clear that similar proportions of children and adults go to the library over 17 times a year. Additionally, children most commonly went to the library between 5 and 8 times, while for adults, this number was between 13 and 16.

Scatter plots

A scatter plot is a chart that shows you the relationship between two or three variables . It’s a visual representation of the strength of a relationship.

In a scatter plot, you plot one variable along the x-axis and another one along the y-axis. Each data point is represented by a point in the chart.

From your scatter plot, you see that as the number of movies seen at movie theaters increases, the number of visits to the library decreases. Based on your visual assessment of a possible linear relationship, you perform further tests of correlation and regression.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Statistical power
• Pearson correlation
• Degrees of freedom
• Statistical significance

Methodology

• Cluster sampling
• Stratified sampling
• Focus group
• Systematic review
• Ethnography
• Double-Barreled Question

Research bias

• Implicit bias
• Publication bias
• Cognitive bias
• Placebo effect
• Pygmalion effect
• Hindsight bias
• Overconfidence bias

Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.

The 3 main types of descriptive statistics concern the frequency distribution, central tendency, and variability of a dataset.

• Distribution refers to the frequencies of different responses.
• Measures of central tendency give you the average for each response.
• Measures of variability show you the spread or dispersion of your dataset.
• Univariate statistics summarize only one variable  at a time.
• Bivariate statistics compare two variables .
• Multivariate statistics compare more than two variables .

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Descriptive Statistics for Summarising Data

Ray w. cooksey.

UNE Business School, University of New England, Armidale, NSW Australia

This chapter discusses and illustrates descriptive statistics . The purpose of the procedures and fundamental concepts reviewed in this chapter is quite straightforward: to facilitate the description and summarisation of data. By ‘describe’ we generally mean either the use of some pictorial or graphical representation of the data (e.g. a histogram, box plot, radar plot, stem-and-leaf display, icon plot or line graph) or the computation of an index or number designed to summarise a specific characteristic of a variable or measurement (e.g., frequency counts, measures of central tendency, variability, standard scores). Along the way, we explore the fundamental concepts of probability and the normal distribution. We seldom interpret individual data points or observations primarily because it is too difficult for the human brain to extract or identify the essential nature, patterns, or trends evident in the data, particularly if the sample is large. Rather we utilise procedures and measures which provide a general depiction of how the data are behaving. These statistical procedures are designed to identify or display specific patterns or trends in the data. What remains after their application is simply for us to interpret and tell the story.

The first broad category of statistics we discuss concerns descriptive statistics . The purpose of the procedures and fundamental concepts in this category is quite straightforward: to facilitate the description and summarisation of data. By ‘describe’ we generally mean either the use of some pictorial or graphical representation of the data or the computation of an index or number designed to summarise a specific characteristic of a variable or measurement.

We seldom interpret individual data points or observations primarily because it is too difficult for the human brain to extract or identify the essential nature, patterns, or trends evident in the data, particularly if the sample is large. Rather we utilise procedures and measures which provide a general depiction of how the data are behaving. These statistical procedures are designed to identify or display specific patterns or trends in the data. What remains after their application is simply for us to interpret and tell the story.

Reflect on the QCI research scenario and the associated data set discussed in Chap. 10.1007/978-981-15-2537-7_4. Consider the following questions that Maree might wish to address with respect to decision accuracy and speed scores:

• What was the typical level of accuracy and decision speed for inspectors in the sample? [see Procedure 5.4 – Assessing central tendency.]
• What was the most common accuracy and speed score amongst the inspectors? [see Procedure 5.4 – Assessing central tendency.]
• What was the range of accuracy and speed scores; the lowest and the highest scores? [see Procedure 5.5 – Assessing variability.]
• How frequently were different levels of inspection accuracy and speed observed? What was the shape of the distribution of inspection accuracy and speed scores? [see Procedure 5.1 – Frequency tabulation, distributions & crosstabulation.]
• What percentage of inspectors would have ‘failed’ to ‘make the cut’ assuming the industry standard for acceptable inspection accuracy and speed combined was set at 95%? [see Procedure 5.7 – Standard ( z ) scores.]
• How variable were the inspectors in their accuracy and speed scores? Were all the accuracy and speed levels relatively close to each other in magnitude or were the scores widely spread out over the range of possible test outcomes? [see Procedure 5.5 – Assessing variability.]
• What patterns might be visually detected when looking at various QCI variables singly and together as a set? [see Procedure 5.2 – Graphical methods for dispaying data, Procedure 5.3 – Multivariate graphs & displays, and Procedure 5.6 – Exploratory data analysis.]

This chapter includes discussions and illustrations of a number of procedures available for answering questions about data like those posed above. In addition, you will find discussions of two fundamental concepts, namely probability and the normal distribution ; concepts that provide building blocks for Chaps. 10.1007/978-981-15-2537-7_6 and 10.1007/978-981-15-2537-7_7.

Procedure 5.1: Frequency Tabulation, Distributions & Crosstabulation

Frequency tabulation and distributions.

Frequency tabulation serves to provide a convenient counting summary for a set of data that facilitates interpretation of various aspects of those data. Basically, frequency tabulation occurs in two stages:

• First, the scores in a set of data are rank ordered from the lowest value to the highest value.
• Second, the number of times each specific score occurs in the sample is counted. This count records the frequency of occurrence for that specific data value.

Consider the overall job satisfaction variable, jobsat , from the QCI data scenario. Performing frequency tabulation across the 112 Quality Control Inspectors on this variable using the SPSS Frequencies procedure (Allen et al. 2019 , ch. 3; George and Mallery 2019 , ch. 6) produces the frequency tabulation shown in Table 5.1 . Note that three of the inspectors in the sample did not provide a rating for jobsat thereby producing three missing values (= 2.7% of the sample of 112) and leaving 109 inspectors with valid data for the analysis.

Frequency tabulation of overall job satisfaction scores

The display of frequency tabulation is often referred to as the frequency distribution for the sample of scores. For each value of a variable, the frequency of its occurrence in the sample of data is reported. It is possible to compute various percentages and percentile values from a frequency distribution.

Table 5.1 shows the ‘Percent’ or relative frequency of each score (the percentage of the 112 inspectors obtaining each score, including those inspectors who were missing scores, which SPSS labels as ‘System’ missing). Table 5.1 also shows the ‘Valid Percent’ which is computed only for those inspectors in the sample who gave a valid or non-missing response.

Finally, it is possible to add up the ‘Valid Percent’ values, starting at the low score end of the distribution, to form the cumulative distribution or ‘Cumulative Percent’ . A cumulative distribution is useful for finding percentiles which reflect what percentage of the sample scored at a specific value or below.

We can see in Table 5.1 that 4 of the 109 valid inspectors (a ‘Valid Percent’ of 3.7%) indicated the lowest possible level of job satisfaction—a value of 1 (Very Low) – whereas 18 of the 109 valid inspectors (a ‘Valid Percent’ of 16.5%) indicated the highest possible level of job satisfaction—a value of 7 (Very High). The ‘Cumulative Percent’ number of 18.3 in the row for the job satisfaction score of 3 can be interpreted as “roughly 18% of the sample of inspectors reported a job satisfaction score of 3 or less”; that is, nearly a fifth of the sample expressed some degree of negative satisfaction with their job as a quality control inspector in their particular company.

If you have a large data set having many different scores for a particular variable, it may be more useful to tabulate frequencies on the basis of intervals of scores.

For the accuracy scores in the QCI database, you could count scores occurring in intervals such as ‘less than 75% accuracy’, ‘between 75% but less than 85% accuracy’, ‘between 85% but less than 95% accuracy’, and ‘95% accuracy or greater’, rather than counting the individual scores themselves. This would yield what is termed a ‘grouped’ frequency distribution since the data have been grouped into intervals or score classes. Producing such an analysis using SPSS would involve extra steps to create the new category or ‘grouping’ system for scores prior to conducting the frequency tabulation.

Crosstabulation

In a frequency crosstabulation , we count frequencies on the basis of two variables simultaneously rather than one; thus we have a bivariate situation.

For example, Maree might be interested in the number of male and female inspectors in the sample of 112 who obtained each jobsat score. Here there are two variables to consider: inspector’s gender and inspector’s j obsat score. Table 5.2 shows such a crosstabulation as compiled by the SPSS Crosstabs procedure (George and Mallery 2019 , ch. 8). Note that inspectors who did not report a score for jobsat and/or gender have been omitted as missing values, leaving 106 valid inspectors for the analysis.

Frequency crosstabulation of jobsat scores by gender category for the QCI data

The crosstabulation shown in Table 5.2 gives a composite picture of the distribution of satisfaction levels for male inspectors and for female inspectors. If frequencies or ‘Counts’ are added across the gender categories, we obtain the numbers in the ‘Total’ column (the percentages or relative frequencies are also shown immediately below each count) for each discrete value of jobsat (note this column of statistics differs from that in Table 5.1 because the gender variable was missing for certain inspectors). By adding down each gender column, we obtain, in the bottom row labelled ‘Total’, the number of males and the number of females that comprised the sample of 106 valid inspectors.

The totals, either across the rows or down the columns of the crosstabulation, are termed the marginal distributions of the table. These marginal distributions are equivalent to frequency tabulations for each of the variables jobsat and gender . As with frequency tabulation, various percentage measures can be computed in a crosstabulation, including the percentage of the sample associated with a specific count within either a row (‘% within jobsat ’) or a column (‘% within gender ’). You can see in Table 5.2 that 18 inspectors indicated a job satisfaction level of 7 (Very High); of these 18 inspectors reported in the ‘Total’ column, 8 (44.4%) were male and 10 (55.6%) were female. The marginal distribution for gender in the ‘Total’ row shows that 57 inspectors (53.8% of the 106 valid inspectors) were male and 49 inspectors (46.2%) were female. Of the 57 male inspectors in the sample, 8 (14.0%) indicated a job satisfaction level of 7 (Very High). Furthermore, we could generate some additional interpretive information of value by adding the ‘% within gender’ values for job satisfaction levels of 5, 6 and 7 (i.e. differing degrees of positive job satisfaction). Here we would find that 68.4% (= 24.6% + 29.8% + 14.0%) of male inspectors indicated some degree of positive job satisfaction compared to 61.2% (= 10.2% + 30.6% + 20.4%) of female inspectors.

This helps to build a picture of the possible relationship between an inspector’s gender and their level of job satisfaction (a relationship that, as we will see later, can be quantified and tested using Procedure 10.1007/978-981-15-2537-7_6#Sec14 and Procedure 10.1007/978-981-15-2537-7_7#Sec17).

It should be noted that a crosstabulation table such as that shown in Table 5.2 is often referred to as a contingency table about which more will be said later (see Procedure 10.1007/978-981-15-2537-7_7#Sec17 and Procedure 10.1007/978-981-15-2537-7_7#Sec115).

Frequency tabulation is useful for providing convenient data summaries which can aid in interpreting trends in a sample, particularly where the number of discrete values for a variable is relatively small. A cumulative percent distribution provides additional interpretive information about the relative positioning of specific scores within the overall distribution for the sample.

Crosstabulation permits the simultaneous examination of the distributions of values for two variables obtained from the same sample of observations. This examination can yield some useful information about the possible relationship between the two variables. More complex crosstabulations can be also done where the values of three or more variables are tracked in a single systematic summary. The use of frequency tabulation or cross-tabulation in conjunction with various other statistical measures, such as measures of central tendency (see Procedure 5.4 ) and measures of variability (see Procedure 5.5 ), can provide a relatively complete descriptive summary of any data set.

Disadvantages

Frequency tabulations can get messy if interval or ratio-level measures are tabulated simply because of the large number of possible data values. Grouped frequency distributions really should be used in such cases. However, certain choices, such as the size of the score interval (group size), must be made, often arbitrarily, and such choices can affect the nature of the final frequency distribution.

Additionally, percentage measures have certain problems associated with them, most notably, the potential for their misinterpretation in small samples. One should be sure to know the sample size on which percentage measures are based in order to obtain an interpretive reference point for the actual percentage values.

For example

In a sample of 10 individuals, 20% represents only two individuals whereas in a sample of 300 individuals, 20% represents 60 individuals. If all that is reported is the 20%, then the mental inference drawn by readers is likely to be that a sizeable number of individuals had a score or scores of a particular value—but what is ‘sizeable’ depends upon the total number of observations on which the percentage is based.

Where Is This Procedure Useful?

Frequency tabulation and crosstabulation are very commonly applied procedures used to summarise information from questionnaires, both in terms of tabulating various demographic characteristics (e.g. gender, age, education level, occupation) and in terms of actual responses to questions (e.g. numbers responding ‘yes’ or ‘no’ to a particular question). They can be particularly useful in helping to build up the data screening and demographic stories discussed in Chap. 10.1007/978-981-15-2537-7_4. Categorical data from observational studies can also be analysed with this technique (e.g. the number of times Suzy talks to Frank, to Billy, and to John in a study of children’s social interactions).

Certain types of experimental research designs may also be amenable to analysis by crosstabulation with a view to drawing inferences about distribution differences across the sets of categories for the two variables being tracked.

You could employ crosstabulation in conjunction with the tests described in Procedure 10.1007/978-981-15-2537-7_7#Sec17 to see if two different styles of advertising campaign differentially affect the product purchasing patterns of male and female consumers.

In the QCI database, Maree could employ crosstabulation to help her answer the question “do different types of electronic manufacturing firms ( company ) differ in terms of their tendency to employ male versus female quality control inspectors ( gender )?”

Software Procedures

Procedure 5.2: graphical methods for displaying data.

Graphical methods for displaying data include bar and pie charts, histograms and frequency polygons, line graphs and scatterplots. It is important to note that what is presented here is a small but representative sampling of the types of simple graphs one can produce to summarise and display trends in data. Generally speaking, SPSS offers the easiest facility for producing and editing graphs, but with a rather limited range of styles and types. SYSTAT, STATGRAPHICS and NCSS offer a much wider range of graphs (including graphs unique to each package), but with the drawback that it takes somewhat more effort to get the graphs in exactly the form you want.

Bar and Pie Charts

These two types of graphs are useful for summarising the frequency of occurrence of various values (or ranges of values) where the data are categorical (nominal or ordinal level of measurement).

• A bar chart uses vertical and horizontal axes to summarise the data. The vertical axis is used to represent frequency (number) of occurrence or the relative frequency (percentage) of occurrence; the horizontal axis is used to indicate the data categories of interest.
• A pie chart gives a simpler visual representation of category frequencies by cutting a circular plot into wedges or slices whose sizes are proportional to the relative frequency (percentage) of occurrence of specific data categories. Some pie charts can have a one or more slices emphasised by ‘exploding’ them out from the rest of the pie.

Consider the company variable from the QCI database. This variable depicts the types of manufacturing firms that the quality control inspectors worked for. Figure 5.1 illustrates a bar chart summarising the percentage of female inspectors in the sample coming from each type of firm. Figure 5.2 shows a pie chart representation of the same data, with an ‘exploded slice’ highlighting the percentage of female inspectors in the sample who worked for large business computer manufacturers – the lowest percentage of the five types of companies. Both graphs were produced using SPSS.

Bar chart: Percentage of female inspectors

Pie chart: Percentage of female inspectors

The pie chart was modified with an option to show the actual percentage along with the label for each category. The bar chart shows that computer manufacturing firms have relatively fewer female inspectors compared to the automotive and electrical appliance (large and small) firms. This trend is less clear from the pie chart which suggests that pie charts may be less visually interpretable when the data categories occur with rather similar frequencies. However, the ‘exploded slice’ option can help interpretation in some circumstances.

Certain software programs, such as SPSS, STATGRAPHICS, NCSS and Microsoft Excel, offer the option of generating 3-dimensional bar charts and pie charts and incorporating other ‘bells and whistles’ that can potentially add visual richness to the graphic representation of the data. However, you should generally be careful with these fancier options as they can produce distortions and create ambiguities in interpretation (e.g. see discussions in Jacoby 1997 ; Smithson 2000 ; Wilkinson 2009 ). Such distortions and ambiguities could ultimately end up providing misinformation to researchers as well as to those who read their research.

Histograms and Frequency Polygons

These two types of graphs are useful for summarising the frequency of occurrence of various values (or ranges of values) where the data are essentially continuous (interval or ratio level of measurement) in nature. Both histograms and frequency polygons use vertical and horizontal axes to summarise the data. The vertical axis is used to represent the frequency (number) of occurrence or the relative frequency (percentage) of occurrences; the horizontal axis is used for the data values or ranges of values of interest. The histogram uses bars of varying heights to depict frequency; the frequency polygon uses lines and points.

There is a visual difference between a histogram and a bar chart: the bar chart uses bars that do not physically touch, signifying the discrete and categorical nature of the data, whereas the bars in a histogram physically touch to signal the potentially continuous nature of the data.

Suppose Maree wanted to graphically summarise the distribution of speed scores for the 112 inspectors in the QCI database. Figure 5.3 (produced using NCSS) illustrates a histogram representation of this variable. Figure 5.3 also illustrates another representational device called the ‘density plot’ (the solid tracing line overlaying the histogram) which gives a smoothed impression of the overall shape of the distribution of speed scores. Figure 5.4 (produced using STATGRAPHICS) illustrates the frequency polygon representation for the same data.

Histogram of the speed variable (with density plot overlaid)

Frequency polygon plot of the speed variable

These graphs employ a grouped format where speed scores which fall within specific intervals are counted as being essentially the same score. The shape of the data distribution is reflected in these plots. Each graph tells us that the inspection speed scores are positively skewed with only a few inspectors taking very long times to make their inspection judgments and the majority of inspectors taking rather shorter amounts of time to make their decisions.

Both representations tell a similar story; the choice between them is largely a matter of personal preference. However, if the number of bars to be plotted in a histogram is potentially very large (and this is usually directly controllable in most statistical software packages), then a frequency polygon would be the preferred representation simply because the amount of visual clutter in the graph will be much reduced.

It is somewhat of an art to choose an appropriate definition for the width of the score grouping intervals (or ‘bins’ as they are often termed) to be used in the plot: choose too many and the plot may look too lumpy and the overall distributional trend may not be obvious; choose too few and the plot will be too coarse to give a useful depiction. Programs like SPSS, SYSTAT, STATGRAPHICS and NCSS are designed to choose an ‘appropriate’ number of bins to be used, but the analyst’s eye is often a better judge than any statistical rule that a software package would use.

There are several interesting variations of the histogram which can highlight key data features or facilitate interpretation of certain trends in the data. One such variation is a graph is called a dual histogram (available in SYSTAT; a variation called a ‘comparative histogram’ can be created in NCSS) – a graph that facilitates visual comparison of the frequency distributions for a specific variable for participants from two distinct groups.

Suppose Maree wanted to graphically compare the distributions of speed scores for inspectors in the two categories of education level ( educlev ) in the QCI database. Figure 5.5 shows a dual histogram (produced using SYSTAT) that accomplishes this goal. This graph still employs the grouped format where speed scores falling within particular intervals are counted as being essentially the same score. The shape of the data distribution within each group is also clearly reflected in this plot. However, the story conveyed by the dual histogram is that, while the inspection speed scores are positively skewed for inspectors in both categories of educlev, the comparison suggests that inspectors with a high school level of education (= 1) tend to take slightly longer to make their inspection decisions than do their colleagues who have a tertiary qualification (= 2).

Dual histogram of speed for the two categories of educlev

Line Graphs

The line graph is similar in style to the frequency polygon but is much more general in its potential for summarising data. In a line graph, we seldom deal with percentage or frequency data. Instead we can summarise other types of information about data such as averages or means (see Procedure 5.4 for a discussion of this measure), often for different groups of participants. Thus, one important use of the line graph is to break down scores on a specific variable according to membership in the categories of a second variable.

In the context of the QCI database, Maree might wish to summarise the average inspection accuracy scores for the inspectors from different types of manufacturing companies. Figure 5.6 was produced using SPSS and shows such a line graph.

Line graph comparison of companies in terms of average inspection accuracy

Note how the trend in performance across the different companies becomes clearer with such a visual representation. It appears that the inspectors from the Large Business Computer and PC manufacturing companies have better average inspection accuracy compared to the inspectors from the remaining three industries.

With many software packages, it is possible to further elaborate a line graph by including error or confidence intervals bars (see Procedure 10.1007/978-981-15-2537-7_8#Sec18). These give some indication of the precision with which the average level for each category in the population has been estimated (narrow bars signal a more precise estimate; wide bars signal a less precise estimate).

Figure 5.7 shows such an elaborated line graph, using 95% confidence interval bars, which can be used to help make more defensible judgments (compared to Fig. 5.6 ) about whether the companies are substantively different from each other in average inspection performance. Companies whose confidence interval bars do not overlap each other can be inferred to be substantively different in performance characteristics.

Line graph using confidence interval bars to compare accuracy across companies

The accuracy confidence interval bars for participants from the Large Business Computer manufacturing firms do not overlap those from the Large or Small Electrical Appliance manufacturers or the Automobile manufacturers.

We might conclude that quality control inspection accuracy is substantially better in the Large Business Computer manufacturing companies than in these other industries but is not substantially better than the PC manufacturing companies. We might also conclude that inspection accuracy in PC manufacturing companies is not substantially different from Small Electrical Appliance manufacturers.

Scatterplots

Scatterplots are useful in displaying the relationship between two interval- or ratio-scaled variables or measures of interest obtained on the same individuals, particularly in correlational research (see Fundamental Concept 10.1007/978-981-15-2537-7_6#Sec1 and Procedure 10.1007/978-981-15-2537-7_6#Sec4).

In a scatterplot, one variable is chosen to be represented on the horizontal axis; the second variable is represented on the vertical axis. In this type of plot, all data point pairs in the sample are graphed. The shape and tilt of the cloud of points in a scatterplot provide visual information about the strength and direction of the relationship between the two variables. A very compact elliptical cloud of points signals a strong relationship; a very loose or nearly circular cloud signals a weak or non-existent relationship. A cloud of points generally tilted upward toward the right side of the graph signals a positive relationship (higher scores on one variable associated with higher scores on the other and vice-versa). A cloud of points generally tilted downward toward the right side of the graph signals a negative relationship (higher scores on one variable associated with lower scores on the other and vice-versa).

Maree might be interested in displaying the relationship between inspection accuracy and inspection speed in the QCI database. Figure 5.8 , produced using SPSS, shows what such a scatterplot might look like. Several characteristics of the data for these two variables can be noted in Fig. 5.8 . The shape of the distribution of data points is evident. The plot has a fan-shaped characteristic to it which indicates that accuracy scores are highly variable (exhibit a very wide range of possible scores) at very fast inspection speeds but get much less variable and tend to be somewhat higher as inspection speed increases (where inspectors take longer to make their quality control decisions). Thus, there does appear to be some relationship between inspection accuracy and inspection speed (a weak positive relationship since the cloud of points tends to be very loose but tilted generally upward toward the right side of the graph – slower speeds tend to be slightly associated with higher accuracy.

Scatterplot relating inspection accuracy to inspection speed

However, it is not the case that the inspection decisions which take longest to make are necessarily the most accurate (see the labelled points for inspectors 7 and 62 in Fig. 5.8 ). Thus, Fig. 5.8 does not show a simple relationship that can be unambiguously summarised by a statement like “the longer an inspector takes to make a quality control decision, the more accurate that decision is likely to be”. The story is more complicated.

Some software packages, such as SPSS, STATGRAPHICS and SYSTAT, offer the option of using different plotting symbols or markers to represent the members of different groups so that the relationship between the two focal variables (the ones anchoring the X and Y axes) can be clarified with reference to a third categorical measure.

Maree might want to see if the relationship depicted in Fig. 5.8 changes depending upon whether the inspector was tertiary-qualified or not (this information is represented in the educlev variable of the QCI database).

Figure 5.9 shows what such a modified scatterplot might look like; the legend in the upper corner of the figure defines the marker symbols for each category of the educlev variable. Note that for both High School only-educated inspectors and Tertiary-qualified inspectors, the general fan-shaped relationship between accuracy and speed is the same. However, it appears that the distribution of points for the High School only-educated inspectors is shifted somewhat upward and toward the right of the plot suggesting that these inspectors tend to be somewhat more accurate as well as slower in their decision processes.

Scatterplot displaying accuracy vs speed conditional on educlev group

There are many other styles of graphs available, often dependent upon the specific statistical package you are using. Interestingly, NCSS and, particularly, SYSTAT and STATGRAPHICS, appear to offer the most variety in terms of types of graphs available for visually representing data. A reading of the user’s manuals for these programs (see the Useful additional readings) would expose you to the great diversity of plotting techniques available to researchers. Many of these techniques go by rather interesting names such as: Chernoff’s faces, radar plots, sunflower plots, violin plots, star plots, Fourier blobs, and dot plots.

These graphical methods provide summary techniques for visually presenting certain characteristics of a set of data. Visual representations are generally easier to understand than a tabular representation and when these plots are combined with available numerical statistics, they can give a very complete picture of a sample of data. Newer methods have become available which permit more complex representations to be depicted, opening possibilities for creatively visually representing more aspects and features of the data (leading to a style of visual data storytelling called infographics ; see, for example, McCandless 2014 ; Toseland and Toseland 2012 ). Many of these newer methods can display data patterns from multiple variables in the same graph (several of these newer graphical methods are illustrated and discussed in Procedure 5.3 ).

Graphs tend to be cumbersome and space consuming if a great many variables need to be summarised. In such cases, using numerical summary statistics (such as means or correlations) in tabular form alone will provide a more economical and efficient summary. Also, it can be very easy to give a misleading picture of data trends using graphical methods by simply choosing the ‘correct’ scaling for maximum effect or choosing a display option (such as a 3-D effect) that ‘looks’ presentable but which actually obscures a clear interpretation (see Smithson 2000 ; Wilkinson 2009 ).

Thus, you must be careful in creating and interpreting visual representations so that the influence of aesthetic choices for sake of appearance do not become more important than obtaining a faithful and valid representation of the data—a very real danger with many of today’s statistical packages where ‘default’ drawing options have been pre-programmed in. No single plot can completely summarise all possible characteristics of a sample of data. Thus, choosing a specific method of graphical display may, of necessity, force a behavioural researcher to represent certain data characteristics (such as frequency) at the expense of others (such as averages).

Virtually any research design which produces quantitative data and statistics (even to the extent of just counting the number of occurrences of several events) provides opportunities for graphical data display which may help to clarify or illustrate important data characteristics or relationships. Remember, graphical displays are communication tools just like numbers—which tool to choose depends upon the message to be conveyed. Visual representations of data are generally more useful in communicating to lay persons who are unfamiliar with statistics. Care must be taken though as these same lay people are precisely the people most likely to misinterpret a graph if it has been incorrectly drawn or scaled.

Procedure 5.3: Multivariate Graphs & Displays

Graphical methods for displaying multivariate data (i.e. many variables at once) include scatterplot matrices, radar (or spider) plots, multiplots, parallel coordinate displays, and icon plots. Multivariate graphs are useful for visualising broad trends and patterns across many variables (Cleveland 1995 ; Jacoby 1998 ). Such graphs typically sacrifice precision in representation in favour of a snapshot pictorial summary that can help you form general impressions of data patterns.

It is important to note that what is presented here is a small but reasonably representative sampling of the types of graphs one can produce to summarise and display trends in multivariate data. Generally speaking, SYSTAT offers the best facilities for producing multivariate graphs, followed by STATGRAPHICS, but with the drawback that it is somewhat tricky to get the graphs in exactly the form you want. SYSTAT also has excellent facilities for creating new forms and combinations of graphs – essentially allowing graphs to be tailor-made for a specific communication purpose. Both SPSS and NCSS offer a more limited range of multivariate graphs, generally restricted to scatterplot matrices and variations of multiplots. Microsoft Excel or STATGRAPHICS are the packages to use if radar or spider plots are desired.

Scatterplot Matrices

A scatterplot matrix is a useful multivariate graph designed to show relationships between pairs of many variables in the same display.

Figure 5.10 illustrates a scatterplot matrix, produced using SYSTAT, for the mentabil , accuracy , speed , jobsat and workcond variables in the QCI database. It is easy to see that all the scatterplot matrix does is stack all pairs of scatterplots into a format where it is easy to pick out the graph for any ‘row’ variable that intersects a column ‘variable’.

Scatterplot matrix relating mentabil , accuracy , speed , jobsat & workcond

In those plots where a ‘row’ variable intersects itself in a column of the matrix (along the so-called ‘diagonal’), SYSTAT permits a range of univariate displays to be shown. Figure 5.10 shows univariate histograms for each variable (recall Procedure 5.2 ). One obvious drawback of the scatterplot matrix is that, if many variables are to be displayed (say ten or more); the graph gets very crowded and becomes very hard to visually appreciate.

Looking at the first column of graphs in Fig. 5.10 , we can see the scatterplot relationships between mentabil and each of the other variables. We can get a visual impression that mentabil seems to be slightly negatively related to accuracy (the cloud of scatter points tends to angle downward to the right, suggesting, very slightly, that higher mentabil scores are associated with lower levels of accuracy ).

Conversely, the visual impression of the relationship between mentabil and speed is that the relationship is slightly positive (higher mentabil scores tend to be associated with higher speed scores = longer inspection times). Similar types of visual impressions can be formed for other parts of Fig. 5.10 . Notice that the histogram plots along the diagonal give a clear impression of the shape of the distribution for each variable.

Radar Plots

The radar plot (also known as a spider graph for obvious reasons) is a simple and effective device for displaying scores on many variables. Microsoft Excel offers a range of options and capabilities for producing radar plots, such as the plot shown in Fig. 5.11 . Radar plots are generally easy to interpret and provide a good visual basis for comparing plots from different individuals or groups, even if a fairly large number of variables (say, up to about 25) are being displayed. Like a clock face, variables are evenly spaced around the centre of the plot in clockwise order starting at the 12 o’clock position. Visual interpretation of a radar plot primarily relies on shape comparisons, i.e. the rise and fall of peaks and valleys along the spokes around the plot. Valleys near the centre display low scores on specific variables, peaks near the outside of the plot display high scores on specific variables. [Note that, technically, radar plots employ polar coordinates.] SYSTAT can draw graphs using polar coordinates but not as easily as Excel can, from the user’s perspective. Radar plots work best if all the variables represented are measured on the same scale (e.g. a 1 to 7 Likert-type scale or 0% to 100% scale). Individuals who are missing any scores on the variables being plotted are typically omitted.

Radar plot comparing attitude ratings for inspectors 66 and 104

The radar plot in Fig. 5.11 , produced using Excel, compares two specific inspectors, 66 and 104, on the nine attitude rating scales. Inspector 66 gave the highest rating (= 7) on the cultqual variable and inspector 104 gave the lowest rating (= 1). The plot shows that inspector 104 tended to provide very low ratings on all nine attitude variables, whereas inspector 66 tended to give very high ratings on all variables except acctrain and trainapp , where the scores were similar to those for inspector 104. Thus, in general, inspector 66 tended to show much more positive attitudes toward their workplace compared to inspector 104.

While Fig. 5.11 was generated to compare the scores for two individuals in the QCI database, it would be just as easy to produce a radar plot that compared the five types of companies in terms of their average ratings on the nine variables, as shown in Fig. 5.12 .

Radar plot comparing average attitude ratings for five types of company

Here we can form the visual impression that the five types of companies differ most in their average ratings of mgmtcomm and least in the average ratings of polsatis . Overall, the average ratings from inspectors from PC manufacturers (black diamonds with solid lines) seem to be generally the most positive as their scores lie on or near the outer ring of scores and those from Automobile manufacturers tend to be least positive on many variables (except the training-related variables).

Extrapolating from Fig. 5.12 , you may rightly conclude that including too many groups and/or too many variables in a radar plot comparison can lead to so much clutter that any visual comparison would be severely degraded. You may have to experiment with using colour-coded lines to represent different groups versus line and marker shape variations (as used in Fig. 5.12 ), because choice of coding method for groups can influence the interpretability of a radar plot.

A multiplot is simply a hybrid style of graph that can display group comparisons across a number of variables. There are a wide variety of possible multiplots one could potentially design (SYSTAT offers great capabilities with respect to multiplots). Figure 5.13 shows a multiplot comprising a side-by-side series of profile-based line graphs – one graph for each type of company in the QCI database.

Multiplot comparing profiles of average attitude ratings for five company types

The multiplot in Fig. 5.13 , produced using SYSTAT, graphs the profile of average attitude ratings for all inspectors within a specific type of company. This multiplot shows the same story as the radar plot in Fig. 5.12 , but in a different graphical format. It is still fairly clear that the average ratings from inspectors from PC manufacturers tend to be higher than for the other types of companies and the profile for inspectors from automobile manufacturers tends to be lower than for the other types of companies.

The profile for inspectors from large electrical appliance manufacturers is the flattest, meaning that their average attitude ratings were less variable than for other types of companies. Comparing the ease with which you can glean the visual impressions from Figs. 5.12 and 5.13 may lead you to prefer one style of graph over another. If you have such preferences, chances are others will also, which may mean you need to carefully consider your options when deciding how best to display data for effect.

Frequently, choice of graph is less a matter of which style is right or wrong, but more a matter of which style will suit specific purposes or convey a specific story, i.e. the choice is often strategic.

Parallel Coordinate Displays

A parallel coordinate display is useful for displaying individual scores on a range of variables, all measured using the same scale. Furthermore, such graphs can be combined side-by-side to facilitate very broad visual comparisons among groups, while retaining individual profile variability in scores. Each line in a parallel coordinate display represents one individual, e.g. an inspector.

The interpretation of a parallel coordinate display, such as the two shown in Fig. 5.14 , depends on visual impressions of the peaks and valleys (highs and lows) in the profiles as well as on the density of similar profile lines. The graph is called ‘parallel coordinate’ simply because it assumes that all variables are measured on the same scale and that scores for each variable can therefore be located along vertical axes that are parallel to each other (imagine vertical lines on Fig. 5.14 running from bottom to top for each variable on the X-axis). The main drawback of this method of data display is that only those individuals in the sample who provided legitimate scores on all of the variables being plotted (i.e. who have no missing scores) can be displayed.

Parallel coordinate displays comparing profiles of average attitude ratings for five company types

The parallel coordinate display in Fig. 5.14 , produced using SYSTAT, graphs the profile of average attitude ratings for all inspectors within two specific types of company: the left graph for inspectors from PC manufacturers and the right graph for automobile manufacturers.

There are fewer lines in each display than the number of inspectors from each type of company simply because several inspectors from each type of company were missing a rating on at least one of the nine attitude variables. The graphs show great variability in scores amongst inspectors within a company type, but there are some overall patterns evident.

For example, inspectors from automobile companies clearly and fairly uniformly rated mgmtcomm toward the low end of the scale, whereas the reverse was generally true for that variable for inspectors from PC manufacturers. Conversely, inspectors from automobile companies tend to rate acctrain and trainapp more toward the middle to high end of the scale, whereas the reverse is generally true for those variables for inspectors from PC manufacturers.

Perhaps the most creative types of multivariate displays are the so-called icon plots . SYSTAT and STATGRAPHICS offer an impressive array of different types of icon plots, including, amongst others, Chernoff’s faces, profile plots, histogram plots, star glyphs and sunray plots (Jacoby 1998 provides a detailed discussion of icon plots).

Icon plots generally use a specific visual construction to represent variables scores obtained by each individual within a sample or group. All icon plots are thus methods for displaying the response patterns for individual members of a sample, as long as those individuals are not missing any scores on the variables to be displayed (note that this is the same limitation as for radar plots and parallel coordinate displays). To illustrate icon plots, without generating too many icons to focus on, Figs. 5.15 , 5.16 , 5.17 and 5.18 present four different icon plots for QCI inspectors classified, using a new variable called BEST_WORST , as either the worst performers (= 1 where their accuracy scores were less than 70%) or the best performers (= 2 where their accuracy scores were 90% or greater).

Chernoff’s faces icon plot comparing individual attitude ratings for best and worst performing inspectors

Profile plot comparing individual attitude ratings for best and worst performing inspectors

Histogram plot comparing individual attitude ratings for best and worst performing inspectors

Sunray plot comparing individual attitude ratings for best and worst performing inspectors

The Chernoff’s faces plot gets its name from the visual icon used to represent variable scores – a cartoon-type face. This icon tries to capitalise on our natural human ability to recognise and differentiate faces. Each feature of the face is controlled by the scores on a single variable. In SYSTAT, up to 20 facial features are controllable; the first five being curvature of mouth, angle of brow, width of nose, length of nose and length of mouth (SYSTAT Software Inc., 2009 , p. 259). The theory behind Chernoff’s faces is that similar patterns of variable scores will produce similar looking faces, thereby making similarities and differences between individuals more apparent.

The profile plot and histogram plot are actually two variants of the same type of icon plot. A profile plot represents individuals’ scores for a set of variables using simplified line graphs, one per individual. The profile is scaled so that the vertical height of the peaks and valleys correspond to actual values for variables where the variables anchor the X-axis in a fashion similar to the parallel coordinate display. So, as you examine a profile from left to right across the X-axis of each graph, you are looking across the set of variables. A histogram plot represents the same information in the same way as for the profile plot but using histogram bars instead.

Figure 5.15 , produced using SYSTAT, shows a Chernoff’s faces plot for the best and worst performing inspectors using their ratings of job satisfaction, working conditions and the nine general attitude statements.

Each face is labelled with the inspector number it represents. The gaps indicate where an inspector had missing data on at least one of the variables, meaning a face could not be generated for them. The worst performers are drawn using red lines; the best using blue lines. The first variable is jobsat and this variable controls mouth curvature; the second variable is workcond and this controls angle of brow, and so on. It seems clear that there are differences in the faces between the best and worst performers with, for example, best performers tending to be more satisfied (smiling) and with higher ratings for working conditions (brow angle).

Beyond a broad visual impression, there is little in terms of precise inferences you can draw from a Chernoff’s faces plot. It really provides a visual sketch, nothing more. The fact that there is no obvious link between facial features, variables and score levels means that the Chernoff’s faces icon plot is difficult to interpret at the level of individual variables – a holistic impression of similarity and difference is what this type of plot facilitates.

Figure 5.16 produced using SYSTAT, shows a profile plot for the best and worst performing inspectors using their ratings of job satisfaction, working conditions and the nine attitude variables.

Like the Chernoff’s faces plot (Fig. 5.15 ), as you read across the rows of the plot from left to right, each plot corresponds respectively to a inspector in the sample who was either in the worst performer (red) or best performer (blue) category. The first attitude variable is jobsat and anchors the left end of each line graph; the last variable is polsatis and anchors the right end of the line graph. The remaining variables are represented in order from left to right across the X-axis of each graph. Figure 5.16 shows that these inspectors are rather different in their attitude profiles, with best performers tending to show taller profiles on the first two variables, for example.

Figure 5.17 produced using SYSTAT, shows a histogram plot for the best and worst performing inspectors based on their ratings of job satisfaction, working conditions and the nine attitude variables. This plot tells the same story as the profile plot, only using histogram bars. Some people would prefer the histogram icon plot to the profile plot because each histogram bar corresponds to one variable, making the visual linking of a specific bar to a specific variable much easier than visually linking a specific position along the profile line to a specific variable.

The sunray plot is actually a simplified adaptation of the radar plot (called a “star glyph”) used to represent scores on a set of variables for each individual within a sample or group. Remember that a radar plot basically arranges the variables around a central point like a clock face; the first variable is represented at the 12 o’clock position and the remaining variables follow around the plot in a clockwise direction.

Unlike a radar plot, while the spokes (the actual ‘star’ of the glyph’s name) of the plot are visible, no interpretive scale is evident. A variable’s score is visually represented by its distance from the central point. Thus, the star glyphs in a sunray plot are designed, like Chernoff’s faces, to provide a general visual impression, based on icon shape. A wide diameter well-rounded plot indicates an individual with high scores on all variables and a small diameter well-rounded plot vice-versa. Jagged plots represent individuals with highly variable scores across the variables. ‘Stars’ of similar size, shape and orientation represent similar individuals.

Figure 5.18 , produced using STATGRAPHICS, shows a sunray plot for the best and worst performing inspectors. An interpretation glyph is also shown in the lower right corner of Fig. 5.18 , where variables are aligned with the spokes of a star (e.g. jobsat is at the 12 o’clock position). This sunray plot could lead you to form the visual impression that the worst performing inspectors (group 1) have rather less rounded rating profiles than do the best performing inspectors (group 2) and that the jobsat and workcond spokes are generally lower for the worst performing inspectors.

Comparatively speaking, the sunray plot makes identifying similar individuals a bit easier (perhaps even easier than Chernoff’s faces) and, when ordered as STATGRAPHICS showed in Fig. 5.18 , permits easier visual comparisons between groups of individuals, but at the expense of precise knowledge about variable scores. Remember, a holistic impression is the goal pursued using a sunray plot.

Multivariate graphical methods provide summary techniques for visually presenting certain characteristics of a complex array of data on variables. Such visual representations are generally better at helping us to form holistic impressions of multivariate data rather than any sort of tabular representation or numerical index. They also allow us to compress many numerical measures into a finite representation that is generally easy to understand. Multivariate graphical displays can add interest to an otherwise dry statistical reporting of numerical data. They are designed to appeal to our pattern recognition skills, focusing our attention on features of the data such as shape, level, variability and orientation. Some multivariate graphs (e.g. radar plots, sunray plots and multiplots) are useful not only for representing score patterns for individuals but also providing summaries of score patterns across groups of individuals.

Multivariate graphs tend to get very busy-looking and are hard to interpret if a great many variables or a large number of individuals need to be displayed (imagine any of the icon plots, for a sample of 200 questionnaire participants, displayed on a A4 page – each icon would be so small that its features could not be easily distinguished, thereby defeating the purpose of the display). In such cases, using numerical summary statistics (such as averages or correlations) in tabular form alone will provide a more economical and efficient summary. Also, some multivariate displays will work better for conveying certain types of information than others.

Information about variable relationships may be better displayed using a scatterplot matrix. Information about individual similarities and difference on a set of variables may be better conveyed using a histogram or sunray plot. Multiplots may be better suited to displaying information about group differences across a set of variables. Information about the overall similarity of individual entities in a sample might best be displayed using Chernoff’s faces.

Because people differ greatly in their visual capacities and preferences, certain types of multivariate displays will work for some people and not others. Sometimes, people will not see what you see in the plots. Some plots, such as Chernoff’s faces, may not strike a reader as a serious statistical procedure and this could adversely influence how convinced they will be by the story the plot conveys. None of the multivariate displays described here provide sufficiently precise information for solid inferences or interpretations; all are designed to simply facilitate the formation of holistic visual impressions. In fact, you may have noticed that some displays (scatterplot matrices and the icon plots, for example) provide no numerical scaling information that would help make precise interpretations. If precision in summary information is desired, the types of multivariate displays discussed here would not be the best strategic choices.

Virtually any research design which produces quantitative data/statistics for multiple variables provides opportunities for multivariate graphical data display which may help to clarify or illustrate important data characteristics or relationships. Thus, for survey research involving many identically-scaled attitudinal questions, a multivariate display may be just the device needed to communicate something about patterns in the data. Multivariate graphical displays are simply specialised communication tools designed to compress a lot of information into a meaningful and efficient format for interpretation—which tool to choose depends upon the message to be conveyed.

Generally speaking, visual representations of multivariate data could prove more useful in communicating to lay persons who are unfamiliar with statistics or who prefer visual as opposed to numerical information. However, these displays would probably require some interpretive discussion so that the reader clearly understands their intent.

Procedure 5.4: Assessing Central Tendency

The three most commonly reported measures of central tendency are the mean, median and mode. Each measure reflects a specific way of defining central tendency in a distribution of scores on a variable and each has its own advantages and disadvantages.

The mean is the most widely used measure of central tendency (also called the arithmetic average). Very simply, a mean is the sum of all the scores for a specific variable in a sample divided by the number of scores used in obtaining the sum. The resulting number reflects the average score for the sample of individuals on which the scores were obtained. If one were asked to predict the score that any single individual in the sample would obtain, the best prediction, in the absence of any other relevant information, would be the sample mean. Many parametric statistical methods (such as Procedures 10.1007/978-981-15-2537-7_7#Sec22 , 10.1007/978-981-15-2537-7_7#Sec32 , 10.1007/978-981-15-2537-7_7#Sec42 and 10.1007/978-981-15-2537-7_7#Sec68) deal with sample means in one way or another. For any sample of data, there is one and only one possible value for the mean in a specific distribution. For most purposes, the mean is the preferred measure of central tendency because it utilises all the available information in a sample.

In the context of the QCI database, Maree could quite reasonably ask what inspectors scored on the average in terms of mental ability ( mentabil ), inspection accuracy ( accuracy ), inspection speed ( speed ), overall job satisfaction ( jobsat ), and perceived quality of their working conditions ( workcond ). Table 5.3 shows the mean scores for the sample of 112 quality control inspectors on each of these variables. The statistics shown in Table 5.3 were computed using the SPSS Frequencies ... procedure. Notice that the table indicates how many of the 112 inspectors had a valid score for each variable and how many were missing a score (e.g. 109 inspectors provided a valid rating for jobsat; 3 inspectors did not).

Measures of central tendency for specific QCI variables

Each mean needs to be interpreted in terms of the original units of measurement for each variable. Thus, the inspectors in the sample showed an average mental ability score of 109.84 (higher than the general population mean of 100 for the test), an average inspection accuracy of 82.14%, and an average speed for making quality control decisions of 4.48 s. Furthermore, in terms of their work context, inspectors reported an average overall job satisfaction of 4.96 (on the 7-point scale, or a level of satisfaction nearly one full scale point above the Neutral point of 4—indicating a generally positive but not strong level of job satisfaction, and an average perceived quality of work conditions of 4.21 (on the 7-point scale which is just about at the level of Stressful but Tolerable.

The mean is sensitive to the presence of extreme values, which can distort its value, giving a biased indication of central tendency. As we will see below, the median is an alternative statistic to use in such circumstances. However, it is also possible to compute what is called a trimmed mean where the mean is calculated after a certain percentage (say, 5% or 10%) of the lowest and highest scores in a distribution have been ignored (a process called ‘trimming’; see, for example, the discussion in Field 2018 , pp. 262–264). This yields a statistic less influenced by extreme scores. The drawbacks are that the decision as to what percentage to trim can be somewhat subjective and trimming necessarily sacrifices information (i.e. the extreme scores) in order to achieve a less biased measure. Some software packages, such as SPSS, SYSTAT or NCSS, can report a specific percentage trimmed mean, if that option is selected for descriptive statistics or exploratory data analysis (see Procedure 5.6 ) procedures. Comparing the original mean with a trimmed mean can provide an indication of the degree to which the original mean has been biased by extreme values.

Very simply, the median is the centre or middle score of a set of scores. By ‘centre’ or ‘middle’ is meant that 50% of the data values are smaller than or equal to the median and 50% of the data values are larger when the entire distribution of scores is rank ordered from the lowest to highest value. Thus, we can say that the median is that score in the sample which occurs at the 50th percentile. [Note that a ‘percentile’ is attached to a specific score that a specific percentage of the sample scored at or below. Thus, a score at the 25th percentile means that 25% of the sample achieved this score or a lower score.] Table 5.3 shows the 25th, 50th and 75th percentile scores for each variable – note how the 50th percentile score is exactly equal to the median in each case .

The median is reported somewhat less frequently than the mean but does have some advantages over the mean in certain circumstances. One such circumstance is when the sample of data has a few extreme values in one direction (either very large or very small relative to all other scores). In this case, the mean would be influenced (biased) to a much greater degree than would the median since all of the data are used to calculate the mean (including the extreme scores) whereas only the single centre score is needed for the median. For this reason, many nonparametric statistical procedures (such as Procedures 10.1007/978-981-15-2537-7_7#Sec27 , 10.1007/978-981-15-2537-7_7#Sec37 and 10.1007/978-981-15-2537-7_7#Sec63) focus on the median as the comparison statistic rather than on the mean.

A discrepancy between the values for the mean and median of a variable provides some insight to the degree to which the mean is being influenced by the presence of extreme data values. In a distribution where there are no extreme values on either side of the distribution (or where extreme values balance each other out on either side of the distribution, as happens in a normal distribution – see Fundamental Concept II ), the mean and the median will coincide at the same value and the mean will not be biased.

For highly skewed distributions, however, the value of the mean will be pulled toward the long tail of the distribution because that is where the extreme values lie. However, in such skewed distributions, the median will be insensitive (statisticians call this property ‘robustness’) to extreme values in the long tail. For this reason, the direction of the discrepancy between the mean and median can give a very rough indication of the direction of skew in a distribution (‘mean larger than median’ signals possible positive skewness; ‘mean smaller than median’ signals possible negative skewness). Like the mean, there is one and only one possible value for the median in a specific distribution.

In Fig. 5.19 , the left graph shows the distribution of speed scores and the right-hand graph shows the distribution of accuracy scores. The speed distribution clearly shows the mean being pulled toward the right tail of the distribution whereas the accuracy distribution shows the mean being just slightly pulled toward the left tail. The effect on the mean is stronger in the speed distribution indicating a greater biasing effect due to some very long inspection decision times.

Effects of skewness in a distribution on the values for the mean and median

If we refer to Table 5.3 , we can see that the median score for each of the five variables has also been computed. Like the mean, the median must be interpreted in the original units of measurement for the variable. We can see that for mentabil , accuracy , and workcond , the value of the median is very close to the value of the mean, suggesting that these distributions are not strongly influenced by extreme data values in either the high or low direction. However, note that the median speed was 3.89 s compared to the mean of 4.48 s, suggesting that the distribution of speed scores is positively skewed (the mean is larger than the median—refer to Fig. 5.19 ). Conversely, the median jobsat score was 5.00 whereas the mean score was 4.96 suggesting very little substantive skewness in the distribution (mean and median are nearly equal).

The mode is the simplest measure of central tendency. It is defined as the most frequently occurring score in a distribution. Put another way, it is the score that more individuals in the sample obtain than any other score. An interesting problem associated with the mode is that there may be more than one in a specific distribution. In the case where multiple modes exist, the issue becomes which value do you report? The answer is that you must report all of them. In a ‘normal’ bell-shaped distribution, there is only one mode and it is indeed at the centre of the distribution, coinciding with both the mean and the median.

Table 5.3 also shows the mode for each of the five variables. For example, more inspectors achieved a mentabil score of 111 more often than any other score and inspectors reported a jobsat rating of 6 more often than any other rating. SPSS only ever reports one mode even if several are present, so one must be careful and look at a histogram plot for each variable to make a final determination of the mode(s) for that variable.

All three measures of central tendency yield information about what is going on in the centre of a distribution of scores. The mean and median provide a single number which can summarise the central tendency in the entire distribution. The mode can yield one or multiple indices. With many measurements on individuals in a sample, it is advantageous to have single number indices which can describe the distributions in summary fashion. In a normal or near-normal distribution of sample data, the mean, the median, and the mode will all generally coincide at the one point. In this instance, all three statistics will provide approximately the same indication of central tendency. Note however that it is seldom the case that all three statistics would yield exactly the same number for any particular distribution. The mean is the most useful statistic, unless the data distribution is skewed by extreme scores, in which case the median should be reported.

While measures of central tendency are useful descriptors of distributions, summarising data using a single numerical index necessarily reduces the amount of information available about the sample. Not only do we need to know what is going on in the centre of a distribution, we also need to know what is going on around the centre of the distribution. For this reason, most social and behavioural researchers report not only measures of central tendency, but also measures of variability (see Procedure 5.5 ). The mode is the least informative of the three statistics because of its potential for producing multiple values.

Measures of central tendency are useful in almost any type of experimental design, survey or interview study, and in any observational studies where quantitative data are available and must be summarised. The decision as to whether the mean or median should be reported depends upon the nature of the data which should ideally be ascertained by visual inspection of the data distribution. Some researchers opt to report both measures routinely. Computation of means is a prelude to many parametric statistical methods (see, for example, Procedure 10.1007/978-981-15-2537-7_7#Sec22 , 10.1007/978-981-15-2537-7_7#Sec32 , 10.1007/978-981-15-2537-7_7#Sec42 , 10.1007/978-981-15-2537-7_7#Sec52 , 10.1007/978-981-15-2537-7_7#Sec68 , 10.1007/978-981-15-2537-7_7#Sec76 and 10.1007/978-981-15-2537-7_7#Sec105); comparison of medians is associated with many nonparametric statistical methods (see, for example, Procedure 10.1007/978-981-15-2537-7_7#Sec27 , 10.1007/978-981-15-2537-7_7#Sec37 , 10.1007/978-981-15-2537-7_7#Sec63 and 10.1007/978-981-15-2537-7_7#Sec81).

Procedure 5.5: Assessing Variability

There are a variety of measures of variability to choose from including the range, interquartile range, variance and standard deviation. Each measure reflects a specific way of defining variability in a distribution of scores on a variable and each has its own advantages and disadvantages. Most measures of variability are associated with a specific measure of central tendency so that researchers are now commonly expected to report both a measure of central tendency and its associated measure of variability whenever they display numerical descriptive statistics on continuous or ranked-ordered variables.

This is the simplest measure of variability for a sample of data scores. The range is merely the largest score in the sample minus the smallest score in the sample. The range is the one measure of variability not explicitly associated with any measure of central tendency. It gives a very rough indication as to the extent of spread in the scores. However, since the range uses only two of the total available scores in the sample, the rest of the scores are ignored, which means that a lot of potentially useful information is being sacrificed. There are also problems if either the highest or lowest (or both) scores are atypical or too extreme in their value (as in highly skewed distributions). When this happens, the range gives a very inflated picture of the typical variability in the scores. Thus, the range tends not be a frequently reported measure of variability.

Table 5.4 shows a set of descriptive statistics, produced by the SPSS Frequencies procedure, for the mentabil, accuracy, speed, jobsat and workcond measures in the QCI database. In the table, you will find three rows labelled ‘Range’, ‘Minimum’ and ‘Maximum’.

Measures of central tendency and variability for specific QCI variables

Using the data from these three rows, we can draw the following descriptive picture. Mentabil scores spanned a range of 50 (from a minimum score of 85 to a maximum score of 135). Speed scores had a range of 16.05 s (from 1.05 s – the fastest quality decision to 17.10 – the slowest quality decision). Accuracy scores had a range of 43 (from 57% – the least accurate inspector to 100% – the most accurate inspector). Both work context measures ( jobsat and workcond ) exhibited a range of 6 – the largest possible range given the 1 to 7 scale of measurement for these two variables.

Interquartile Range

The Interquartile Range ( IQR ) is a measure of variability that is specifically designed to be used in conjunction with the median. The IQR also takes care of the extreme data problem which typically plagues the range measure. The IQR is defined as the range that is covered by the middle 50% of scores in a distribution once the scores have been ranked in order from lowest value to highest value. It is found by locating the value in the distribution at or below which 25% of the sample scored and subtracting this number from the value in the distribution at or below which 75% of the sample scored. The IQR can also be thought of as the range one would compute after the bottom 25% of scores and the top 25% of scores in the distribution have been ‘chopped off’ (or ‘trimmed’ as statisticians call it).

The IQR gives a much more stable picture of the variability of scores and, like the median, is relatively insensitive to the biasing effects of extreme data values. Some behavioural researchers prefer to divide the IQR in half which gives a measure called the Semi-Interquartile Range ( S-IQR ) . The S-IQR can be interpreted as the distance one must travel away from the median, in either direction, to reach the value which separates the top (or bottom) 25% of scores in the distribution from the remaining 75%.

The IQR or S-IQR is typically not produced by descriptive statistics procedures by default in many computer software packages; however, it can usually be requested as an optional statistic to report or it can easily be computed by hand using percentile scores. Both the median and the IQR figure prominently in Exploratory Data Analysis, particularly in the production of boxplots (see Procedure 5.6 ).

Figure 5.20 illustrates the conceptual nature of the IQR and S-IQR compared to that of the range. Assume that 100% of data values are covered by the distribution curve in the figure. It is clear that these three measures would provide very different values for a measure of variability. Your choice would depend on your purpose. If you simply want to signal the overall span of scores between the minimum and maximum, the range is the measure of choice. But if you want to signal the variability around the median, the IQR or S-IQR would be the measure of choice.

How the range, IQR and S-IQR measures of variability conceptually differ

Note: Some behavioural researchers refer to the IQR as the hinge-spread (or H-spread ) because of its use in the production of boxplots:

• the 25th percentile data value is referred to as the ‘lower hinge’;
• the 75th percentile data value is referred to as the ‘upper hinge’; and
• their difference gives the H-spread.

Midspread is another term you may see used as a synonym for interquartile range.

Referring back to Table 5.4 , we can find statistics reported for the median and for the ‘quartiles’ (25th, 50th and 75th percentile scores) for each of the five variables of interest. The ‘quartile’ values are useful for finding the IQR or S-IQR because SPSS does not report these measures directly. The median clearly equals the 50th percentile data value in the table.

If we focus, for example, on the speed variable, we could find its IQR by subtracting the 25th percentile score of 2.19 s from the 75th percentile score of 5.71 s to give a value for the IQR of 3.52 s (the S-IQR would simply be 3.52 divided by 2 or 1.76 s). Thus, we could report that the median decision speed for inspectors was 3.89 s and that the middle 50% of inspectors showed scores spanning a range of 3.52 s. Alternatively, we could report that the median decision speed for inspectors was 3.89 s and that the middle 50% of inspectors showed scores which ranged 1.76 s either side of the median value.

Note: We could compare the ‘Minimum’ or ‘Maximum’ scores to the 25th percentile score and 75th percentile score respectively to get a feeling for whether the minimum or maximum might be considered extreme or uncharacteristic data values.

The variance uses information from every individual in the sample to assess the variability of scores relative to the sample mean. Variance assesses the average squared deviation of each score from the mean of the sample. Deviation refers to the difference between an observed score value and the mean of the sample—they are squared simply because adding them up in their naturally occurring unsquared form (where some differences are positive and others are negative) always gives a total of zero, which is useless for an index purporting to measure something.

If many scores are quite different from the mean, we would expect the variance to be large. If all the scores lie fairly close to the sample mean, we would expect a small variance. If all scores exactly equal the mean (i.e. all the scores in the sample have the same value), then we would expect the variance to be zero.

Figure 5.21 illustrates some possibilities regarding variance of a distribution of scores having a mean of 100. The very tall curve illustrates a distribution with small variance. The distribution of medium height illustrates a distribution with medium variance and the flattest distribution ia a distribution with large variance.

The concept of variance

If we had a distribution with no variance, the curve would simply be a vertical line at a score of 100 (meaning that all scores were equal to the mean). You can see that as variance increases, the tails of the distribution extend further outward and the concentration of scores around the mean decreases. You may have noticed that variance and range (as well as the IQR) will be related, since the range focuses on the difference between the ends of the two tails in the distribution and larger variances extend the tails. So, a larger variance will generally be associated with a larger range and IQR compared to a smaller variance.

It is generally difficult to descriptively interpret the variance measure in a meaningful fashion since it involves squared deviations around the sample mean. [Note: If you look back at Table 5.4 , you will see the variance listed for each of the variables (e.g. the variance of accuracy scores is 84.118), but the numbers themselves make little sense and do not relate to the original measurement scale for the variables (which, for the accuracy variable, went from 0% to 100% accuracy).] Instead, we use the variance as a steppingstone for obtaining a measure of variability that we can clearly interpret, namely the standard deviation . However, you should know that variance is an important concept in its own right simply because it provides the statistical foundation for many of the correlational procedures and statistical inference procedures described in Chaps. 10.1007/978-981-15-2537-7_6 , 10.1007/978-981-15-2537-7_7 and 10.1007/978-981-15-2537-7_8.

When considering either correlations or tests of statistical hypotheses, we frequently speak of one variable explaining or sharing variance with another (see Procedure 10.1007/978-981-15-2537-7_6#Sec27 and 10.1007/978-981-15-2537-7_7#Sec47 ). In doing so, we are invoking the concept of variance as set out here—what we are saying is that variability in the behaviour of scores on one particular variable may be associated with or predictive of variability in scores on another variable of interest (e.g. it could explain why those scores have a non-zero variance).

Standard Deviation

The standard deviation (often abbreviated as SD, sd or Std. Dev.) is the most commonly reported measure of variability because it has a meaningful interpretation and is used in conjunction with reports of sample means. Variance and standard deviation are closely related measures in that the standard deviation is found by taking the square root of the variance. The standard deviation, very simply, is a summary number that reflects the ‘average distance of each score from the mean of the sample’. In many parametric statistical methods, both the sample mean and sample standard deviation are employed in some form. Thus, the standard deviation is a very important measure, not only for data description, but also for hypothesis testing and the establishment of relationships as well.

Referring again back to Table 5.4 , we’ll focus on the results for the speed variable for discussion purposes. Table 5.4 shows that the mean inspection speed for the QCI sample was 4.48 s. We can also see that the standard deviation (in the row labelled ‘Std Deviation’) for speed was 2.89 s.

This standard deviation has a straightforward interpretation: we would say that ‘on the average, an inspector’s quality inspection decision speed differed from the mean of the sample by about 2.89 s in either direction’. In a normal distribution of scores (see Fundamental Concept II ), we would expect to see about 68% of all inspectors having decision speeds between 1.59 s (the mean minus one amount of the standard deviation) and 7.37 s (the mean plus one amount of the standard deviation).

We noted earlier that the range of the speed scores was 16.05 s. However, the fact that the maximum speed score was 17.1 s compared to the 75th percentile score of just 5.71 s seems to suggest that this maximum speed might be rather atypically large compared to the bulk of speed scores. This means that the range is likely to be giving us a false impression of the overall variability of the inspectors’ decision speeds.

Furthermore, given that the mean speed score was higher than the median speed score, suggesting that speed scores were positively skewed (this was confirmed by the histogram for speed shown in Fig. 5.19 in Procedure 5.4 ), we might consider emphasising the median and its associated IQR or S-IQR rather than the mean and standard deviation. Of course, similar diagnostic and interpretive work could be done for each of the other four variables in Table 5.4 .

Measures of variability (particularly the standard deviation) provide a summary measure that gives an indication of how variable (spread out) a particular sample of scores is. When used in conjunction with a relevant measure of central tendency (particularly the mean), a reasonable yet economical description of a set of data emerges. When there are extreme data values or severe skewness is present in the data, the IQR (or S-IQR) becomes the preferred measure of variability to be reported in conjunction with the sample median (or 50th percentile value). These latter measures are much more resistant (‘robust’) to influence by data anomalies than are the mean and standard deviation.

As mentioned above, the range is a very cursory index of variability, thus, it is not as useful as variance or standard deviation. Variance has little meaningful interpretation as a descriptive index; hence, standard deviation is most often reported. However, the standard deviation (or IQR) has little meaning if the sample mean (or median) is not reported along with it.

Knowing that the standard deviation for accuracy is 9.17 tells you little unless you know the mean accuracy (82.14) that it is the standard deviation from.

Like the sample mean, the standard deviation can be strongly biased by the presence of extreme data values or severe skewness in a distribution in which case the median and IQR (or S-IQR) become the preferred measures. The biasing effect will be most noticeable in samples which are small in size (say, less than 30 individuals) and far less noticeable in large samples (say, in excess of 200 or 300 individuals). [Note that, in a manner similar to a trimmed mean, it is possible to compute a trimmed standard deviation to reduce the biasing effect of extreme data values, see Field 2018 , p. 263.]

It is important to realise that the resistance of the median and IQR (or S-IQR) to extreme values is only gained by deliberately sacrificing a good deal of the information available in the sample (nothing is obtained without a cost in statistics). What is sacrificed is information from all other members of the sample other than those members who scored at the median and 25th and 75th percentile points on a variable of interest; information from all members of the sample would automatically be incorporated in mean and standard deviation for that variable.

Any investigation where you might report on or read about measures of central tendency on certain variables should also report measures of variability. This is particularly true for data from experiments, quasi-experiments, observational studies and questionnaires. It is important to consider measures of central tendency and measures of variability to be inextricably linked—one should never report one without the other if an adequate descriptive summary of a variable is to be communicated.

Other descriptive measures, such as those for skewness and kurtosis 1 may also be of interest if a more complete description of any variable is desired. Most good statistical packages can be instructed to report these additional descriptive measures as well.

Of all the statistics you are likely to encounter in the business, behavioural and social science research literature, means and standard deviations will dominate as measures for describing data. Additionally, these statistics will usually be reported when any parametric tests of statistical hypotheses are presented as the mean and standard deviation provide an appropriate basis for summarising and evaluating group differences.

Fundamental Concept I: Basic Concepts in Probability

The concept of simple probability.

In Procedures 5.1 and 5.2 , you encountered the idea of the frequency of occurrence of specific events such as particular scores within a sample distribution. Furthermore, it is a simple operation to convert the frequency of occurrence of a specific event into a number representing the relative frequency of that event. The relative frequency of an observed event is merely the number of times the event is observed divided by the total number of times one makes an observation. The resulting number ranges between 0 and 1 but we typically re-express this number as a percentage by multiplying it by 100%.

In the QCI database, Maree Lakota observed data from 112 quality control inspectors of which 58 were male and 51 were female (gender indications were missing for three inspectors). The statistics 58 and 51 are thus the frequencies of occurrence for two specific types of research participant, a male inspector or a female inspector.

If she divided each frequency by the total number of observations (i.e. 112), whe would obtain .52 for males and .46 for females (leaving .02 of observations with unknown gender). These statistics are relative frequencies which indicate the proportion of times that Maree obtained data from a male or female inspector. Multiplying each relative frequency by 100% would yield 52% and 46% which she could interpret as indicating that 52% of her sample was male and 46% was female (leaving 2% of the sample with unknown gender).

It does not take much of a leap in logic to move from the concept of ‘relative frequency’ to the concept of ‘probability’. In our discussion above, we focused on relative frequency as indicating the proportion or percentage of times a specific category of participant was obtained in a sample. The emphasis here is on data from a sample.

Imagine now that Maree had infinite resources and research time and was able to obtain ever larger samples of quality control inspectors for her study. She could still compute the relative frequencies for obtaining data from males and females in her sample but as her sample size grew larger and larger, she would notice these relative frequencies converging toward some fixed values.

If, by some miracle, Maree could observe all of the quality control inspectors on the planet today, she would have measured the entire population and her computations of relative frequency for males and females would yield two precise numbers, each indicating the proportion of the population of inspectors that was male and the proportion that was female.

If Maree were then to list all of these inspectors and randomly choose one from the list, the chances that she would choose a male inspector would be equal to the proportion of the population of inspectors that was male and this logic extends to choosing a female inspector. The number used to quantify this notion of ‘chances’ is called a probability. Maree would therefore have established the probability of randomly observing a male or a female inspector in the population on any specific occasion.

Probability is expressed on a 0.0 (the observation or event will certainly not be seen) to 1.0 (the observation or event will certainly be seen) scale where values close to 0.0 indicate observations that are less certain to be seen and values close to 1.0 indicate observations that are more certain to be seen (a value of .5 indicates an even chance that an observation or event will or will not be seen – a state of maximum uncertainty). Statisticians often interpret a probability as the likelihood of observing an event or type of individual in the population.

In the QCI database, we noted that the relative frequency of observing males was .52 and for females was .46. If we take these relative frequencies as estimates of the proportions of each gender in the population of inspectors, then .52 and .46 represent the probability of observing a male or female inspector, respectively.

Statisticians would state this as “the probability of observing a male quality control inspector is .52” or in a more commonly used shorthand code, the likelihood of observing a male quality control inspector is p = .52 (p for probability). For some, probabilities make more sense if they are converted to percentages (by multiplying by 100%). Thus, p = .52 can also understood as a 52% chance of observing a male quality control inspector.

We have seen that relative frequency is a sample statistic that can be used to estimate the population probability. Our estimate will get more precise as we use larger and larger samples (technically, as the size of our samples more closely approximates the size of our population). In most behavioural research, we never have access to entire populations so we must always estimate our probabilities.

In some very special populations, having a known number of fixed possible outcomes, such as results of coin tosses or rolls of a die, we can analytically establish event probabilities without doing an infinite number of observations; all we must do is assume that we have a fair coin or die. Thus, with a fair coin, the probability of observing a H or a T on any single coin toss is ½ or .5 or 50%; the probability of observing a 6 on any single throw of a die is 1/6 or .16667 or 16.667%. With behavioural data, though, we can never measure all possible behavioural outcomes, which thereby forces researchers to depend on samples of observations in order to make estimates of population values.

The concept of probability is central to much of what is done in the statistical analysis of behavioural data. Whenever a behavioural scientist wishes to establish whether a particular relationship exists between variables or whether two groups, treated differently, actually show different behaviours, he/she is playing a probability game. Given a sample of observations, the behavioural scientist must decide whether what he/she has observed is providing sufficient information to conclude something about the population from which the sample was drawn.

This decision always has a non-zero probability of being in error simply because in samples that are much smaller than the population, there is always the chance or probability that we are observing something rare and atypical instead of something which is indicative of a consistent population trend. Thus, the concept of probability forms the cornerstone for statistical inference about which we will have more to say later (see Fundamental Concept 10.1007/978-981-15-2537-7_7#Sec6). Probability also plays an important role in helping us to understand theoretical statistical distributions (e.g. the normal distribution) and what they can tell us about our observations. We will explore this idea further in Fundamental Concept II .

The Concept of Conditional Probability

It is important to understand that the concept of probability as described above focuses upon the likelihood or chances of observing a specific event or type of observation for a specific variable relative to a population or sample of observations. However, many important behavioural research issues may focus on the question of the probability of observing a specific event given that the researcher has knowledge that some other event has occurred or been observed (this latter event is usually measured by a second variable). Here, the focus is on the potential relationship or link between two variables or two events.

With respect to the QCI database, Maree could ask the quite reasonable question “what is the probability (estimated in the QCI sample by a relative frequency) of observing an inspector being female given that she knows that an inspector works for a Large Business Computer manufacturer.

To address this question, all she needs to know is:

• how many inspectors from Large Business Computer manufacturers are in the sample ( 22 ); and
• how many of those inspectors were female ( 7 ) (inspectors who were missing a score for either company or gender have been ignored here).

If she divides 7 by 22, she would obtain the probability that an inspector is female given that they work for a Large Business Computer manufacturer – that is, p = .32 .

This type of question points to the important concept of conditional probability (‘conditional’ because we are asking “what is the probability of observing one event conditional upon our knowledge of some other event”).

Continuing with the previous example, Maree would say that the conditional probability of observing a female inspector working for a Large Business Computer manufacturer is .32 or, equivalently, a 32% chance. Compare this conditional probability of p  = .32 to the overall probability of observing a female inspector in the entire sample ( p  = .46 as shown above).

This means that there is evidence for a connection or relationship between gender and the type of company an inspector works for. That is, the chances are lower for observing a female inspector from a Large Business Computer manufacturer than they are for simply observing a female inspector at all.

Maree therefore has evidence suggesting that females may be relatively under-represented in Large Business Computer manufacturing companies compared to the overall population. Knowing something about the company an inspector works for therefore can help us make a better prediction about their likely gender.

Suppose, however, that Maree’s conditional probability had been exactly equal to p  = .46. This would mean that there was exactly the same chance of observing a female inspector working for a Large Business Computer manufacturer as there was of observing a female inspector in the general population. Here, knowing something about the company an inspector works doesn’t help Maree make any better prediction about their likely gender. This would mean that the two variables are statistically independent of each other.

A classic case of events that are statistically independent is two successive throws of a fair die: rolling a six on the first throw gives us no information for predicting how likely it will be that we would roll a six on the second throw. The conditional probability of observing a six on the second throw given that I have observed a six on the first throw is 0.16667 (= 1 divided by 6) which is the same as the simple probability of observing a six on any specific throw. This statistical independence also means that if we wanted to know what the probability of throwing two sixes on two successive throws of a fair die, we would just multiply the probabilities for each independent event (i.e., throw) together; that is, .16667 × .16667 = .02789 (this is known as the multiplication rule of probability, see, for example, Smithson 2000 , p. 114).

Finally, you should know that conditional probabilities are often asymmetric. This means that for many types of behavioural variables, reversing the conditional arrangement will change the story about the relationship. Bayesian statistics (see Fundamental Concept 10.1007/978-981-15-2537-7_7#Sec73) relies heavily upon this asymmetric relationship between conditional probabilities.

Maree has already learned that the conditional probability that an inspector is female given that they worked for a Large Business Computer manufacturer is p = .32. She could easily turn the conditional relationship around and ask what is the conditional probability that an inspector works for a Large Business Computer manufacturer given that the inspector is female?

From the QCI database, she can find that 51 inspectors in her total sample were female and of those 51, 7 worked for a Large Business Computer manufacturer. If she divided 7 by 51, she would get p = .14 (did you notice that all that changed was the number she divided by?). Thus, there is only a 14% chance of observing an inspector working for a Large Business Computer manufacturer given that the inspector is female – a rather different probability from p = .32, which tells a different story.

As you will see in Procedures 10.1007/978-981-15-2537-7_6#Sec14 and 10.1007/978-981-15-2537-7_7#Sec17, conditional relationships between categorical variables are precisely what crosstabulation contingency tables are designed to reveal.

Procedure 5.6: Exploratory Data Analysis

There are a variety of visual display methods for EDA, including stem & leaf displays, boxplots and violin plots. Each method reflects a specific way of displaying features of a distribution of scores or measurements and, of course, each has its own advantages and disadvantages. In addition, EDA displays are surprisingly flexible and can combine features in various ways to enhance the story conveyed by the plot.

Stem & Leaf Displays

The stem & leaf display is a simple data summary technique which not only rank orders the data points in a sample but presents them visually so that the shape of the data distribution is reflected. Stem & leaf displays are formed from data scores by splitting each score into two parts: the first part of each score serving as the ‘stem’, the second part as the ‘leaf’ (e.g. for 2-digit data values, the ‘stem’ is the number in the tens position; the ‘leaf’ is the number in the ones position). Each stem is then listed vertically, in ascending order, followed horizontally by all the leaves in ascending order associated with it. The resulting display thus shows all of the scores in the sample, but reorganised so that a rough idea of the shape of the distribution emerges. As well, extreme scores can be easily identified in a stem & leaf display.

Consider the accuracy and speed scores for the 112 quality control inspectors in the QCI sample. Figure 5.22 (produced by the R Commander Stem-and-leaf display … procedure) shows the stem & leaf displays for inspection accuracy (left display) and speed (right display) data.

Stem & leaf displays produced by R Commander

[The first six lines reflect information from R Commander about each display: lines 1 and 2 show the actual R command used to produce the plot (the variable name has been highlighted in bold); line 3 gives a warning indicating that inspectors with missing values (= NA in R ) on the variable have been omitted from the display; line 4 shows how the stems and leaves have been defined; line 5 indicates what a leaf unit represents in value; and line 6 indicates the total number (n) of inspectors included in the display).] In Fig. 5.22 , for the accuracy display on the left-hand side, the ‘stems’ have been split into ‘half-stems’—one (which is starred) associated with the ‘leaves’ 0 through 4 and the other associated with the ‘leaves’ 5 through 9—a strategy that gives the display better balance and visual appeal.

Notice how the left stem & leaf display conveys a fairly clear (yet sideways) picture of the shape of the distribution of accuracy scores. It has a rather symmetrical bell-shape to it with only a slight suggestion of negative skewness (toward the extreme score at the top). The right stem & leaf display clearly depicts the highly positively skewed nature of the distribution of speed scores. Importantly, we could reconstruct the entire sample of scores for each variable using its display, which means that unlike most other graphical procedures, we didn’t have to sacrifice any information to produce the visual summary.

Some programs, such as SYSTAT, embellish their stem & leaf displays by indicating in which stem or half-stem the ‘median’ (50th percentile), the ‘upper hinge score’ (75th percentile), and ‘lower hinge score’ (25th percentile) occur in the distribution (recall the discussion of interquartile range in Procedure 5.5 ). This is shown in Fig. 5.23 , produced by SYSTAT, where M and H indicate the stem locations for the median and hinge points, respectively. This stem & leaf display labels a single extreme accuracy score as an ‘outside value’ and clearly shows that this actual score was 57.

Stem & leaf display, produced by SYSTAT, of the accuracy QCI variable

Another important EDA technique is the boxplot or, as it is sometimes known, the box-and-whisker plot . This plot provides a symbolic representation that preserves less of the original nature of the data (compared to a stem & leaf display) but typically gives a better picture of the distributional characteristics. The basic boxplot, shown in Fig. 5.24 , utilises information about the median (50th percentile score) and the upper (75th percentile score) and lower (25th percentile score) hinge points in the construction of the ‘box’ portion of the graph (the ‘median’ defines the centre line in the box; the ‘upper’ and ‘lower hinge values’ define the end boundaries of the box—thus the box encompasses the middle 50% of data values).

Boxplots for the accuracy and speed QCI variables

Additionally, the boxplot utilises the IQR (recall Procedure 5.5 ) as a way of defining what are called ‘fences’ which are used to indicate score boundaries beyond which we would consider a score in a distribution to be an ‘outlier’ (or an extreme or unusual value). In SPSS, the inner fence is typically defined as 1.5 times the IQR in each direction and a ‘far’ outlier or extreme case is typically defined as 3 times the IQR in either direction (Field 2018 , p. 193). The ‘whiskers’ in a boxplot extend out to the data values which are closest to the upper and lower inner fences (in most cases, the vast majority of data values will be contained within the fences). Outliers beyond these ‘whiskers’ are then individually listed. ‘Near’ outliers are those lying just beyond the inner fences and ‘far’ outliers lie well beyond the inner fences.

Figure 5.24 shows two simple boxplots (produced using SPSS), one for the accuracy QCI variable and one for the speed QCI variable. The accuracy plot shows a median value of about 83, roughly 50% of the data fall between about 77 and 89 and there is one outlier, inspector 83, in the lower ‘tail’ of the distribution. The accuracy boxplot illustrates data that are relatively symmetrically distributed without substantial skewness. Such data will tend to have their median in the middle of the box, whiskers of roughly equal length extending out from the box and few or no outliers.

The speed plot shows a median value of about 4 s, roughly 50% of the data fall between 2 s and 6 s and there are four outliers, inspectors 7, 62, 65 and 75 (although inspectors 65 and 75 fall at the same place and are rather difficult to read), all falling in the slow speed ‘tail’ of the distribution. Inspectors 65, 75 and 7 are shown as ‘near’ outliers (open circles) whereas inspector 62 is shown as a ‘far’ outlier (asterisk). The speed boxplot illustrates data which are asymmetrically distributed because of skewness in one direction. Such data may have their median offset from the middle of the box and/or whiskers of unequal length extending out from the box and outliers in the direction of the longer whisker. In the speed boxplot, the data are clearly positively skewed (the longer whisker and extreme values are in the slow speed ‘tail’).

Boxplots are very versatile representations in that side-by-side displays for sub-groups of data within a sample can permit easy visual comparisons of groups with respect to central tendency and variability. Boxplots can also be modified to incorporate information about error bands associated with the median producing what is called a ‘notched boxplot’. This helps in the visual detection of meaningful subgroup differences, where boxplot ‘notches’ don’t overlap.

Figure 5.25 (produced using NCSS), compares the distributions of accuracy and speed scores for QCI inspectors from the five types of companies, plotted side-by-side.

Comparisons of the accuracy (regular boxplots) and speed (notched boxplots) QCI variables for different types of companies

Focus first on the left graph in Fig. 5.25 which plots the distribution of accuracy scores broken down by company using regular boxplots. This plot clearly shows the differing degree of skewness in each type of company (indicated by one or more outliers in one ‘tail’, whiskers which are not the same length and/or the median line being offset from the centre of a box), the differing variability of scores within each type of company (indicated by the overall length of each plot—box and whiskers), and the differing central tendency in each type of company (the median lines do not all fall at the same level of accuracy score). From the left graph in Fig. 5.25 , we could conclude that: inspection accuracy scores are most variable in PC and Large Electrical Appliance manufacturing companies and least variable in the Large Business Computer manufacturing companies; Large Business Computer and PC manufacturing companies have the highest median level of inspection accuracy; and inspection accuracy scores tend to be negatively skewed (many inspectors toward higher levels, relatively fewer who are poorer in inspection performance) in the Automotive manufacturing companies. One inspector, working for an Automotive manufacturing company, shows extremely poor inspection accuracy performance.

The right display compares types of companies in terms of their inspection speed scores, using’ notched’ boxplots. The notches define upper and lower error limits around each median. Aside from the very obvious positive skewness for speed scores (with a number of slow speed outliers) in every type of company (least so for Large Electrical Appliance manufacturing companies), the story conveyed by this comparison is that inspectors from Large Electrical Appliance and Automotive manufacturing companies have substantially faster median decision speeds compared to inspectors from Large Business Computer and PC manufacturing companies (i.e. their ‘notches’ do not overlap, in terms of speed scores, on the display).

Boxplots can also add interpretive value to other graphical display methods through the creation of hybrid displays. Such displays might combine a standard histogram with a boxplot along the X-axis to provide an enhanced picture of the data distribution as illustrated for the mentabil variable in Fig. 5.26 (produced using NCSS). This hybrid plot also employs a data ‘smoothing’ method called a density trace to outline an approximate overall shape for the data distribution. Any one graphical method would tell some of the story, but combined in the hybrid display, the story of a relatively symmetrical set of mentabil scores becomes quite visually compelling.

A hybrid histogram-density-boxplot of the mentabil QCI variable

Violin Plots

Violin plots are a more recent and interesting EDA innovation, implemented in the NCSS software package (Hintze 2012 ). The violin plot gets its name from the rough shape that the plots tend to take on. Violin plots are another type of hybrid plot, this time combining density traces (mirror-imaged right and left so that the plots have a sense of symmetry and visual balance) with boxplot-type information (median, IQR and upper and lower inner ‘fences’, but not outliers). The goal of the violin plot is to provide a quick visual impression of the shape, central tendency and variability of a distribution (the length of the violin conveys a sense of the overall variability whereas the width of the violin conveys a sense of the frequency of scores occurring in a specific region).

Figure 5.27 (produced using NCSS), compares the distributions of speed scores for QCI inspectors across the five types of companies, plotted side-by-side. The violin plot conveys a similar story to the boxplot comparison for speed in the right graph of Fig. 5.25 . However, notice that with the violin plot, unlike with a boxplot, you also get a sense of distributions that have ‘clumps’ of scores in specific areas. Some violin plots, like that for Automobile manufacturing companies in Fig. 5.27 , have a shape suggesting a multi-modal distribution (recall Procedure 5.4 and the discussion of the fact that a distribution may have multiple modes). The violin plot in Fig. 5.27 has also been produced to show where the median (solid line) and mean (dashed line) would fall within each violin. This facilitates two interpretations: (1) a relative comparison of central tendency across the five companies and (2) relative degree of skewness in the distribution for each company (indicated by the separation of the two lines within a violin; skewness is particularly bad for the Large Business Computer manufacturing companies).

Violin plot comparisons of the speed QCI variable for different types of companies

EDA methods (of which we have illustrated only a small subset; we have not reviewed dot density diagrams, for example) provide summary techniques for visually displaying certain characteristics of a set of data. The advantage of the EDA methods over more traditional graphing techniques such as those described in Procedure 5.2 is that as much of the original integrity of the data is maintained as possible while maximising the amount of summary information available about distributional characteristics.

Stem & leaf displays maintain the data in as close to their original form as possible whereas boxplots and violin plots provide more symbolic and flexible representations. EDA methods are best thought of as communication devices designed to facilitate quick visual impressions and they can add interest to any statistical story being conveyed about a sample of data. NCSS, SYSTAT, STATGRAPHICS and R Commander generally offer more options and flexibility in the generation of EDA displays than SPSS.

EDA methods tend to get cumbersome if a great many variables or groups need to be summarised. In such cases, using numerical summary statistics (such as means and standard deviations) will provide a more economical and efficient summary. Boxplots or violin plots are generally more space efficient summary techniques than stem & leaf displays.

Often, EDA techniques are used as data screening devices, which are typically not reported in actual write-ups of research (we will discuss data screening in more detail in Procedure 10.1007/978-981-15-2537-7_8#Sec11). This is a perfectly legitimate use for the methods although there is an argument for researchers to put these techniques to greater use in published literature.

Software packages may use different rules for constructing EDA plots which means that you might get rather different looking plots and different information from different programs (you saw some evidence of this in Figs. 5.22 and 5.23 ). It is important to understand what the programs are using as decision rules for locating fences and outliers so that you are clear on how best to interpret the resulting plot—such information is generally contained in the user’s guides or manuals for NCSS (Hintze 2012 ), SYSTAT (SYSTAT Inc. 2009a , b ), STATGRAPHICS (StatPoint Technologies Inc. 2010 ) and SPSS (Norušis 2012 ).

Virtually any research design which produces numerical measures (even to the extent of just counting the number of occurrences of several events) provides opportunities for employing EDA displays which may help to clarify data characteristics or relationships. One extremely important use of EDA methods is as data screening devices for detecting outliers and other data anomalies, such as non-normality and skewness, before proceeding to parametric statistical analyses. In some cases, EDA methods can help the researcher to decide whether parametric or nonparametric statistical tests would be best to apply to his or her data because critical data characteristics such as distributional shape and spread are directly reflected.

Procedure 5.7: Standard ( z ) Scores

In certain practical situations in behavioural research, it may be desirable to know where a specific individual’s score lies relative to all other scores in a distribution. A convenient measure is to observe how many standard deviations (see Procedure 5.5 ) above or below the sample mean a specific score lies. This measure is called a standard score or z -score . Very simply, any raw score can be converted to a z -score by subtracting the sample mean from the raw score and dividing that result by the sample’s standard deviation. z -scores can be positive or negative and their sign simply indicates whether the score lies above (+) or below (−) the mean in value. A z -score has a very simple interpretation: it measures the number of standard deviations above or below the sample mean a specific raw score lies.

In the QCI database, we have a sample mean for speed scores of 4.48 s, a standard deviation for speed scores of 2.89 s (recall Table 5.4 in Procedure 5.5 ). If we are interested in the z -score for Inspector 65’s raw speed score of 11.94 s, we would obtain a z -score of +2.58 using the method described above (subtract 4.48 from 11.94 and divide the result by 2.89). The interpretation of this number is that a raw decision speed score of 11.94 s lies about 2.9 standard deviations above the mean decision speed for the sample.

z -scores have some interesting properties. First, if one converts (statisticians would say ‘transforms’) every available raw score in a sample to z -scores, the mean of these z -scores will always be zero and the standard deviation of these z -scores will always be 1.0. These two facts about z -scores (mean = 0; standard deviation = 1) will be true no matter what sample you are dealing with and no matter what the original units of measurement are (e.g. seconds, percentages, number of widgets assembled, amount of preference for a product, attitude rating, amount of money spent). This is because transforming raw scores to z -scores automatically changes the measurement units from whatever they originally were to a new system of measurements expressed in standard deviation units.

Suppose Maree was interested in the performance statistics for the top 25% most accurate quality control inspectors in the sample. Given a sample size of 112, this would mean finding the top 28 inspectors in terms of their accuracy scores. Since Maree is interested in performance statistics, speed scores would also be of interest. Table 5.5 (generated using the SPSS Descriptives … procedure, listed using the Case Summaries … procedure and formatted for presentation using Excel) shows accuracy and speed scores for the top 28 inspectors in descending order of accuracy scores. The z -score transformation for each of these scores is also shown (last two columns) as are the type of company, education level and gender for each inspector.

Listing of the 28 (top 25%) most accurate QCI inspectors’ accuracy and speed scores as well as standard ( z ) score transformations for each score

There are three inspectors (8, 9 and 14) who scored maximum accuracy of 100%. Such accuracy converts to a z -score of +1.95. Thus 100% accuracy is 1.95 standard deviations above the sample’s mean accuracy level. Interestingly, all three inspectors worked for PC manufacturers and all three had only high school-level education. The least accurate inspector in the top 25% had a z -score for accuracy that was .75 standard deviations above the sample mean.

Interestingly, the top three inspectors in terms of accuracy had decision speeds that fell below the sample’s mean speed; inspector 8 was the fastest inspector of the three with a speed just over 1 standard deviation ( z  = −1.03) below the sample mean. The slowest inspector in the top 25% was inspector 75 (case #28 in the list) with a speed z -score of +2.62; i.e., he was over two and a half standard deviations slower in making inspection decisions relative to the sample’s mean speed.

The fact that z -scores always have a common measurement scale having a mean of 0 and a standard deviation of 1.0 leads to an interesting application of standard scores. Suppose we focus on inspector number 65 (case #8 in the list) in Table 5.5 . It might be of interest to compare this inspector’s quality control performance in terms of both his decision accuracy and decision speed. Such a comparison is impossible using raw scores since the inspector’s accuracy score and speed scores are different measures which have differing means and standard deviations expressed in fundamentally different units of measurement (percentages and seconds). However, if we are willing to assume that the score distributions for both variables are approximately the same shape and that both accuracy and speed are measured with about the same level of reliability or consistency (see Procedure 10.1007/978-981-15-2537-7_8#Sec1), we can compare the inspector’s two scores by first converting them to z -scores within their own respective distributions as shown in Table 5.5 .

Inspector 65 looks rather anomalous in that he demonstrated a relatively high level of accuracy (raw score = 94%; z  = +1.29) but took a very long time to make those accurate decisions (raw score = 11.94 s; z  = +2.58). Contrast this with inspector 106 (case #17 in the list) who demonstrated a similar level of accuracy (raw score = 92%; z  = +1.08) but took a much shorter time to make those accurate decisions (raw score = 1.70 s; z  = −.96). In terms of evaluating performance, from a company perspective, we might conclude that inspector 106 is performing at an overall higher level than inspector 65 because he can achieve a very high level of accuracy but much more quickly; accurate and fast is more cost effective and efficient than accurate and slow.

Note: We should be cautious here since we know from our previous explorations of the speed variable in Procedure 5.6 , that accuracy scores look fairly symmetrical and speed scores are positively skewed, so assuming that the two variables have the same distribution shape, so that z -score comparisons are permitted, would be problematic.

You might have noticed that as you scanned down the two columns of z -scores in Table 5.5 , there was a suggestion of a pattern between the signs attached to the respective z -scores for each person. There seems to be a very slight preponderance of pairs of z -scores where the signs are reversed (12 out of 22 pairs). This observation provides some very preliminary evidence to suggest that there may be a relationship between inspection accuracy and decision speed, namely that a more accurate decision tends to be associated with a faster decision speed. Of course, this pattern would be better verified using the entire sample rather than the top 25% of inspectors. However, you may find it interesting to learn that it is precisely this sort of suggestive evidence (about agreement or disagreement between z -score signs for pairs of variable scores throughout a sample) that is captured and summarised by a single statistical indicator called a ‘correlation coefficient’ (see Fundamental Concept 10.1007/978-981-15-2537-7_6#Sec1 and Procedure 10.1007/978-981-15-2537-7_6#Sec4).

z -scores are not the only type of standard score that is commonly used. Three other types of standard scores are: stanines (standard nines), IQ scores and T-scores (not to be confused with the t -test described in Procedure 10.1007/978-981-15-2537-7_7#Sec22). These other types of scores have the advantage of producing only positive integer scores rather than positive and negative decimal scores. This makes interpretation somewhat easier for certain applications. However, you should know that almost all other types of standard scores come from a specific transformation of z -scores. This is because once you have converted raw scores into z -scores, they can then be quite readily transformed into any other system of measurement by simply multiplying a person’s z -score by the new desired standard deviation for the measure and adding to that product the new desired mean for the measure.

T-scores are simply z-scores transformed to have a mean of 50.0 and a standard deviation of 10.0; IQ scores are simply z-scores transformed to have a mean of 100 and a standard deviation of 15 (or 16 in some systems). For more information, see Fundamental Concept II .

Standard scores are useful for representing the position of each raw score within a sample distribution relative to the mean of that distribution. The unit of measurement becomes the number of standard deviations a specific score is away from the sample mean. As such, z -scores can permit cautious comparisons across samples or across different variables having vastly differing means and standard deviations within the constraints of the comparison samples having similarly shaped distributions and roughly equivalent levels of measurement reliability. z -scores also form the basis for establishing the degree of correlation between two variables. Transforming raw scores into z -scores does not change the shape of a distribution or rank ordering of individuals within that distribution. For this reason, a z -score is referred to as a linear transformation of a raw score. Interestingly, z -scores provide an important foundational element for more complex analytical procedures such as factor analysis ( Procedure 10.1007/978-981-15-2537-7_6#Sec36), cluster analysis ( Procedure 10.1007/978-981-15-2537-7_6#Sec41) and multiple regression analysis (see, for example, Procedure 10.1007/978-981-15-2537-7_6#Sec27 and 10.1007/978-981-15-2537-7_7#Sec86).

While standard scores are useful indices, they are subject to restrictions if used to compare scores across samples or across different variables. The samples must have similar distribution shapes for the comparisons to be meaningful and the measures must have similar levels of reliability in each sample. The groups used to generate the z -scores should also be similar in composition (with respect to age, gender distribution, and so on). Because z -scores are not an intuitively meaningful way of presenting scores to lay-persons, many other types of standard score schemes have been devised to improve interpretability. However, most of these schemes produce scores that run a greater risk of facilitating lay-person misinterpretations simply because their connection with z -scores is hidden or because the resulting numbers ‘look’ like a more familiar type of score which people do intuitively understand.

It is extremely rare for a T-score to exceed 100 or go below 0 because this would mean that the raw score was in excess of 5 standard deviations away from the sample mean. This unfortunately means that T-scores are often misinterpreted as percentages because they typically range between 0 and 100 and therefore ‘look’ like percentages. However, T-scores are definitely not percentages.

Finally, a common misunderstanding of z -scores is that transforming raw scores into z -scores makes them follow a normal distribution (see Fundamental Concept II ). This is not the case. The distribution of z -scores will have exactly the same shape as that for the raw scores; if the raw scores are positively skewed, then the corresponding z -scores will also be positively skewed.

z -scores are particularly useful in evaluative studies where relative performance indices are of interest. Whenever you compute a correlation coefficient ( Procedure 10.1007/978-981-15-2537-7_6#Sec4), you are implicitly transforming the two variables involved into z -scores (which equates the variables in terms of mean and standard deviation), so that only the patterning in the relationship between the variables is represented. z -scores are also useful as a preliminary step to more advanced parametric statistical methods when variables differing in scale, range and/or measurement units must be equated for means and standard deviations prior to analysis.

Fundamental Concept II: The Normal Distribution

Arguably the most fundamental distribution used in the statistical analysis of quantitative data in the behavioural and social sciences is the normal distribution (also known as the Gaussian or bell-shaped distribution ). Many behavioural phenomena, if measured on a large enough sample of people, tend to produce ‘normally distributed’ variable scores. This includes most measures of ability, performance and productivity, personality characteristics and attitudes. The normal distribution is important because it is the one form of distribution that you must assume describes the scores of a variable in the population when parametric tests of statistical inference are undertaken. The standard normal distribution is defined as having a population mean of 0.0 and a population standard deviation of 1.0. The normal distribution is also important as a means of interpreting various types of scoring systems.

Figure 5.28 displays the standard normal distribution (mean = 0; standard deviation = 1.0) and shows that there is a clear link between z -scores and the normal distribution. Statisticians have analytically calculated the probability (also expressed as percentages or percentiles) that observations will fall above or below any specific z -score in the theoretical standard normal distribution. Thus, a z -score of +1.0 in the standard normal distribution will have 84.13% (equals a probability of .8413) of observations in the population falling at or below one standard deviation above the mean and 15.87% falling above that point. A z -score of −2.0 will have 2.28% of observations falling at that point or below and 97.72% of observations falling above that point. It is clear then that, in a standard normal distribution, z -scores have a direct relationship with percentiles .

The normal (bell-shaped or Gaussian) distribution

Figure 5.28 also shows how T-scores relate to the standard normal distribution and to z -scores. The mean T-score falls at 50 and each increment or decrement of 10 T-score units means a movement of another standard deviation away from this mean of 50. Thus, a T-score of 80 corresponds to a z -score of +3.0—a score 3 standard deviations higher than the mean of 50.

Of special interest to behavioural researchers are the values for z -scores in a standard normal distribution that encompass 90% of observations ( z  = ±1.645—isolating 5% of the distribution in each tail), 95% of observations ( z  = ±1.96—isolating 2.5% of the distribution in each tail), and 99% of observations ( z  = ±2.58—isolating 0.5% of the distribution in each tail).

Depending upon the degree of certainty required by the researcher, these bands describe regions outside of which one might define an observation as being atypical or as perhaps not belonging to a distribution being centred at a mean of 0.0. Most often, what is taken as atypical or rare in the standard normal distribution is a score at least two standard deviations away from the mean, in either direction. Why choose two standard deviations? Since in the standard normal distribution, only about 5% of observations will fall outside a band defined by z -scores of ±1.96 (rounded to 2 for simplicity), this equates to data values that are 2 standard deviations away from their mean. This can give us a defensible way to identify outliers or extreme values in a distribution.

Thinking ahead to what you will encounter in Chap. 10.1007/978-981-15-2537-7_7, this ‘banding’ logic can be extended into the world of statistics (like means and percentages) as opposed to just the world of observations. You will frequently hear researchers speak of some statistic estimating a specific value (a parameter ) in a population, plus or minus some other value.

A survey organisation might report political polling results in terms of a percentage and an error band, e.g. 59% of Australians indicated that they would vote Labour at the next federal election, plus or minus 2%.

Most commonly, this error band (±2%) is defined by possible values for the population parameter that are about two standard deviations (or two standard errors—a concept discussed further in Fundamental Concept 10.1007/978-981-15-2537-7_7#Sec14) away from the reported or estimated statistical value. In effect, the researcher is saying that on 95% of the occasions he/she would theoretically conduct his/her study, the population value estimated by the statistic being reported would fall between the limits imposed by the endpoints of the error band (the official name for this error band is a confidence interval ; see Procedure 10.1007/978-981-15-2537-7_8#Sec18). The well-understood mathematical properties of the standard normal distribution are what make such precise statements about levels of error in statistical estimates possible.

Checking for Normality

It is important to understand that transforming the raw scores for a variable to z -scores (recall Procedure 5.7 ) does not produce z -scores which follow a normal distribution; rather they will have the same distributional shape as the original scores. However, if you are willing to assume that the normal distribution is the correct reference distribution in the population, then you are justified is interpreting z -scores in light of the known characteristics of the normal distribution.

In order to justify this assumption, not only to enhance the interpretability of z -scores but more generally to enhance the integrity of parametric statistical analyses, it is helpful to actually look at the sample frequency distributions for variables (using a histogram (illustrated in Procedure 5.2 ) or a boxplot (illustrated in Procedure 5.6 ), for example), since non-normality can often be visually detected. It is important to note that in the social and behavioural sciences as well as in economics and finance, certain variables tend to be non-normal by their very nature. This includes variables that measure time taken to complete a task, achieve a goal or make decisions and variables that measure, for example, income, occurrence of rare or extreme events or organisational size. Such variables tend to be positively skewed in the population, a pattern that can often be confirmed by graphing the distribution.

If you cannot justify an assumption of ‘normality’, you may be able to force the data to be normally distributed by using what is called a ‘normalising transformation’. Such transformations will usually involve a nonlinear mathematical conversion (such as computing the logarithm, square root or reciprocal) of the raw scores. Such transformations will force the data to take on a more normal appearance so that the assumption of ‘normality’ can be reasonably justified, but at the cost of creating a new variable whose units of measurement and interpretation are more complicated. [For some non-normal variables, such as the occurrence of rare, extreme or catastrophic events (e.g. a 100-year flood or forest fire, coronavirus pandemic, the Global Financial Crisis or other type of financial crisis, man-made or natural disaster), the distributions cannot be ‘normalised’. In such cases, the researcher needs to model the distribution as it stands. For such events, extreme value theory (e.g. see Diebold et al. 2000 ) has proven very useful in recent years. This theory uses a variation of the Pareto or Weibull distribution as a reference, rather than the normal distribution, when making predictions.]

Figure 5.29 displays before and after pictures of the effects of a logarithmic transformation on the positively skewed speed variable from the QCI database. Each graph, produced using NCSS, is of the hybrid histogram-density trace-boxplot type first illustrated in Procedure 5.6 . The left graph clearly shows the strong positive skew in the speed scores and the right graph shows the result of taking the log 10 of each raw score.

Combined histogram-density trace-boxplot graphs displaying the before and after effects of a ‘normalising’ log 10 transformation of the speed variable

Notice how the long tail toward slow speed scores is pulled in toward the mean and the very short tail toward fast speed scores is extended away from the mean. The result is a more ‘normal’ appearing distribution. The assumption would then be that we could assume normality of speed scores, but only in a log 10 format (i.e. it is the log of speed scores that we assume is normally distributed in the population). In general, taking the logarithm of raw scores provides a satisfactory remedy for positively skewed distributions (but not for negatively skewed ones). Furthermore, anything we do with the transformed speed scores now has to be interpreted in units of log 10 (seconds) which is a more complex interpretation to make.

Another visual method for detecting non-normality is to graph what is called a normal Q-Q plot (the Q-Q stands for Quantile-Quantile). This plots the percentiles for the observed data against the percentiles for the standard normal distribution (see Cleveland 1995 for more detailed discussion; also see Lane 2007 , http://onlinestatbook.com/2/advanced_graphs/ q-q_plots.html) . If the pattern for the observed data follows a normal distribution, then all the points on the graph will fall approximately along a diagonal line.

Figure 5.30 shows the normal Q-Q plots for the original speed variable and the transformed log-speed variable, produced using the SPSS Explore... procedure. The diagnostic diagonal line is shown on each graph. In the left-hand plot, for speed , the plot points clearly deviate from the diagonal in a way that signals positive skewness. The right-hand plot, for log_speed, shows the plot points generally falling along the diagonal line thereby conforming much more closely to what is expected in a normal distribution.

Normal Q-Q plots for the original speed variable and the new log_speed variable

In addition to visual ways of detecting non-normality, there are also numerical ways. As highlighted in Chap. 10.1007/978-981-15-2537-7_1, there are two additional characteristics of any distribution, namely skewness (asymmetric distribution tails) and kurtosis (peakedness of the distribution). Both have an associated statistic that provides a measure of that characteristic, similar to the mean and standard deviation statistics. In a normal distribution, the values for the skewness and kurtosis statistics are both zero (skewness = 0 means a symmetric distribution; kurtosis = 0 means a mesokurtic distribution). The further away each statistic is from zero, the more the distribution deviates from a normal shape. Both the skewness statistic and the kurtosis statistic have standard errors (see Fundamental Concept 10.1007/978-981-15-2537-7_7#Sec14) associated with them (which work very much like the standard deviation, only for a statistic rather than for observations); these can be routinely computed by almost any statistical package when you request a descriptive analysis. Without going into the logic right now (this will come in Fundamental Concept 10.1007/978-981-15-2537-7_7#Sec1), a rough rule of thumb you can use to check for normality using the skewness and kurtosis statistics is to do the following:

• Prepare : Take the standard error for the statistic and multiply it by 2 (or 3 if you want to be more conservative).
• Interval : Add the result from the Prepare step to the value of the statistic and subtract the result from the value of the statistic. You will end up with two numbers, one low - one high, that define the ends of an interval (what you have just created approximates what is called a ‘confidence interval’, see Procedure 10.1007/978-981-15-2537-7_8#Sec18).
• Check : If zero falls inside of this interval (i.e. between the low and high endpoints from the Interval step), then there is likely to be no significant issue with that characteristic of the distribution. If zero falls outside of the interval (i.e. lower than the low value endpoint or higher than the high value endpoint), then you likely have an issue with non-normality with respect to that characteristic.

Visually, we saw in the left graph in Fig. 5.29 that the speed variable was highly positively skewed. What if Maree wanted to check some numbers to support this judgment? She could ask SPSS to produce the skewness and kurtosis statistics for both the original speed variable and the new log_speed variable using the Frequencies... or the Explore... procedure. Table 5.6 shows what SPSS would produce if the Frequencies ... procedure were used.

Skewness and kurtosis statistics and their standard errors for both the original speed variable and the new log_speed variable

Using the 3-step check rule described above, Maree could roughly evaluate the normality of the two variables as follows:

• skewness : [Prepare] 2 × .229 = .458 ➔ [Interval] 1.487 − .458 = 1.029 and 1.487 + .458 = 1.945 ➔ [Check] zero does not fall inside the interval bounded by 1.029 and 1.945, so there appears to be a significant problem with skewness. Since the value for the skewness statistic (1.487) is positive, this means the problem is positive skewness, confirming what the left graph in Fig. 5.29 showed.
• kurtosis : [Prepare] 2 × .455 = .91 ➔ [Interval] 3.071 − .91 = 2.161 and 3.071 + .91 = 3.981 ➔ [Check] zero does not fall in interval bounded by 2.161 and 3.981, so there appears to be a significant problem with kurtosis. Since the value for the kurtosis statistic (1.487) is positive, this means the problem is leptokurtosis—the peakedness of the distribution is too tall relative to what is expected in a normal distribution.
• skewness : [Prepare] 2 × .229 = .458 ➔ [Interval] −.050 − .458 = −.508 and −.050 + .458 = .408 ➔ [Check] zero falls within interval bounded by −.508 and .408, so there appears to be no problem with skewness. The log transform appears to have corrected the problem, confirming what the right graph in Fig. 5.29 showed.
• kurtosis : [Prepare] 2 × .455 = .91 ➔ [Interval] −.672 – .91 = −1.582 and −.672 + .91 = .238 ➔ [Check] zero falls within interval bounded by −1.582 and .238, so there appears to be no problem with kurtosis. The log transform appears to have corrected this problem as well, rendering the distribution more approximately mesokurtic (i.e. normal) in shape.

There are also more formal tests of significance (see Fundamental Concept 10.1007/978-981-15-2537-7_7#Sec1) that one can use to numerically evaluate normality, such as the Kolmogorov-Smirnov test and the Shapiro-Wilk’s test . Each of these tests, for example, can be produced by SPSS on request, via the Explore... procedure.

1 For more information, see Chap. 10.1007/978-981-15-2537-7_1 – The language of statistics .

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• Published: 26 March 2024

Predicting and improving complex beer flavor through machine learning

• Michiel Schreurs   ORCID: orcid.org/0000-0002-9449-5619 1 , 2 , 3   na1 ,
• Supinya Piampongsant 1 , 2 , 3   na1 ,
• Miguel Roncoroni   ORCID: orcid.org/0000-0001-7461-1427 1 , 2 , 3   na1 ,
• Lloyd Cool   ORCID: orcid.org/0000-0001-9936-3124 1 , 2 , 3 , 4 ,
• Beatriz Herrera-Malaver   ORCID: orcid.org/0000-0002-5096-9974 1 , 2 , 3 ,
• Christophe Vanderaa   ORCID: orcid.org/0000-0001-7443-5427 4 ,
• Florian A. Theßeling 1 , 2 , 3 ,
• Łukasz Kreft   ORCID: orcid.org/0000-0001-7620-4657 5 ,
• Alexander Botzki   ORCID: orcid.org/0000-0001-6691-4233 5 ,
• Philippe Malcorps 6 ,
• Luk Daenen 6 ,
• Tom Wenseleers   ORCID: orcid.org/0000-0002-1434-861X 4 &
• Kevin J. Verstrepen   ORCID: orcid.org/0000-0002-3077-6219 1 , 2 , 3  

Nature Communications volume  15 , Article number:  2368 ( 2024 ) Cite this article

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• Chemical engineering
• Gas chromatography
• Machine learning
• Metabolomics
• Taste receptors

The perception and appreciation of food flavor depends on many interacting chemical compounds and external factors, and therefore proves challenging to understand and predict. Here, we combine extensive chemical and sensory analyses of 250 different beers to train machine learning models that allow predicting flavor and consumer appreciation. For each beer, we measure over 200 chemical properties, perform quantitative descriptive sensory analysis with a trained tasting panel and map data from over 180,000 consumer reviews to train 10 different machine learning models. The best-performing algorithm, Gradient Boosting, yields models that significantly outperform predictions based on conventional statistics and accurately predict complex food features and consumer appreciation from chemical profiles. Model dissection allows identifying specific and unexpected compounds as drivers of beer flavor and appreciation. Adding these compounds results in variants of commercial alcoholic and non-alcoholic beers with improved consumer appreciation. Together, our study reveals how big data and machine learning uncover complex links between food chemistry, flavor and consumer perception, and lays the foundation to develop novel, tailored foods with superior flavors.

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Introduction

Predicting and understanding food perception and appreciation is one of the major challenges in food science. Accurate modeling of food flavor and appreciation could yield important opportunities for both producers and consumers, including quality control, product fingerprinting, counterfeit detection, spoilage detection, and the development of new products and product combinations (food pairing) 1 , 2 , 3 , 4 , 5 , 6 . Accurate models for flavor and consumer appreciation would contribute greatly to our scientific understanding of how humans perceive and appreciate flavor. Moreover, accurate predictive models would also facilitate and standardize existing food assessment methods and could supplement or replace assessments by trained and consumer tasting panels, which are variable, expensive and time-consuming 7 , 8 , 9 . Lastly, apart from providing objective, quantitative, accurate and contextual information that can help producers, models can also guide consumers in understanding their personal preferences 10 .

Despite the myriad of applications, predicting food flavor and appreciation from its chemical properties remains a largely elusive goal in sensory science, especially for complex food and beverages 11 , 12 . A key obstacle is the immense number of flavor-active chemicals underlying food flavor. Flavor compounds can vary widely in chemical structure and concentration, making them technically challenging and labor-intensive to quantify, even in the face of innovations in metabolomics, such as non-targeted metabolic fingerprinting 13 , 14 . Moreover, sensory analysis is perhaps even more complicated. Flavor perception is highly complex, resulting from hundreds of different molecules interacting at the physiochemical and sensorial level. Sensory perception is often non-linear, characterized by complex and concentration-dependent synergistic and antagonistic effects 15 , 16 , 17 , 18 , 19 , 20 , 21 that are further convoluted by the genetics, environment, culture and psychology of consumers 22 , 23 , 24 . Perceived flavor is therefore difficult to measure, with problems of sensitivity, accuracy, and reproducibility that can only be resolved by gathering sufficiently large datasets 25 . Trained tasting panels are considered the prime source of quality sensory data, but require meticulous training, are low throughput and high cost. Public databases containing consumer reviews of food products could provide a valuable alternative, especially for studying appreciation scores, which do not require formal training 25 . Public databases offer the advantage of amassing large amounts of data, increasing the statistical power to identify potential drivers of appreciation. However, public datasets suffer from biases, including a bias in the volunteers that contribute to the database, as well as confounding factors such as price, cult status and psychological conformity towards previous ratings of the product.

Classical multivariate statistics and machine learning methods have been used to predict flavor of specific compounds by, for example, linking structural properties of a compound to its potential biological activities or linking concentrations of specific compounds to sensory profiles 1 , 26 . Importantly, most previous studies focused on predicting organoleptic properties of single compounds (often based on their chemical structure) 27 , 28 , 29 , 30 , 31 , 32 , 33 , thus ignoring the fact that these compounds are present in a complex matrix in food or beverages and excluding complex interactions between compounds. Moreover, the classical statistics commonly used in sensory science 34 , 35 , 36 , 37 , 38 , 39 require a large sample size and sufficient variance amongst predictors to create accurate models. They are not fit for studying an extensive set of hundreds of interacting flavor compounds, since they are sensitive to outliers, have a high tendency to overfit and are less suited for non-linear and discontinuous relationships 40 .

In this study, we combine extensive chemical analyses and sensory data of a set of different commercial beers with machine learning approaches to develop models that predict taste, smell, mouthfeel and appreciation from compound concentrations. Beer is particularly suited to model the relationship between chemistry, flavor and appreciation. First, beer is a complex product, consisting of thousands of flavor compounds that partake in complex sensory interactions 41 , 42 , 43 . This chemical diversity arises from the raw materials (malt, yeast, hops, water and spices) and biochemical conversions during the brewing process (kilning, mashing, boiling, fermentation, maturation and aging) 44 , 45 . Second, the advent of the internet saw beer consumers embrace online review platforms, such as RateBeer (ZX Ventures, Anheuser-Busch InBev SA/NV) and BeerAdvocate (Next Glass, inc.). In this way, the beer community provides massive data sets of beer flavor and appreciation scores, creating extraordinarily large sensory databases to complement the analyses of our professional sensory panel. Specifically, we characterize over 200 chemical properties of 250 commercial beers, spread across 22 beer styles, and link these to the descriptive sensory profiling data of a 16-person in-house trained tasting panel and data acquired from over 180,000 public consumer reviews. These unique and extensive datasets enable us to train a suite of machine learning models to predict flavor and appreciation from a beer’s chemical profile. Dissection of the best-performing models allows us to pinpoint specific compounds as potential drivers of beer flavor and appreciation. Follow-up experiments confirm the importance of these compounds and ultimately allow us to significantly improve the flavor and appreciation of selected commercial beers. Together, our study represents a significant step towards understanding complex flavors and reinforces the value of machine learning to develop and refine complex foods. In this way, it represents a stepping stone for further computer-aided food engineering applications 46 .

To generate a comprehensive dataset on beer flavor, we selected 250 commercial Belgian beers across 22 different beer styles (Supplementary Fig.  S1 ). Beers with ≤ 4.2% alcohol by volume (ABV) were classified as non-alcoholic and low-alcoholic. Blonds and Tripels constitute a significant portion of the dataset (12.4% and 11.2%, respectively) reflecting their presence on the Belgian beer market and the heterogeneity of beers within these styles. By contrast, lager beers are less diverse and dominated by a handful of brands. Rare styles such as Brut or Faro make up only a small fraction of the dataset (2% and 1%, respectively) because fewer of these beers are produced and because they are dominated by distinct characteristics in terms of flavor and chemical composition.

Extensive analysis identifies relationships between chemical compounds in beer

For each beer, we measured 226 different chemical properties, including common brewing parameters such as alcohol content, iso-alpha acids, pH, sugar concentration 47 , and over 200 flavor compounds (Methods, Supplementary Table  S1 ). A large portion (37.2%) are terpenoids arising from hopping, responsible for herbal and fruity flavors 16 , 48 . A second major category are yeast metabolites, such as esters and alcohols, that result in fruity and solvent notes 48 , 49 , 50 . Other measured compounds are primarily derived from malt, or other microbes such as non- Saccharomyces yeasts and bacteria (‘wild flora’). Compounds that arise from spices or staling are labeled under ‘Others’. Five attributes (caloric value, total acids and total ester, hop aroma and sulfur compounds) are calculated from multiple individually measured compounds.

As a first step in identifying relationships between chemical properties, we determined correlations between the concentrations of the compounds (Fig.  1 , upper panel, Supplementary Data  1 and 2 , and Supplementary Fig.  S2 . For the sake of clarity, only a subset of the measured compounds is shown in Fig.  1 ). Compounds of the same origin typically show a positive correlation, while absence of correlation hints at parameters varying independently. For example, the hop aroma compounds citronellol, and alpha-terpineol show moderate correlations with each other (Spearman’s rho=0.39 and 0.57), but not with the bittering hop component iso-alpha acids (Spearman’s rho=0.16 and −0.07). This illustrates how brewers can independently modify hop aroma and bitterness by selecting hop varieties and dosage time. If hops are added early in the boiling phase, chemical conversions increase bitterness while aromas evaporate, conversely, late addition of hops preserves aroma but limits bitterness 51 . Similarly, hop-derived iso-alpha acids show a strong anti-correlation with lactic acid and acetic acid, likely reflecting growth inhibition of lactic acid and acetic acid bacteria, or the consequent use of fewer hops in sour beer styles, such as West Flanders ales and Fruit beers, that rely on these bacteria for their distinct flavors 52 . Finally, yeast-derived esters (ethyl acetate, ethyl decanoate, ethyl hexanoate, ethyl octanoate) and alcohols (ethanol, isoamyl alcohol, isobutanol, and glycerol), correlate with Spearman coefficients above 0.5, suggesting that these secondary metabolites are correlated with the yeast genetic background and/or fermentation parameters and may be difficult to influence individually, although the choice of yeast strain may offer some control 53 .

Spearman rank correlations are shown. Descriptors are grouped according to their origin (malt (blue), hops (green), yeast (red), wild flora (yellow), Others (black)), and sensory aspect (aroma, taste, palate, and overall appreciation). Please note that for the chemical compounds, for the sake of clarity, only a subset of the total number of measured compounds is shown, with an emphasis on the key compounds for each source. For more details, see the main text and Methods section. Chemical data can be found in Supplementary Data  1 , correlations between all chemical compounds are depicted in Supplementary Fig.  S2 and correlation values can be found in Supplementary Data  2 . See Supplementary Data  4 for sensory panel assessments and Supplementary Data  5 for correlation values between all sensory descriptors.

Interestingly, different beer styles show distinct patterns for some flavor compounds (Supplementary Fig.  S3 ). These observations agree with expectations for key beer styles, and serve as a control for our measurements. For instance, Stouts generally show high values for color (darker), while hoppy beers contain elevated levels of iso-alpha acids, compounds associated with bitter hop taste. Acetic and lactic acid are not prevalent in most beers, with notable exceptions such as Kriek, Lambic, Faro, West Flanders ales and Flanders Old Brown, which use acid-producing bacteria ( Lactobacillus and Pediococcus ) or unconventional yeast ( Brettanomyces ) 54 , 55 . Glycerol, ethanol and esters show similar distributions across all beer styles, reflecting their common origin as products of yeast metabolism during fermentation 45 , 53 . Finally, low/no-alcohol beers contain low concentrations of glycerol and esters. This is in line with the production process for most of the low/no-alcohol beers in our dataset, which are produced through limiting fermentation or by stripping away alcohol via evaporation or dialysis, with both methods having the unintended side-effect of reducing the amount of flavor compounds in the final beer 56 , 57 .

Besides expected associations, our data also reveals less trivial associations between beer styles and specific parameters. For example, geraniol and citronellol, two monoterpenoids responsible for citrus, floral and rose flavors and characteristic of Citra hops, are found in relatively high amounts in Christmas, Saison, and Brett/co-fermented beers, where they may originate from terpenoid-rich spices such as coriander seeds instead of hops 58 .

Tasting panel assessments reveal sensorial relationships in beer

To assess the sensory profile of each beer, a trained tasting panel evaluated each of the 250 beers for 50 sensory attributes, including different hop, malt and yeast flavors, off-flavors and spices. Panelists used a tasting sheet (Supplementary Data  3 ) to score the different attributes. Panel consistency was evaluated by repeating 12 samples across different sessions and performing ANOVA. In 95% of cases no significant difference was found across sessions ( p  > 0.05), indicating good panel consistency (Supplementary Table  S2 ).

Aroma and taste perception reported by the trained panel are often linked (Fig.  1 , bottom left panel and Supplementary Data  4 and 5 ), with high correlations between hops aroma and taste (Spearman’s rho=0.83). Bitter taste was found to correlate with hop aroma and taste in general (Spearman’s rho=0.80 and 0.69), and particularly with “grassy” noble hops (Spearman’s rho=0.75). Barnyard flavor, most often associated with sour beers, is identified together with stale hops (Spearman’s rho=0.97) that are used in these beers. Lactic and acetic acid, which often co-occur, are correlated (Spearman’s rho=0.66). Interestingly, sweetness and bitterness are anti-correlated (Spearman’s rho = −0.48), confirming the hypothesis that they mask each other 59 , 60 . Beer body is highly correlated with alcohol (Spearman’s rho = 0.79), and overall appreciation is found to correlate with multiple aspects that describe beer mouthfeel (alcohol, carbonation; Spearman’s rho= 0.32, 0.39), as well as with hop and ester aroma intensity (Spearman’s rho=0.39 and 0.35).

Similar to the chemical analyses, sensorial analyses confirmed typical features of specific beer styles (Supplementary Fig.  S4 ). For example, sour beers (Faro, Flanders Old Brown, Fruit beer, Kriek, Lambic, West Flanders ale) were rated acidic, with flavors of both acetic and lactic acid. Hoppy beers were found to be bitter and showed hop-associated aromas like citrus and tropical fruit. Malt taste is most detected among scotch, stout/porters, and strong ales, while low/no-alcohol beers, which often have a reputation for being ‘worty’ (reminiscent of unfermented, sweet malt extract) appear in the middle. Unsurprisingly, hop aromas are most strongly detected among hoppy beers. Like its chemical counterpart (Supplementary Fig.  S3 ), acidity shows a right-skewed distribution, with the most acidic beers being Krieks, Lambics, and West Flanders ales.

Tasting panel assessments of specific flavors correlate with chemical composition

We find that the concentrations of several chemical compounds strongly correlate with specific aroma or taste, as evaluated by the tasting panel (Fig.  2 , Supplementary Fig.  S5 , Supplementary Data  6 ). In some cases, these correlations confirm expectations and serve as a useful control for data quality. For example, iso-alpha acids, the bittering compounds in hops, strongly correlate with bitterness (Spearman’s rho=0.68), while ethanol and glycerol correlate with tasters’ perceptions of alcohol and body, the mouthfeel sensation of fullness (Spearman’s rho=0.82/0.62 and 0.72/0.57 respectively) and darker color from roasted malts is a good indication of malt perception (Spearman’s rho=0.54).

Heatmap colors indicate Spearman’s Rho. Axes are organized according to sensory categories (aroma, taste, mouthfeel, overall), chemical categories and chemical sources in beer (malt (blue), hops (green), yeast (red), wild flora (yellow), Others (black)). See Supplementary Data  6 for all correlation values.

Interestingly, for some relationships between chemical compounds and perceived flavor, correlations are weaker than expected. For example, the rose-smelling phenethyl acetate only weakly correlates with floral aroma. This hints at more complex relationships and interactions between compounds and suggests a need for a more complex model than simple correlations. Lastly, we uncovered unexpected correlations. For instance, the esters ethyl decanoate and ethyl octanoate appear to correlate slightly with hop perception and bitterness, possibly due to their fruity flavor. Iron is anti-correlated with hop aromas and bitterness, most likely because it is also anti-correlated with iso-alpha acids. This could be a sign of metal chelation of hop acids 61 , given that our analyses measure unbound hop acids and total iron content, or could result from the higher iron content in dark and Fruit beers, which typically have less hoppy and bitter flavors 62 .

Public consumer reviews complement expert panel data

To complement and expand the sensory data of our trained tasting panel, we collected 180,000 reviews of our 250 beers from the online consumer review platform RateBeer. This provided numerical scores for beer appearance, aroma, taste, palate, overall quality as well as the average overall score.

Public datasets are known to suffer from biases, such as price, cult status and psychological conformity towards previous ratings of a product. For example, prices correlate with appreciation scores for these online consumer reviews (rho=0.49, Supplementary Fig.  S6 ), but not for our trained tasting panel (rho=0.19). This suggests that prices affect consumer appreciation, which has been reported in wine 63 , while blind tastings are unaffected. Moreover, we observe that some beer styles, like lagers and non-alcoholic beers, generally receive lower scores, reflecting that online reviewers are mostly beer aficionados with a preference for specialty beers over lager beers. In general, we find a modest correlation between our trained panel’s overall appreciation score and the online consumer appreciation scores (Fig.  3 , rho=0.29). Apart from the aforementioned biases in the online datasets, serving temperature, sample freshness and surroundings, which are all tightly controlled during the tasting panel sessions, can vary tremendously across online consumers and can further contribute to (among others, appreciation) differences between the two categories of tasters. Importantly, in contrast to the overall appreciation scores, for many sensory aspects the results from the professional panel correlated well with results obtained from RateBeer reviews. Correlations were highest for features that are relatively easy to recognize even for untrained tasters, like bitterness, sweetness, alcohol and malt aroma (Fig.  3 and below).

RateBeer text mining results can be found in Supplementary Data  7 . Rho values shown are Spearman correlation values, with asterisks indicating significant correlations ( p  < 0.05, two-sided). All p values were smaller than 0.001, except for Esters aroma (0.0553), Esters taste (0.3275), Esters aroma—banana (0.0019), Coriander (0.0508) and Diacetyl (0.0134).

Besides collecting consumer appreciation from these online reviews, we developed automated text analysis tools to gather additional data from review texts (Supplementary Data  7 ). Processing review texts on the RateBeer database yielded comparable results to the scores given by the trained panel for many common sensory aspects, including acidity, bitterness, sweetness, alcohol, malt, and hop tastes (Fig.  3 ). This is in line with what would be expected, since these attributes require less training for accurate assessment and are less influenced by environmental factors such as temperature, serving glass and odors in the environment. Consumer reviews also correlate well with our trained panel for 4-vinyl guaiacol, a compound associated with a very characteristic aroma. By contrast, correlations for more specific aromas like ester, coriander or diacetyl are underrepresented in the online reviews, underscoring the importance of using a trained tasting panel and standardized tasting sheets with explicit factors to be scored for evaluating specific aspects of a beer. Taken together, our results suggest that public reviews are trustworthy for some, but not all, flavor features and can complement or substitute taste panel data for these sensory aspects.

Models can predict beer sensory profiles from chemical data

The rich datasets of chemical analyses, tasting panel assessments and public reviews gathered in the first part of this study provided us with a unique opportunity to develop predictive models that link chemical data to sensorial features. Given the complexity of beer flavor, basic statistical tools such as correlations or linear regression may not always be the most suitable for making accurate predictions. Instead, we applied different machine learning models that can model both simple linear and complex interactive relationships. Specifically, we constructed a set of regression models to predict (a) trained panel scores for beer flavor and quality and (b) public reviews’ appreciation scores from beer chemical profiles. We trained and tested 10 different models (Methods), 3 linear regression-based models (simple linear regression with first-order interactions (LR), lasso regression with first-order interactions (Lasso), partial least squares regressor (PLSR)), 5 decision tree models (AdaBoost regressor (ABR), extra trees (ET), gradient boosting regressor (GBR), random forest (RF) and XGBoost regressor (XGBR)), 1 support vector regression (SVR), and 1 artificial neural network (ANN) model.

To compare the performance of our machine learning models, the dataset was randomly split into a training and test set, stratified by beer style. After a model was trained on data in the training set, its performance was evaluated on its ability to predict the test dataset obtained from multi-output models (based on the coefficient of determination, see Methods). Additionally, individual-attribute models were ranked per descriptor and the average rank was calculated, as proposed by Korneva et al. 64 . Importantly, both ways of evaluating the models’ performance agreed in general. Performance of the different models varied (Table  1 ). It should be noted that all models perform better at predicting RateBeer results than results from our trained tasting panel. One reason could be that sensory data is inherently variable, and this variability is averaged out with the large number of public reviews from RateBeer. Additionally, all tree-based models perform better at predicting taste than aroma. Linear models (LR) performed particularly poorly, with negative R 2 values, due to severe overfitting (training set R 2  = 1). Overfitting is a common issue in linear models with many parameters and limited samples, especially with interaction terms further amplifying the number of parameters. L1 regularization (Lasso) successfully overcomes this overfitting, out-competing multiple tree-based models on the RateBeer dataset. Similarly, the dimensionality reduction of PLSR avoids overfitting and improves performance, to some extent. Still, tree-based models (ABR, ET, GBR, RF and XGBR) show the best performance, out-competing the linear models (LR, Lasso, PLSR) commonly used in sensory science 65 .

GBR models showed the best overall performance in predicting sensory responses from chemical information, with R 2 values up to 0.75 depending on the predicted sensory feature (Supplementary Table  S4 ). The GBR models predict consumer appreciation (RateBeer) better than our trained panel’s appreciation (R 2 value of 0.67 compared to R 2 value of 0.09) (Supplementary Table  S3 and Supplementary Table  S4 ). ANN models showed intermediate performance, likely because neural networks typically perform best with larger datasets 66 . The SVR shows intermediate performance, mostly due to the weak predictions of specific attributes that lower the overall performance (Supplementary Table  S4 ).

Model dissection identifies specific, unexpected compounds as drivers of consumer appreciation

Next, we leveraged our models to infer important contributors to sensory perception and consumer appreciation. Consumer preference is a crucial sensory aspects, because a product that shows low consumer appreciation scores often does not succeed commercially 25 . Additionally, the requirement for a large number of representative evaluators makes consumer trials one of the more costly and time-consuming aspects of product development. Hence, a model for predicting chemical drivers of overall appreciation would be a welcome addition to the available toolbox for food development and optimization.

Since GBR models on our RateBeer dataset showed the best overall performance, we focused on these models. Specifically, we used two approaches to identify important contributors. First, rankings of the most important predictors for each sensorial trait in the GBR models were obtained based on impurity-based feature importance (mean decrease in impurity). High-ranked parameters were hypothesized to be either the true causal chemical properties underlying the trait, to correlate with the actual causal properties, or to take part in sensory interactions affecting the trait 67 (Fig.  4A ). In a second approach, we used SHAP 68 to determine which parameters contributed most to the model for making predictions of consumer appreciation (Fig.  4B ). SHAP calculates parameter contributions to model predictions on a per-sample basis, which can be aggregated into an importance score.

A The impurity-based feature importance (mean deviance in impurity, MDI) calculated from the Gradient Boosting Regression (GBR) model predicting RateBeer appreciation scores. The top 15 highest ranked chemical properties are shown. B SHAP summary plot for the top 15 parameters contributing to our GBR model. Each point on the graph represents a sample from our dataset. The color represents the concentration of that parameter, with bluer colors representing low values and redder colors representing higher values. Greater absolute values on the horizontal axis indicate a higher impact of the parameter on the prediction of the model. C Spearman correlations between the 15 most important chemical properties and consumer overall appreciation. Numbers indicate the Spearman Rho correlation coefficient, and the rank of this correlation compared to all other correlations. The top 15 important compounds were determined using SHAP (panel B).

Both approaches identified ethyl acetate as the most predictive parameter for beer appreciation (Fig.  4 ). Ethyl acetate is the most abundant ester in beer with a typical ‘fruity’, ‘solvent’ and ‘alcoholic’ flavor, but is often considered less important than other esters like isoamyl acetate. The second most important parameter identified by SHAP is ethanol, the most abundant beer compound after water. Apart from directly contributing to beer flavor and mouthfeel, ethanol drastically influences the physical properties of beer, dictating how easily volatile compounds escape the beer matrix to contribute to beer aroma 69 . Importantly, it should also be noted that the importance of ethanol for appreciation is likely inflated by the very low appreciation scores of non-alcoholic beers (Supplementary Fig.  S4 ). Despite not often being considered a driver of beer appreciation, protein level also ranks highly in both approaches, possibly due to its effect on mouthfeel and body 70 . Lactic acid, which contributes to the tart taste of sour beers, is the fourth most important parameter identified by SHAP, possibly due to the generally high appreciation of sour beers in our dataset.

Interestingly, some of the most important predictive parameters for our model are not well-established as beer flavors or are even commonly regarded as being negative for beer quality. For example, our models identify methanethiol and ethyl phenyl acetate, an ester commonly linked to beer staling 71 , as a key factor contributing to beer appreciation. Although there is no doubt that high concentrations of these compounds are considered unpleasant, the positive effects of modest concentrations are not yet known 72 , 73 .

To compare our approach to conventional statistics, we evaluated how well the 15 most important SHAP-derived parameters correlate with consumer appreciation (Fig.  4C ). Interestingly, only 6 of the properties derived by SHAP rank amongst the top 15 most correlated parameters. For some chemical compounds, the correlations are so low that they would have likely been considered unimportant. For example, lactic acid, the fourth most important parameter, shows a bimodal distribution for appreciation, with sour beers forming a separate cluster, that is missed entirely by the Spearman correlation. Additionally, the correlation plots reveal outliers, emphasizing the need for robust analysis tools. Together, this highlights the need for alternative models, like the Gradient Boosting model, that better grasp the complexity of (beer) flavor.

Finally, to observe the relationships between these chemical properties and their predicted targets, partial dependence plots were constructed for the six most important predictors of consumer appreciation 74 , 75 , 76 (Supplementary Fig.  S7 ). One-way partial dependence plots show how a change in concentration affects the predicted appreciation. These plots reveal an important limitation of our models: appreciation predictions remain constant at ever-increasing concentrations. This implies that once a threshold concentration is reached, further increasing the concentration does not affect appreciation. This is false, as it is well-documented that certain compounds become unpleasant at high concentrations, including ethyl acetate (‘nail polish’) 77 and methanethiol (‘sulfury’ and ‘rotten cabbage’) 78 . The inability of our models to grasp that flavor compounds have optimal levels, above which they become negative, is a consequence of working with commercial beer brands where (off-)flavors are rarely too high to negatively impact the product. The two-way partial dependence plots show how changing the concentration of two compounds influences predicted appreciation, visualizing their interactions (Supplementary Fig.  S7 ). In our case, the top 5 parameters are dominated by additive or synergistic interactions, with high concentrations for both compounds resulting in the highest predicted appreciation.

To assess the robustness of our best-performing models and model predictions, we performed 100 iterations of the GBR, RF and ET models. In general, all iterations of the models yielded similar performance (Supplementary Fig.  S8 ). Moreover, the main predictors (including the top predictors ethanol and ethyl acetate) remained virtually the same, especially for GBR and RF. For the iterations of the ET model, we did observe more variation in the top predictors, which is likely a consequence of the model’s inherent random architecture in combination with co-correlations between certain predictors. However, even in this case, several of the top predictors (ethanol and ethyl acetate) remain unchanged, although their rank in importance changes (Supplementary Fig.  S8 ).

Next, we investigated if a combination of RateBeer and trained panel data into one consolidated dataset would lead to stronger models, under the hypothesis that such a model would suffer less from bias in the datasets. A GBR model was trained to predict appreciation on the combined dataset. This model underperformed compared to the RateBeer model, both in the native case and when including a dataset identifier (R 2  = 0.67, 0.26 and 0.42 respectively). For the latter, the dataset identifier is the most important feature (Supplementary Fig.  S9 ), while most of the feature importance remains unchanged, with ethyl acetate and ethanol ranking highest, like in the original model trained only on RateBeer data. It seems that the large variation in the panel dataset introduces noise, weakening the models’ performances and reliability. In addition, it seems reasonable to assume that both datasets are fundamentally different, with the panel dataset obtained by blind tastings by a trained professional panel.

Lastly, we evaluated whether beer style identifiers would further enhance the model’s performance. A GBR model was trained with parameters that explicitly encoded the styles of the samples. This did not improve model performance (R2 = 0.66 with style information vs R2 = 0.67). The most important chemical features are consistent with the model trained without style information (eg. ethanol and ethyl acetate), and with the exception of the most preferred (strong ale) and least preferred (low/no-alcohol) styles, none of the styles were among the most important features (Supplementary Fig.  S9 , Supplementary Table  S5 and S6 ). This is likely due to a combination of style-specific chemical signatures, such as iso-alpha acids and lactic acid, that implicitly convey style information to the original models, as well as the low number of samples belonging to some styles, making it difficult for the model to learn style-specific patterns. Moreover, beer styles are not rigorously defined, with some styles overlapping in features and some beers being misattributed to a specific style, all of which leads to more noise in models that use style parameters.

Model validation

To test if our predictive models give insight into beer appreciation, we set up experiments aimed at improving existing commercial beers. We specifically selected overall appreciation as the trait to be examined because of its complexity and commercial relevance. Beer flavor comprises a complex bouquet rather than single aromas and tastes 53 . Hence, adding a single compound to the extent that a difference is noticeable may lead to an unbalanced, artificial flavor. Therefore, we evaluated the effect of combinations of compounds. Because Blond beers represent the most extensive style in our dataset, we selected a beer from this style as the starting material for these experiments (Beer 64 in Supplementary Data  1 ).

In the first set of experiments, we adjusted the concentrations of compounds that made up the most important predictors of overall appreciation (ethyl acetate, ethanol, lactic acid, ethyl phenyl acetate) together with correlated compounds (ethyl hexanoate, isoamyl acetate, glycerol), bringing them up to 95 th percentile ethanol-normalized concentrations (Methods) within the Blond group (‘Spiked’ concentration in Fig.  5A ). Compared to controls, the spiked beers were found to have significantly improved overall appreciation among trained panelists, with panelist noting increased intensity of ester flavors, sweetness, alcohol, and body fullness (Fig.  5B ). To disentangle the contribution of ethanol to these results, a second experiment was performed without the addition of ethanol. This resulted in a similar outcome, including increased perception of alcohol and overall appreciation.

Adding the top chemical compounds, identified as best predictors of appreciation by our model, into poorly appreciated beers results in increased appreciation from our trained panel. Results of sensory tests between base beers and those spiked with compounds identified as the best predictors by the model. A Blond and Non/Low-alcohol (0.0% ABV) base beers were brought up to 95th-percentile ethanol-normalized concentrations within each style. B For each sensory attribute, tasters indicated the more intense sample and selected the sample they preferred. The numbers above the bars correspond to the p values that indicate significant changes in perceived flavor (two-sided binomial test: alpha 0.05, n  = 20 or 13).

In a last experiment, we tested whether using the model’s predictions can boost the appreciation of a non-alcoholic beer (beer 223 in Supplementary Data  1 ). Again, the addition of a mixture of predicted compounds (omitting ethanol, in this case) resulted in a significant increase in appreciation, body, ester flavor and sweetness.

Predicting flavor and consumer appreciation from chemical composition is one of the ultimate goals of sensory science. A reliable, systematic and unbiased way to link chemical profiles to flavor and food appreciation would be a significant asset to the food and beverage industry. Such tools would substantially aid in quality control and recipe development, offer an efficient and cost-effective alternative to pilot studies and consumer trials and would ultimately allow food manufacturers to produce superior, tailor-made products that better meet the demands of specific consumer groups more efficiently.

A limited set of studies have previously tried, to varying degrees of success, to predict beer flavor and beer popularity based on (a limited set of) chemical compounds and flavors 79 , 80 . Current sensitive, high-throughput technologies allow measuring an unprecedented number of chemical compounds and properties in a large set of samples, yielding a dataset that can train models that help close the gaps between chemistry and flavor, even for a complex natural product like beer. To our knowledge, no previous research gathered data at this scale (250 samples, 226 chemical parameters, 50 sensory attributes and 5 consumer scores) to disentangle and validate the chemical aspects driving beer preference using various machine-learning techniques. We find that modern machine learning models outperform conventional statistical tools, such as correlations and linear models, and can successfully predict flavor appreciation from chemical composition. This could be attributed to the natural incorporation of interactions and non-linear or discontinuous effects in machine learning models, which are not easily grasped by the linear model architecture. While linear models and partial least squares regression represent the most widespread statistical approaches in sensory science, in part because they allow interpretation 65 , 81 , 82 , modern machine learning methods allow for building better predictive models while preserving the possibility to dissect and exploit the underlying patterns. Of the 10 different models we trained, tree-based models, such as our best performing GBR, showed the best overall performance in predicting sensory responses from chemical information, outcompeting artificial neural networks. This agrees with previous reports for models trained on tabular data 83 . Our results are in line with the findings of Colantonio et al. who also identified the gradient boosting architecture as performing best at predicting appreciation and flavor (of tomatoes and blueberries, in their specific study) 26 . Importantly, besides our larger experimental scale, we were able to directly confirm our models’ predictions in vivo.

Our study confirms that flavor compound concentration does not always correlate with perception, suggesting complex interactions that are often missed by more conventional statistics and simple models. Specifically, we find that tree-based algorithms may perform best in developing models that link complex food chemistry with aroma. Furthermore, we show that massive datasets of untrained consumer reviews provide a valuable source of data, that can complement or even replace trained tasting panels, especially for appreciation and basic flavors, such as sweetness and bitterness. This holds despite biases that are known to occur in such datasets, such as price or conformity bias. Moreover, GBR models predict taste better than aroma. This is likely because taste (e.g. bitterness) often directly relates to the corresponding chemical measurements (e.g., iso-alpha acids), whereas such a link is less clear for aromas, which often result from the interplay between multiple volatile compounds. We also find that our models are best at predicting acidity and alcohol, likely because there is a direct relation between the measured chemical compounds (acids and ethanol) and the corresponding perceived sensorial attribute (acidity and alcohol), and because even untrained consumers are generally able to recognize these flavors and aromas.

The predictions of our final models, trained on review data, hold even for blind tastings with small groups of trained tasters, as demonstrated by our ability to validate specific compounds as drivers of beer flavor and appreciation. Since adding a single compound to the extent of a noticeable difference may result in an unbalanced flavor profile, we specifically tested our identified key drivers as a combination of compounds. While this approach does not allow us to validate if a particular single compound would affect flavor and/or appreciation, our experiments do show that this combination of compounds increases consumer appreciation.

It is important to stress that, while it represents an important step forward, our approach still has several major limitations. A key weakness of the GBR model architecture is that amongst co-correlating variables, the largest main effect is consistently preferred for model building. As a result, co-correlating variables often have artificially low importance scores, both for impurity and SHAP-based methods, like we observed in the comparison to the more randomized Extra Trees models. This implies that chemicals identified as key drivers of a specific sensory feature by GBR might not be the true causative compounds, but rather co-correlate with the actual causative chemical. For example, the high importance of ethyl acetate could be (partially) attributed to the total ester content, ethanol or ethyl hexanoate (rho=0.77, rho=0.72 and rho=0.68), while ethyl phenylacetate could hide the importance of prenyl isobutyrate and ethyl benzoate (rho=0.77 and rho=0.76). Expanding our GBR model to include beer style as a parameter did not yield additional power or insight. This is likely due to style-specific chemical signatures, such as iso-alpha acids and lactic acid, that implicitly convey style information to the original model, as well as the smaller sample size per style, limiting the power to uncover style-specific patterns. This can be partly attributed to the curse of dimensionality, where the high number of parameters results in the models mainly incorporating single parameter effects, rather than complex interactions such as style-dependent effects 67 . A larger number of samples may overcome some of these limitations and offer more insight into style-specific effects. On the other hand, beer style is not a rigid scientific classification, and beers within one style often differ a lot, which further complicates the analysis of style as a model factor.

Our study is limited to beers from Belgian breweries. Although these beers cover a large portion of the beer styles available globally, some beer styles and consumer patterns may be missing, while other features might be overrepresented. For example, many Belgian ales exhibit yeast-driven flavor profiles, which is reflected in the chemical drivers of appreciation discovered by this study. In future work, expanding the scope to include diverse markets and beer styles could lead to the identification of even more drivers of appreciation and better models for special niche products that were not present in our beer set.

In addition to inherent limitations of GBR models, there are also some limitations associated with studying food aroma. Even if our chemical analyses measured most of the known aroma compounds, the total number of flavor compounds in complex foods like beer is still larger than the subset we were able to measure in this study. For example, hop-derived thiols, that influence flavor at very low concentrations, are notoriously difficult to measure in a high-throughput experiment. Moreover, consumer perception remains subjective and prone to biases that are difficult to avoid. It is also important to stress that the models are still immature and that more extensive datasets will be crucial for developing more complete models in the future. Besides more samples and parameters, our dataset does not include any demographic information about the tasters. Including such data could lead to better models that grasp external factors like age and culture. Another limitation is that our set of beers consists of high-quality end-products and lacks beers that are unfit for sale, which limits the current model in accurately predicting products that are appreciated very badly. Finally, while models could be readily applied in quality control, their use in sensory science and product development is restrained by their inability to discern causal relationships. Given that the models cannot distinguish compounds that genuinely drive consumer perception from those that merely correlate, validation experiments are essential to identify true causative compounds.

Despite the inherent limitations, dissection of our models enabled us to pinpoint specific molecules as potential drivers of beer aroma and consumer appreciation, including compounds that were unexpected and would not have been identified using standard approaches. Important drivers of beer appreciation uncovered by our models include protein levels, ethyl acetate, ethyl phenyl acetate and lactic acid. Currently, many brewers already use lactic acid to acidify their brewing water and ensure optimal pH for enzymatic activity during the mashing process. Our results suggest that adding lactic acid can also improve beer appreciation, although its individual effect remains to be tested. Interestingly, ethanol appears to be unnecessary to improve beer appreciation, both for blond beer and alcohol-free beer. Given the growing consumer interest in alcohol-free beer, with a predicted annual market growth of >7% 84 , it is relevant for brewers to know what compounds can further increase consumer appreciation of these beers. Hence, our model may readily provide avenues to further improve the flavor and consumer appreciation of both alcoholic and non-alcoholic beers, which is generally considered one of the key challenges for future beer production.

Whereas we see a direct implementation of our results for the development of superior alcohol-free beverages and other food products, our study can also serve as a stepping stone for the development of novel alcohol-containing beverages. We want to echo the growing body of scientific evidence for the negative effects of alcohol consumption, both on the individual level by the mutagenic, teratogenic and carcinogenic effects of ethanol 85 , 86 , as well as the burden on society caused by alcohol abuse and addiction. We encourage the use of our results for the production of healthier, tastier products, including novel and improved beverages with lower alcohol contents. Furthermore, we strongly discourage the use of these technologies to improve the appreciation or addictive properties of harmful substances.

The present work demonstrates that despite some important remaining hurdles, combining the latest developments in chemical analyses, sensory analysis and modern machine learning methods offers exciting avenues for food chemistry and engineering. Soon, these tools may provide solutions in quality control and recipe development, as well as new approaches to sensory science and flavor research.

Beer selection

250 commercial Belgian beers were selected to cover the broad diversity of beer styles and corresponding diversity in chemical composition and aroma. See Supplementary Fig.  S1 .

Chemical dataset

Sample preparation.

Beers within their expiration date were purchased from commercial retailers. Samples were prepared in biological duplicates at room temperature, unless explicitly stated otherwise. Bottle pressure was measured with a manual pressure device (Steinfurth Mess-Systeme GmbH) and used to calculate CO 2 concentration. The beer was poured through two filter papers (Macherey-Nagel, 500713032 MN 713 ¼) to remove carbon dioxide and prevent spontaneous foaming. Samples were then prepared for measurements by targeted Headspace-Gas Chromatography-Flame Ionization Detector/Flame Photometric Detector (HS-GC-FID/FPD), Headspace-Solid Phase Microextraction-Gas Chromatography-Mass Spectrometry (HS-SPME-GC-MS), colorimetric analysis, enzymatic analysis, Near-Infrared (NIR) analysis, as described in the sections below. The mean values of biological duplicates are reported for each compound.

HS-GC-FID/FPD

HS-GC-FID/FPD (Shimadzu GC 2010 Plus) was used to measure higher alcohols, acetaldehyde, esters, 4-vinyl guaicol, and sulfur compounds. Each measurement comprised 5 ml of sample pipetted into a 20 ml glass vial containing 1.75 g NaCl (VWR, 27810.295). 100 µl of 2-heptanol (Sigma-Aldrich, H3003) (internal standard) solution in ethanol (Fisher Chemical, E/0650DF/C17) was added for a final concentration of 2.44 mg/L. Samples were flushed with nitrogen for 10 s, sealed with a silicone septum, stored at −80 °C and analyzed in batches of 20.

The GC was equipped with a DB-WAXetr column (length, 30 m; internal diameter, 0.32 mm; layer thickness, 0.50 µm; Agilent Technologies, Santa Clara, CA, USA) to the FID and an HP-5 column (length, 30 m; internal diameter, 0.25 mm; layer thickness, 0.25 µm; Agilent Technologies, Santa Clara, CA, USA) to the FPD. N 2 was used as the carrier gas. Samples were incubated for 20 min at 70 °C in the headspace autosampler (Flow rate, 35 cm/s; Injection volume, 1000 µL; Injection mode, split; Combi PAL autosampler, CTC analytics, Switzerland). The injector, FID and FPD temperatures were kept at 250 °C. The GC oven temperature was first held at 50 °C for 5 min and then allowed to rise to 80 °C at a rate of 5 °C/min, followed by a second ramp of 4 °C/min until 200 °C kept for 3 min and a final ramp of (4 °C/min) until 230 °C for 1 min. Results were analyzed with the GCSolution software version 2.4 (Shimadzu, Kyoto, Japan). The GC was calibrated with a 5% EtOH solution (VWR International) containing the volatiles under study (Supplementary Table  S7 ).

HS-SPME-GC-MS

HS-SPME-GC-MS (Shimadzu GCMS-QP-2010 Ultra) was used to measure additional volatile compounds, mainly comprising terpenoids and esters. Samples were analyzed by HS-SPME using a triphase DVB/Carboxen/PDMS 50/30 μm SPME fiber (Supelco Co., Bellefonte, PA, USA) followed by gas chromatography (Thermo Fisher Scientific Trace 1300 series, USA) coupled to a mass spectrometer (Thermo Fisher Scientific ISQ series MS) equipped with a TriPlus RSH autosampler. 5 ml of degassed beer sample was placed in 20 ml vials containing 1.75 g NaCl (VWR, 27810.295). 5 µl internal standard mix was added, containing 2-heptanol (1 g/L) (Sigma-Aldrich, H3003), 4-fluorobenzaldehyde (1 g/L) (Sigma-Aldrich, 128376), 2,3-hexanedione (1 g/L) (Sigma-Aldrich, 144169) and guaiacol (1 g/L) (Sigma-Aldrich, W253200) in ethanol (Fisher Chemical, E/0650DF/C17). Each sample was incubated at 60 °C in the autosampler oven with constant agitation. After 5 min equilibration, the SPME fiber was exposed to the sample headspace for 30 min. The compounds trapped on the fiber were thermally desorbed in the injection port of the chromatograph by heating the fiber for 15 min at 270 °C.

The GC-MS was equipped with a low polarity RXi-5Sil MS column (length, 20 m; internal diameter, 0.18 mm; layer thickness, 0.18 µm; Restek, Bellefonte, PA, USA). Injection was performed in splitless mode at 320 °C, a split flow of 9 ml/min, a purge flow of 5 ml/min and an open valve time of 3 min. To obtain a pulsed injection, a programmed gas flow was used whereby the helium gas flow was set at 2.7 mL/min for 0.1 min, followed by a decrease in flow of 20 ml/min to the normal 0.9 mL/min. The temperature was first held at 30 °C for 3 min and then allowed to rise to 80 °C at a rate of 7 °C/min, followed by a second ramp of 2 °C/min till 125 °C and a final ramp of 8 °C/min with a final temperature of 270 °C.

Mass acquisition range was 33 to 550 amu at a scan rate of 5 scans/s. Electron impact ionization energy was 70 eV. The interface and ion source were kept at 275 °C and 250 °C, respectively. A mix of linear n-alkanes (from C7 to C40, Supelco Co.) was injected into the GC-MS under identical conditions to serve as external retention index markers. Identification and quantification of the compounds were performed using an in-house developed R script as described in Goelen et al. and Reher et al. 87 , 88 (for package information, see Supplementary Table  S8 ). Briefly, chromatograms were analyzed using AMDIS (v2.71) 89 to separate overlapping peaks and obtain pure compound spectra. The NIST MS Search software (v2.0 g) in combination with the NIST2017, FFNSC3 and Adams4 libraries were used to manually identify the empirical spectra, taking into account the expected retention time. After background subtraction and correcting for retention time shifts between samples run on different days based on alkane ladders, compound elution profiles were extracted and integrated using a file with 284 target compounds of interest, which were either recovered in our identified AMDIS list of spectra or were known to occur in beer. Compound elution profiles were estimated for every peak in every chromatogram over a time-restricted window using weighted non-negative least square analysis after which peak areas were integrated 87 , 88 . Batch effect correction was performed by normalizing against the most stable internal standard compound, 4-fluorobenzaldehyde. Out of all 284 target compounds that were analyzed, 167 were visually judged to have reliable elution profiles and were used for final analysis.

Discrete photometric and enzymatic analysis

Discrete photometric and enzymatic analysis (Thermo Scientific TM Gallery TM Plus Beermaster Discrete Analyzer) was used to measure acetic acid, ammonia, beta-glucan, iso-alpha acids, color, sugars, glycerol, iron, pH, protein, and sulfite. 2 ml of sample volume was used for the analyses. Information regarding the reagents and standard solutions used for analyses and calibrations is included in Supplementary Table  S7 and Supplementary Table  S9 .

NIR analyses

NIR analysis (Anton Paar Alcolyzer Beer ME System) was used to measure ethanol. Measurements comprised 50 ml of sample, and a 10% EtOH solution was used for calibration.

Correlation calculations

Pairwise Spearman Rank correlations were calculated between all chemical properties.

Sensory dataset

Trained panel.

Our trained tasting panel consisted of volunteers who gave prior verbal informed consent. All compounds used for the validation experiment were of food-grade quality. The tasting sessions were approved by the Social and Societal Ethics Committee of the KU Leuven (G-2022-5677-R2(MAR)). All online reviewers agreed to the Terms and Conditions of the RateBeer website.

Sensory analysis was performed according to the American Society of Brewing Chemists (ASBC) Sensory Analysis Methods 90 . 30 volunteers were screened through a series of triangle tests. The sixteen most sensitive and consistent tasters were retained as taste panel members. The resulting panel was diverse in age [22–42, mean: 29], sex [56% male] and nationality [7 different countries]. The panel developed a consensus vocabulary to describe beer aroma, taste and mouthfeel. Panelists were trained to identify and score 50 different attributes, using a 7-point scale to rate attributes’ intensity. The scoring sheet is included as Supplementary Data  3 . Sensory assessments took place between 10–12 a.m. The beers were served in black-colored glasses. Per session, between 5 and 12 beers of the same style were tasted at 12 °C to 16 °C. Two reference beers were added to each set and indicated as ‘Reference 1 & 2’, allowing panel members to calibrate their ratings. Not all panelists were present at every tasting. Scores were scaled by standard deviation and mean-centered per taster. Values are represented as z-scores and clustered by Euclidean distance. Pairwise Spearman correlations were calculated between taste and aroma sensory attributes. Panel consistency was evaluated by repeating samples on different sessions and performing ANOVA to identify differences, using the ‘stats’ package (v4.2.2) in R (for package information, see Supplementary Table  S8 ).

Online reviews from a public database

The ‘scrapy’ package in Python (v3.6) (for package information, see Supplementary Table  S8 ). was used to collect 232,288 online reviews (mean=922, min=6, max=5343) from RateBeer, an online beer review database. Each review entry comprised 5 numerical scores (appearance, aroma, taste, palate and overall quality) and an optional review text. The total number of reviews per reviewer was collected separately. Numerical scores were scaled and centered per rater, and mean scores were calculated per beer.

For the review texts, the language was estimated using the packages ‘langdetect’ and ‘langid’ in Python. Reviews that were classified as English by both packages were kept. Reviewers with fewer than 100 entries overall were discarded. 181,025 reviews from >6000 reviewers from >40 countries remained. Text processing was done using the ‘nltk’ package in Python. Texts were corrected for slang and misspellings; proper nouns and rare words that are relevant to the beer context were specified and kept as-is (‘Chimay’,’Lambic’, etc.). A dictionary of semantically similar sensorial terms, for example ‘floral’ and ‘flower’, was created and collapsed together into one term. Words were stemmed and lemmatized to avoid identifying words such as ‘acid’ and ‘acidity’ as separate terms. Numbers and punctuation were removed.

Sentences from up to 50 randomly chosen reviews per beer were manually categorized according to the aspect of beer they describe (appearance, aroma, taste, palate, overall quality—not to be confused with the 5 numerical scores described above) or flagged as irrelevant if they contained no useful information. If a beer contained fewer than 50 reviews, all reviews were manually classified. This labeled data set was used to train a model that classified the rest of the sentences for all beers 91 . Sentences describing taste and aroma were extracted, and term frequency–inverse document frequency (TFIDF) was implemented to calculate enrichment scores for sensorial words per beer.

The sex of the tasting subject was not considered when building our sensory database. Instead, results from different panelists were averaged, both for our trained panel (56% male, 44% female) and the RateBeer reviews (70% male, 30% female for RateBeer as a whole).

Beer price collection and processing

Beer prices were collected from the following stores: Colruyt, Delhaize, Total Wine, BeerHawk, The Belgian Beer Shop, The Belgian Shop, and Beer of Belgium. Where applicable, prices were converted to Euros and normalized per liter. Spearman correlations were calculated between these prices and mean overall appreciation scores from RateBeer and the taste panel, respectively.

Pairwise Spearman Rank correlations were calculated between all sensory properties.

Machine learning models

Predictive modeling of sensory profiles from chemical data.

Regression models were constructed to predict (a) trained panel scores for beer flavors and quality from beer chemical profiles and (b) public reviews’ appreciation scores from beer chemical profiles. Z-scores were used to represent sensory attributes in both data sets. Chemical properties with log-normal distributions (Shapiro-Wilk test, p  <  0.05 ) were log-transformed. Missing chemical measurements (0.1% of all data) were replaced with mean values per attribute. Observations from 250 beers were randomly separated into a training set (70%, 175 beers) and a test set (30%, 75 beers), stratified per beer style. Chemical measurements (p = 231) were normalized based on the training set average and standard deviation. In total, three linear regression-based models: linear regression with first-order interaction terms (LR), lasso regression with first-order interaction terms (Lasso) and partial least squares regression (PLSR); five decision tree models, Adaboost regressor (ABR), Extra Trees (ET), Gradient Boosting regressor (GBR), Random Forest (RF) and XGBoost regressor (XGBR); one support vector machine model (SVR) and one artificial neural network model (ANN) were trained. The models were implemented using the ‘scikit-learn’ package (v1.2.2) and ‘xgboost’ package (v1.7.3) in Python (v3.9.16). Models were trained, and hyperparameters optimized, using five-fold cross-validated grid search with the coefficient of determination (R 2 ) as the evaluation metric. The ANN (scikit-learn’s MLPRegressor) was optimized using Bayesian Tree-Structured Parzen Estimator optimization with the ‘Optuna’ Python package (v3.2.0). Individual models were trained per attribute, and a multi-output model was trained on all attributes simultaneously.

Model dissection

GBR was found to outperform other methods, resulting in models with the highest average R 2 values in both trained panel and public review data sets. Impurity-based rankings of the most important predictors for each predicted sensorial trait were obtained using the ‘scikit-learn’ package. To observe the relationships between these chemical properties and their predicted targets, partial dependence plots (PDP) were constructed for the six most important predictors of consumer appreciation 74 , 75 .

The ‘SHAP’ package in Python (v0.41.0) was implemented to provide an alternative ranking of predictor importance and to visualize the predictors’ effects as a function of their concentration 68 .

Validation of causal chemical properties

To validate the effects of the most important model features on predicted sensory attributes, beers were spiked with the chemical compounds identified by the models and descriptive sensory analyses were carried out according to the American Society of Brewing Chemists (ASBC) protocol 90 .

Compound spiking was done 30 min before tasting. Compounds were spiked into fresh beer bottles, that were immediately resealed and inverted three times. Fresh bottles of beer were opened for the same duration, resealed, and inverted thrice, to serve as controls. Pairs of spiked samples and controls were served simultaneously, chilled and in dark glasses as outlined in the Trained panel section above. Tasters were instructed to select the glass with the higher flavor intensity for each attribute (directional difference test 92 ) and to select the glass they prefer.

The final concentration after spiking was equal to the within-style average, after normalizing by ethanol concentration. This was done to ensure balanced flavor profiles in the final spiked beer. The same methods were applied to improve a non-alcoholic beer. Compounds were the following: ethyl acetate (Merck KGaA, W241415), ethyl hexanoate (Merck KGaA, W243906), isoamyl acetate (Merck KGaA, W205508), phenethyl acetate (Merck KGaA, W285706), ethanol (96%, Colruyt), glycerol (Merck KGaA, W252506), lactic acid (Merck KGaA, 261106).

Significant differences in preference or perceived intensity were determined by performing the two-sided binomial test on each attribute.

Reporting summary

Further information on research design is available in the  Nature Portfolio Reporting Summary linked to this article.

Data availability

The data that support the findings of this work are available in the Supplementary Data files and have been deposited to Zenodo under accession code 10653704 93 . The RateBeer scores data are under restricted access, they are not publicly available as they are property of RateBeer (ZX Ventures, USA). Access can be obtained from the authors upon reasonable request and with permission of RateBeer (ZX Ventures, USA).  Source data are provided with this paper.

Code availability

The code for training the machine learning models, analyzing the models, and generating the figures has been deposited to Zenodo under accession code 10653704 93 .

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Acknowledgements

We thank all lab members for their discussions and thank all tasting panel members for their contributions. Special thanks go out to Dr. Karin Voordeckers for her tremendous help in proofreading and improving the manuscript. M.S. was supported by a Baillet-Latour fellowship, L.C. acknowledges financial support from KU Leuven (C16/17/006), F.A.T. was supported by a PhD fellowship from FWO (1S08821N). Research in the lab of K.J.V. is supported by KU Leuven, FWO, VIB, VLAIO and the Brewing Science Serves Health Fund. Research in the lab of T.W. is supported by FWO (G.0A51.15) and KU Leuven (C16/17/006).

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These authors contributed equally: Michiel Schreurs, Supinya Piampongsant, Miguel Roncoroni.

Authors and Affiliations

VIB—KU Leuven Center for Microbiology, Gaston Geenslaan 1, B-3001, Leuven, Belgium

Michiel Schreurs, Supinya Piampongsant, Miguel Roncoroni, Lloyd Cool, Beatriz Herrera-Malaver, Florian A. Theßeling & Kevin J. Verstrepen

CMPG Laboratory of Genetics and Genomics, KU Leuven, Gaston Geenslaan 1, B-3001, Leuven, Belgium

Leuven Institute for Beer Research (LIBR), Gaston Geenslaan 1, B-3001, Leuven, Belgium

Laboratory of Socioecology and Social Evolution, KU Leuven, Naamsestraat 59, B-3000, Leuven, Belgium

Lloyd Cool, Christophe Vanderaa & Tom Wenseleers

VIB Bioinformatics Core, VIB, Rijvisschestraat 120, B-9052, Ghent, Belgium

Łukasz Kreft & Alexander Botzki

AB InBev SA/NV, Brouwerijplein 1, B-3000, Leuven, Belgium

Philippe Malcorps & Luk Daenen

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Contributions

S.P., M.S. and K.J.V. conceived the experiments. S.P., M.S. and K.J.V. designed the experiments. S.P., M.S., M.R., B.H. and F.A.T. performed the experiments. S.P., M.S., L.C., C.V., L.K., A.B., P.M., L.D., T.W. and K.J.V. contributed analysis ideas. S.P., M.S., L.C., C.V., T.W. and K.J.V. analyzed the data. All authors contributed to writing the manuscript.

Corresponding author

Correspondence to Kevin J. Verstrepen .

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Competing interests.

K.J.V. is affiliated with bar.on. The other authors declare no competing interests.

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Schreurs, M., Piampongsant, S., Roncoroni, M. et al. Predicting and improving complex beer flavor through machine learning. Nat Commun 15 , 2368 (2024). https://doi.org/10.1038/s41467-024-46346-0

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Received : 30 October 2023

Accepted : 21 February 2024

Published : 26 March 2024

DOI : https://doi.org/10.1038/s41467-024-46346-0

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