Introduction-to-statistics-printable-guide-UniSkills.pdf

Introduction to statistics

Table of Contents

Introduction

Welcome to the Introduction to statistics module. This module introduces key ideas that will help you understand statistics used in journal articles and other academic literature, and that you can use to analyse your own data. Note that if you are not planning to do either of these things, and instead only want to learn how to interpret a small number of common graphs and tables and calculate a few important values, you may find the Statistics and probability page of the Numeracy fundamentals module more suitable.

You can work through this module from beginning to end in the order provided, or you can move directly to specific pages or sections based on your needs or preferences. Additionally, if you are interested in learning how to carry out the statistical tests discussed in this module using the statistical software SPSS or Stata, you may also wish to complete the Introduction to SPSS or Introduction to Stata module. Note that all graphs and tables included in this module have been created using SPSS, except the tables in the Interpreting statistics page which have been taken from journal articles.

Before starting the module, you may also find it helpful to read the How to get confident with statistics post on The Research Whisperer blog. This post provides practical advice on building confidence in statistics and may help you make better use of this module.

Your feedback on this module is welcome at any time and can be provided through the feedback page. If you have questions about the module, please contact Library-UniSkills@curtin.edu.au

What you will learn

What data and variables are, and how to identify different types, including categorical and continuous data and independent and dependent variables.
How to interpret and critically evaluate statistics published in academic sources, including how to tell the difference between descriptive and inferential statistics and between statistical and practical significance.
What descriptive statistics are, and how to display data and calculate descriptive statistics for one variable or two variables at a time, for both categorical and continuous data.
What the normal distribution and sampling distributions are, and how to determine whether a variable is normally distributed or whether a sample size is large enough for the sampling distribution of the mean to be treated as a normal distribution.
What inferential statistics are, including how to distinguish between estimation and hypothesis testing, how to calculate confidence intervals, and how to interpret confidence intervals and \(p\) values.
How to compare the means of continuous variables using \(t\)-tests or one-way analysis of variance (ANOVA).
How to test for relationships between pairs of variables using the chi-square test of independence or Pearson’s correlation.
How to use linear and logistic regression to understand and explain relationships between variables.Data & variable types

This page explains key concepts that will be used in later pages. It introduces what data and variables are, and how to tell the difference between common types.

In brief, it covers the following:

A definition of data
A definition of a variable
How to distinguish between categorical and continuous data
How to distinguish between independent and dependent variables

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What is data?

Data refers to observations and measurements that have been obtained or collected, often through research. It can be recorded in different forms:

quantitative data is recorded as numbers and measures quantities
qualitative data is recorded as text and describes qualities.

Quantitative data can be analysed using statistics, and so can qualitative data that is grouped into categories (for example, hair colour, country of birth, or marital status). However, qualitative data that records thoughts, feelings, and opinions is usually analysed in a different way.

In this module, you will work with quantitative data and categorical qualitative data. These types of data use specific statistical terms, which will be introduced shortly.

What is a variable?

A variable is a characteristic or attribute that you observe, measure, or record. Some examples include height, weight, eye colour, dog breed, climate, electrical conductivity, customer service satisfaction and class attendance.

A variable can have different values for different subjects (a subject can be a person, place, or thing). For example:

the variable ‘height’ might be \(170\textrm{cm}\) for one person, \(163\textrm{cm}\) for another, and \(154\textrm{cm}\) for another
the variable ‘climate’ might be arid for one city, tropical for another, and Mediterranean for another
the variable ‘class attendance’ might be \(17\) for one class, \(25\) for another, and \(32\) for another.

These differences show how the value of a variable can change from one subject to another.

Categorical and continuous data

Choosing the correct statistic or statistical test depends on the type of data you have, and therefore the type of variable. For this reason, it is very important to be able to tell the difference between types of data.

Most of the time, you will classify your data (and variables) as either categorical or continuous. Each type can also be divided into smaller groups. These definitions and classifications are explained below.

Categorical data is data that is grouped into categories. For example, this includes variables such as gender or smoking status. It can be divided into two types:

Nominal data: The categories do not have an order. For example, marital status. If there are only two categories, the terms binary or dichotomous are sometimes used.
Ordinal data: The categories do have an order. For example, satisfaction level (such as low, medium, high).

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Continuous data is measured on a continuous numerical scale and can take many possible values. For example, this includes variables such as weight or distance. It can be divided into two types:

Interval data: There is no true zero, and negative numbers have meaning. For example, temperature in degrees Celsius.
Ratio data: There is a true zero, and negative numbers do not have meaning. For example, height.

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One other type of data is discrete data, which can be defined as follows:

Discrete data measures counts or the number of events. For example, class attendance. Although it is numerical, it is not measured on a continuous scale. Because of this, it does not fit neatly into either categorical or continuous data.

Instead, you can think of discrete data as a special type of data that can be treated as either categorical or continuous, depending on how many possible values there are. It is usually treated as continuous, but if there are only a small number of possible values (such as the number of units studied in a semester), it may be more useful to treat it as categorical.

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It is also important to understand that continuous data can be turned into categorical data by creating categories from it. This can be useful if you want to use statistical tests that are designed for categorical data. For example, continuous data for a weight variable could be grouped into categories such as different weight ranges.

However, it is not possible to turn categorical data into continuous data. For this reason, if you have a choice, it is usually better to collect continuous data because it gives you more flexibility when analysing your data.

If you would like to practise classifying variables, try one or both of the following activities.

Activity: Decide whether you would treat each variable as categorical or continuous for the purpose of analysis and drag each variable into the correct box. Click on the ‘i’ symbol if you need a tip. Note that although you can always create categories from continuous data, you should place variables in the continuous box if it is possible to collect continuous data for them.

H5P interactive activity

Activity: Decide whether the variables shown contain data that is nominal, ordinal, discrete or continuous and drag each variable into the correct box. Click on the ‘i’ symbol if you need a tip.

H5P interactive activity

Independent and dependent variables

When you use bivariate analysis to test relationships between two variables, you will usually have one independent variable and one dependent variable. These are defined below.

The independent variable (also called the predictor variable) is the variable that may influence, affect, or predict another variable. For example, if you are investigating whether age influences income, then age is the independent variable.

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The dependent variable (also called the outcome variable) is the variable that may be influenced, affected, or predicted. For example, if you are investigating whether income can be predicted by age, then income is the dependent variable.

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It is important to be able to tell the difference between these two types of variables, because it determines where each variable is placed in tables and graphs and how it is used in statistical tests. However, it is also important to understand that the role of a variable depends on the context. Some variables (such as age) are always independent. Other variables (such as smoking status) can be either independent or dependent, depending on what you are trying to investigate.

If you would like to practise identifying independent and dependent variables, try the following activity.

H5P interactive activity
Interpreting statistics

Even if you are not carrying out statistical analysis yourself, it is important to be able to interpret and critically evaluate statistics that have been published as this will help you get the most out of journal articles and other academic sources that you read. This page explains some commonly reported statistics and how to interpret them.

In brief, it covers the following:

How to interpret descriptive statistics
How to interpret inferential statistics, including statistical and practical significance where required

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Note that if you would like extra support with interpreting graphs and tables, or with performing key calculations, you may find the Statistics and probability page of the Numeracy fundamentals module useful. If you would like to carry out your own statistical analysis, information on how to do this is provided in later pages of this module.

Interpreting descriptive statistics

Descriptive statistics are used to summarise and describe the data you have collected. This may be:

data for the whole population of interest (for example, everyone who attends a particular school), or
data collected from a random sample of a larger population (for example, a random sample of people who were patients at a particular hospital over a two-year period).

The key difference is that when you use data from a sample (which is most common), descriptive statistics cannot be used to draw conclusions about the wider population. For this reason, they are usually used together with other types of statistics. These will be explained shortly, but first we will look at the descriptive statistics in Table 1.

Table 1
Participant Characteristics

From “Examining Associations between Health, Wellbeing and Social Capital: Findings from a Survey Developed and Conducted using Participatory Action Research,” by S. Visram, S. Smith, N. Connor, G. Greig, & C. Scorer, 2018, Journal of Public Mental Health , 17 (3), p. 127 (https://doi.org/10.1108/JPMH-09-2017-0035). Copyright 2018 by Emerald Publishing Limited.

This data was collected from a sample of people who live in a particular city. The sample size is \(233\) (as shown by the \(n = 233\) at the bottom of the table), and there are two types of variables summarised:

categorical variables, which group data into categories (for example, the ‘Gender’ variable), and
continuous variables, which involve numerical data (for example, the ‘Age in years’ variable).

These two types of variables need to be analysed in different ways, and this has been done in the table. Let’s look at the categorical variables first.

Interpreting descriptive statistics for categorical variables

The categorical variables in this example have been summarised using percentages, which show the percentage of the sample that falls into each category for a variable. For example, the descriptive statistics for the ‘Gender’ variable show that \(65.7\%\) of the sample in Table 1 are women and \(34.3\%\) are men. Note that sometimes frequencies are used instead of percentages, and these show how many observations are in each category instead.

If you would like to learn more about descriptive statistics for categorical variables, including how to calculate them, please refer to the Descriptive statistics for one categorical variable section of this module. Alternatively, to practise interpreting descriptive statistics for categorical variables, try the following question.

What percentage of the sample described in Table 1 are in each of the categories for the ‘Living situation’ variable?

ANSWER: \(48.7\%\) own their own home; \(30\%\) rent from a housing association; \(12.6\%\) rent from a landlord; \(7\%\) live with parents/family.

Interpreting descriptive statistics for continuous variables

In this example, the continuous variables are summarised using two measures. The first measure is the mean, also called the arithmetic average. This is the most commonly used measure of central tendency, and is a single value that shows the central value of the data set. For example, in Table 1 the mean for the ‘Age in years’ variable is \(47.3\), which shows that the average (or central) age of the sample of \(233\) people is \(47.3\). The second measure is the standard deviation (SD), which is the most commonly used measure of dispersion. It is a single value that shows how varied (or spread out) the data is. In Table 1, the standard deviation for the ‘Age in years’ variable is \(17.4\). This is quite large compared to the mean, which shows that the ages in the sample of \(233\) people vary quite a lot.

Using both a measure of central tendency and a measure of dispersion gives a better summary of a continuous variable, as one measure on its own does not fully describe the data. For example, if we only knew that the mean age was \(47.3\), we would not know if all values were close to \(47.3\) or spread out across a wide range. Similarly, if we only knew that the standard deviation was \(17.4\), we would not know the central value of the data.

It is also important to understand that the mean and standard deviation are not always the best measures to use, as in some situations they do not describe the data accurately. This happens when:

the data is skewed (not normally distributed, which will be explained later), or
there are outliers (values that are much higher or lower than most of the data).

For example, suppose most people in Table 1 are aged between \(40\) and \(45\), but a small number are in their \(80\)s. These higher values would be outliers, and would increase the mean so it would no longer represent the central value accurately.

In these situations, the median is a more suitable measure of central tendency because it is not affected by outliers or skew, and the interquartile range is then used as the matching measure of dispersion. So when a variable is skewed or contains outliers, good research will use the median and interquartile range instead of the mean and standard deviation.

If you would like to learn more about descriptive statistics for continuous variables, including how to calculate them, refer to the Descriptive statistics for one continuous variable section of this module. Alternatively, to practise interpreting descriptive statistics for continuous variables, try the following question.

Which other continuous variable (aside from ‘Age in years’) has been summarised in Table 1, and what is its mean and standard deviation?

ANSWER: ‘Years lived in Shildon’; mean is \(32.9\) years and standard deviation is \(20.1\) years.

Interpreting inferential statistics

Inferential statistics are used to draw inferences about a wider population when data has been collected from a sample, rather than the whole population. This is usually necessary because collecting data from an entire population is often not practical. There are many different inferential statistical tests, used for different types of analysis and different types of variables. For example, different tests are used for continuous and categorical variables, and different tests may also be used for continuous variables depending on the sample size or the distribution (the pattern or spread of the data).

One type of distribution that can affect the choice of test is the normal distribution, which is a special type of distribution that large sets of naturally occurring continuous data often follow. It has two key properties:

The mean, median and mode are all equal.
Fixed proportions of the data fall within certain numbers of standard deviations from the mean (\(68\%\) within one SD, \(95\%\) within two SDs and \(99.7\%\) within three SDs).

When the sample size is small (typically fewer than about \(30\)), and sometimes even when it is larger, researchers may check whether continuous variables follow a normal distribution. This is not always reported in journal articles, but when it is, the Shapiro–Wilk test is often used to assess normality.

If an article states that variables pass this test (that is, the \(p\) value for the test is above \(.05\)), it means that normality can be assumed. For example, Power et al. (2019) refer to this when they state that their “outcomes were assessed for normality using Shapiro-Wilk” (p. 5).

In some cases, an article may state that variables have been transformed (for example, using a natural logarithm). This means the data was not originally normally distributed, but a mathematical change has been applied to make it follow a normal distribution. If you would like more information about the normal distribution and how to assess it, refer to The normal distribution page of this module.

If continuous variables are normally distributed, or can be transformed to be normally distributed, or if the sample size is large enough that this is not an issue, parametric tests are used. If the sample size is small and the variables are not normally distributed, or if the variables are categorical, nonparametric tests are used instead. Examples of parametric tests include \(t\) tests, ANOVA and Pearson’s correlation, while examples of nonparametric tests include chi-square tests, the Mann-Whitney U Test, the Wilcoxon Signed Rank Test, the Kruskal-Wallis One-Way ANOVA and Spearman’s rho.

For each of these tests, a good research article should report statistics that show whether any differences or relationships between variables are statistically significant and/or practically significant for the population. These terms, the relevant statistics, and how to interpret them are explained below. Alternatively, for more information on inferential statistics, including how to calculate statistical and practical significance, refer to the Inferential statistics page of this module.

Interpreting statistical significance

Statistical significance refers to how likely it is that what has been observed in a sample (for example, a difference in means or a relationship between variables) could have happened by random chance alone. If it is very unlikely that the result is due to chance, it is considered statistically significant in the population. To measure this, we use the \(p\) value.

The \(p\) value is the probability that the result occurred due to random chance. The smaller the \(p\) value, the less likely it is that the result is due to chance. To decide whether a \(p\) value is small enough, researchers use a level of significance, written as \(\alpha\). Different studies may use different levels, but a common value is \(.05\). In this case, a \(p\) value less than or equal to \(.05\) means that the result is considered statistically significant. Other common levels of significance used are \(.01\) and \(.001\), and articles may report the level at which results are significant (for example, significant at the \(.05\) or \(.01\) level).

Articles often include confidence intervals as well as \(p\) values. These are ranges of values that a population value is expected to fall within, with a certain level of confidence. This is usually \(95\%\), and is called a \(95\%\) confidence interval. Confidence intervals can also be used to assess statistical significance (at the \(.05\) level for a \(95\%\) confidence interval). In addition, they provide extra information such as the size and direction of a difference.

Table 2 shows descriptive statistics along with \(p\) values and confidence intervals for a set of variables. These statistics allow us to decide whether there is a statistically significant difference in means between two groups (adolescents with and without cerebral palsy) at the \(.05\) level of significance.

Table 2
Descriptive statistics, \(95\%\) CIs and \(p\) values for a study investigating mental and physical health of adults with cerebral palsy

From “Health-Related Quality of Life and Mental Health of Adolescents with Cerebral Palsy in Rural Bangladesh,” by R. Power, M. Muhit, E. Heanoy, T. Karim, N. Badawi, R. Akhter, & G. Khandaker, 2019, PLOS One , 14 (6), p. 9 (https://doi.org/10.1371/journal.pone.0217675). CC-BY.

Consider the ‘Total score’ variable, which measures overall health-related quality of life. The mean difference of \(11.9\) shows the difference between the two groups in the sample and is a descriptive statistic. The \(95\%\) confidence interval from \(10.1\) to \(13.7\) shows that we are \(95\%\) confident that the true difference in the population lies within this range, and because this interval does not include \(0\) (which would mean no difference), the result is significant at the \(.05\) level.

This result is also supported by the \(p\) value, as the fact that \(p < .0001\) shows that the result is statistically significant, not only at the \(.05\) level but even at the \(.0001\) level. The other variables in the table can be interpreted in the same way. To practise doing this, have a go at the following questions.

Within which range is the difference in means for the ‘Peer and social support’ variable expected to lie with \(95\%\) confidence? Does this show statistical significance at the \(.05\) level?

ANSWER: The difference is expected to lie between \(3.7\) and \(8.6\). Because this range does not include \(0\), the result is statistically significant at the \(5\%\) level.

Is the difference in means for the ‘School environment’ variable statistically significant at the \(.05\) level? What about the \(.01\) level?

ANSWER: The result is statistically significant at the \(.05\) level because \(p < .05\) and the confidence interval does not include \(0\). However, it is not statistically significant at the \(.01\) level because \(p > .01\).

Interpreting practical significance

Practical significance refers to whether a result observed in a sample (for example, a difference in means or a relationship between variables) is meaningful in a real-life context. It is measured using an effect size, which is different for different statistical tests. For example:

Cohen’s \(d\) is used with \(t\) tests
Pearson’s correlation coefficient is used with Pearson’s correlation
Phi, Cramer’s V, odds ratio and relative risk are commonly used with the chi-square test of independence.

Including practical significance in research provides important additional information, as statistical significance alone does not show whether a result is meaningful in practice because it is affected by sample size:

In very large samples, very small differences that are not meaningful in real life can still be statistically significant.
In very small samples, large and meaningful differences may not be statistically significant.

This idea is related to statistical power, which is the probability that a test will detect an effect of a certain size as statistically significant. Researchers can calculate how large a sample needs to be to detect an effect of a certain size at a chosen level of significance, for a given level of power (usually \(0.8\) or higher). For this reason, research articles may also report the power of a test.

Returning to measures of effect size, and Table 3 presents odds ratios along with \(95\%\) confidence intervals and \(p\) values. The confidence intervals and \(p\) values show the statistical significance of the relationship between each exposure variable and the outcome variable (L.longbeachae infection), while the odds ratios show the practical significance of these relationships.

Table 3
Odds ratios, \(95\%\) CIs and \(p\) values for a study investigating L.longbeachae infection exposures involving potting mix

From “Does using Potting Mix make you Sick? Results from a Legionella Longbeachae Case-Control Study in South Australia,” by B. A. O’Connor, J. Carman, K. Eckert, G. Tucker, R. Givney, & S. Cameron, 2007, Epidemiology and Infection , 135 (1), p. 36 (https://doi.org/10.1017/S095026880600656X). Copyright 2006 by Cambridge University Press.

For each exposure variable, the odds ratio compares the odds of exposure in the group with L. longbeachae infection and the odds of exposure in the group without the infection. In other words, the odds ratio shows how many times more or less likely the group with the infection is to have the exposure compared to the group without the infection.

An odds ratio less than \(1\) means the infected group is less likely to have the exposure.
An odds ratio equal to \(1\) means both groups are equally likely to have the exposure.
An odds ratio greater than \(1\) means the infected group is more likely to have the exposure.

For example, consider the first odds ratio which has a value of \(4.74\). This shows that people with L.longbeachae were \(4.74\) times more likely to have used potting mix in the last four weeks than those without the infection. Based on common guidelines for interpreting effect size, this represents a medium to large effect. (However, practical significance should always be interpreted in context, rather than relying only on cut-off values.)

Furthermore, the \(95\%\) confidence interval for this result is \(1.65\) to \(13.55\). This means we are \(95\%\) confident that the true value in the population lies within this range. Because this range does not include \(1\) (which would mean equal odds), the result is statistically significant at the \(.05\) level. This is also supported by the \(p\) value of \(.004\).

Research articles may also report adjusted odds ratios. These take into account other variables (called confounding variables) that may affect the outcome. Adjusted odds ratios are interpreted in the same way as the odds ratios described above.

If you would like to practise interpreting some of the other odds ratios in the table, have a go at the following questions.

How many times more or less likely were those with L.longbeachae infection to be near dripping hanging pots, compared to those without infection? Is this statistically significant at the \(.05\) level?

ANSWER: Those with the infection were \(2.79\) times more likely to be near dripping hanging pots. This is statistically significant at the \(.05\) level because the \(95\%\) confidence interval does not include \(1\) and the \(p\) value is less than \(.05\).

How many times more or less likely were those with L.longbeachae infection to water down potting mix before use, compared to those without infection? Is this statistically significant at the \(.05\) level?

ANSWER: Those with the infection were \(0.14\) times as likely to water down potting mix before use. This is often expressed as \(86\%\) less likely (since \(1 - 0.14 = 0.86\)). This is not statistically significant at the \(.05\) level because the \(95\%\) confidence interval includes \(1\) and the \(p\) value is greater than \(.05\).

Further example: interpreting one-way ANOVA

As another example of interpreting statistical and practical significance, consider Tables 4 and 5. These show the results of two one-way ANOVAs, which test for differences in mean ‘school attachment scores’:

between students of different ages, and
between students of different socio-economic groups.

Table 4
Descriptive statistics and one-way ANOVA statistics for a study investigating differences in school attachment scores between age groups

Table 5
Descriptive statistics and one-way ANOVA statistics for a study investigating differences in school attachment scores between socio-economic groups

From “School Attachment and Video Game Addiction of Adolescents with Divorced vs. Married Parents,” by B. TAS, 2019, TOJET: The Turkish Online Journal of Educational Technology , 18 (2), pp. 99-100 (https://files.eric.ed.gov/fulltext/EJ1211160.pdf). Copyright 2019 by The Turkish Online Journal of Educational Technology.

The \(p\) value in each table shows whether there is a statistically significant difference between the group means. If \(p\) is less than or equal to the level of significance, there is a statistically significant difference between the means of at least two groups. If \(p\) is greater than the level of significance, there is no statistically significant difference between any of the group means.

If a one-way ANOVA is statistically significant, more information is needed to identify which groups are different from each other. This information is usually provided through planned comparisons or post hoc tests. These produce additional \(p\) values, which are interpreted in the same way.

The tables also include an \(F\) value, which is the test statistic for a one-way ANOVA. Other statistical tests also have test statistics (for example, a \(t\) value for a \(t\) test or a chi-square value for a chi-square test). The \(F\) value compares the variability between groups and the variability within each group, and is calculated as a ratio of these two types of variability.

For example, an \(F\) value of \(0.818\) for the ‘Age’ variable means that the variability in school attachment scores between the four age groups is \(0.818\) times the variability within the groups. This shows that the variability between groups is smaller than the variability within groups. In general, the larger the \(F\) value (especially when it is greater than \(1\)), the more likely it is that there is a difference between the group means. However, the \(p\) value must be used to decide whether this difference is statistically significant.

Lastly, the tables also report eta-squared (\(\eta^2\)), which is a measure of effect size. Eta-squared shows the percentage of variability in the outcome (school attachment scores) that is explained by the variable (for example, age or socio-economic group). For example, the \(\eta^2\) value of \(0.024\) means that \(2.4\%\) of the variability in school attachment scores is explained by age. Guidelines suggested by Cohen (1988) state that \(.01\) is a small effect, \(.059\) is a medium effect and \(.138\) is a large effect.

Use the information above to answer the following questions:

Is there a statistically significant difference between the mean school attachment scores for the four age groups at the \(.05\) level?

ANSWER: No, because the \(p\) value is greater than \(.05\) (\(p = .48\)).

How many times greater is the variability in school attachment scores between the socio-economic groups compared to within each group?

ANSWER: The variability between the groups is \(2.666\) times the variability within the groups.

What percentage of the variability in school attachment scores is explained by socio-economic group?

ANSWER: \(5.1\%\) of the variability in school attachment scores is explained by socio-economic group.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates, Inc.

Connor, B.A., Carman, J., Eckert, K., Tucker, G., Givney, R., & Cameron, S. (2007). Does using potting mix make you sick? Results from a Legionella longbeachae case-control study in South Australia. Epidemiology and Infection, 135 (1), 34-39. https://doi.org/10.1017/S095026880600656X

Power, R., Muhit, M., Heanoy, E., Karim, T., Badawi, N., Akhter, R., & Khandaker, G. (2019). Health-related quality of life and mental health of adolescents with cerebral palsy in rural Bangladesh. PLoS One, 14 (6), 1-17. https://doi.org/10.1371/journal.pone.0217675

TAS, B. (2019). School attachment and video game addiction of adolescents with divorced vs. married parents. TOJET : The Turkish Online Journal of Educational Technology , 18 (2), 93-106. https://files.eric.ed.gov/fulltext/EJ1211160.pdf

Visram, S., Smith, S., Connor, N., Greig, G., & Scorer, C. (2018). Examining associations between health, wellbeing and social capital: Findings from a survey developed and conducted using participatory action research. Journal of Public Mental Health, 17 (3), 122-134. https://doi.org/10.1108/JPMH-09-2017-0035 Descriptive statistics

Descriptive statistics are used to summarise and describe one or more variables in a sample of data. They do not draw conclusions about a larger population (this is covered in the Inferential statistics page). For example, statistics such as the mean (\(\bar{x}\)) and standard deviation (\(s\)) are often used to summarise and describe continuous variables.

Descriptive statistics can be used in different ways, including to summarise one variable at a time (univariate analysis), to analyse relationships between two variables (bivariate analysis) and to describe patterns across three or more variables (multivariate analysis, which is not covered here).

This page explains how to display data and how to use descriptive statistics to carry out univariate and bivariate analysis for both categorical and continuous data.

In brief, it covers the following:

How to display data for one categorical variable
How to use descriptive statistics to analyse data for one categorical variable
How to display data for one continuous variable
How to use descriptive statistics to analyse data for one continuous variable
How to compare means of a continuous variable between different groups
How to display data for two categorical variables
How to use descriptive statistics to analyse data for two categorical variables
How to display data for two continuous variables
How to use descriptive statistics to analyse data for two continuous variables

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Displaying data for one categorical variable

One way to display data for a single categorical variable is by using a table, specifically a frequency distribution table. This table lists the different categories for a variable, along with the frequency (how often each category appears in the data) and usually the percentage for each category. Sometimes, cumulative percentages are also included. These show the total percentage up to and including each row of the table.

For example, a frequency distribution table could show the number and percentage of people in each marital status category in a sample of \(80\) people:

Another way to display categorical data for a single variable is with a column graph or bar chart. These use rectangles of equal width that do not touch, with each rectangle representing a category. Column graphs display the rectangles vertically, while bar charts display them horizontally. In both cases, the height or length of each rectangle shows the size of the category, usually as a count or percentage, making it easy to compare categories. For example:

Categorical data for a single variable can also be displayed in a pie chart, which shows how the whole sample is divided into different categories:

Descriptive statistics for one categorical variable

Descriptive statistics used to analyse a single categorical variable include frequencies, percentages, fractions, and relative frequencies (which are obtained by dividing the frequency by the sample size).

For example, the data for the marital status variable from the previous section can be described using key values such as:

\(91.3\%\) of the sample are either single or married
\(7\) people in the sample identified their marital status as ‘other’
the relative frequency of people in the sample who are married is \(0.36\)
more than \(\frac{1}{2}\) of the sample are single.

If you would like to practise analysing data for a single categorical variable, have a go at the following activity.

H5P interactive activity

Displaying data for one continuous variable

Continuous data for a single variable can also be displayed in a frequency distribution table, but the data must first be grouped into bins (also known as class intervals). These bins should be chosen based on the data. In general, they should be the same size, and there should not be too many.

For example, the following table shows the frequency of ages in a sample of 80 people after they have been grouped into bins:

One of the most common graphs used to display continuous data is a histogram. Histograms have several key features:

the bars touch (unless a bin has zero frequency)
the bars are vertical
the vertical axis starts at zero and has no breaks
the area of each bar represents the frequency of the bin.

Different histograms can be created for the same variable depending on the bin (class interval) width. For example, the first histogram below uses bins of width one, while the second uses the same bins as the frequency distribution table above. The software you use will usually choose the bin width for you, so this is not something to worry about too much. However, if you create your own histogram, you can choose the bin width based on how much detail you want to show:

Another graph used to display continuous data is a box plot (also called a box-and-whisker plot). This shows how the data is distributed. For example, a box plot for the Age variable is shown below. Click on the plus signs to explore its key features, some of which are explained in more detail in the next section.

H5P interactive activity

Descriptive statistics for one continuous variable

Continuous data for a single variable is usually analysed using two types of descriptive statistics: measures of central tendency and measures of dispersion.

Measures of central tendency describe the average or typical value (for example, the mean, median, and mode).
Measures of dispersion describe how spread out the data is (for example, the range, interquartile range, variance, and standard deviation).

It is important to understand how to calculate and interpret these measures, and also when it is appropriate to use each one. This is explained in the following sections.

Measures of central tendency

The most common measure of central tendency is the mean (also called the arithmetic average). It is calculated by adding all values and dividing by the number of values. The sample mean is written as \(\bar{x}\), and for a sample of size \(n\) it is calculated using the following formula (note that \(\sum\) represents the sum of all the values):

\[\bar{x} = \frac{\sum x}{n}\]

For example, the mean of the following ages:

\[19, 25, 28, 28, 23, 15, 28, 22, 24, 21\]

is:

\[\begin{aligned} \bar{x} &= \frac{19+25+28+28+23+15+28+22+24+21}{10} \\\ &= 23.3 \end{aligned}\]

The mean is usually used for continuous data, but it may not be suitable if the data is skewed, contains outliers, or is censored (not fully observed). In these cases, the median is more appropriate. The median is the middle value of the data when it is arranged from smallest to largest. If there is an even number of values, it is the mean of the middle two values. For example, ordering the ages data gives:

\[15, 19, 21, 22, 23, 24, 25, 28, 28, 28\]

and the median is then the mean of the middle two:

\[\frac{(23+24)}{2} = 23.5\]

Finally, the mode is the most frequently occurring value (or values) in the data. If two or more values occur equally often, all of them are reported. The mode is not always near the centre of the data, so it is not always a measure of “central” location. However, it is useful when you want to know the most common value. For example, the mode of the ages data:

\[19, 25, 28, 28, 23, 15, 28, 22, 24, 21\]

is \(28\).

The mode can also be used, and is more commonly used, for categorical data.

If you would like to test your understanding of the mean, median, and mode, have a go at one or both of the following activities.

H5P interactive activity

Measures of dispersion

The simplest measure of dispersion is the range, which is the difference between the smallest and the largest values in a data set. For example, the range of the ages in the previous section is \(13\) (that is, \(28 - 15\)). While the range is easy to calculate, it is limited because it only uses two values.

Another way to describe spread is by using percentiles. Percentiles show the position of a value within an ordered data set. They are useful for comparing a value to the rest of the sample. For example, if a child is in the \(97\)th percentile for height, this means the child is taller than \(97\%\) of others in the sample.

Because the sample of ages is small, it is useful for demonstrating percentiles. You can explore some examples in the diagram below:

H5P interactive activity

A specific type of percentile is quartiles, which divide the data into four equal parts:

the \(25\)th percentile is the first (lower) quartile
the \(50\)th percentile is the median
the \(75\)th percentile is the third (upper) quartile.

The quartiles for the age data are shown below:

H5P interactive activity

The interquartile range (IQR) is the difference between the first and third quartiles. In this example, it is \(7\). The IQR is often used along with the median when the data is skewed or contains outliers.

In other situations, the mean and standard deviation are usually used together. The standard deviation describes how spread out the data is and takes into account all values in the sample. It is calculated by finding how far each value is from the mean, squaring these differences, adding them together, and dividing by one less than the sample size. This result is called the variance, and taking the square root of the variance gives the standard deviation, which is usually preferred because it is in the same units as the original data. The formula for the sample standard deviation (\(s\)) of a sample of size \(n\) is:

\[s =\sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}\]

To calculate the standard deviation for the age data:

\[19, 25, 28, 28, 23, 15, 28, 22, 24, 21\]

we first use the mean \(\bar{x} = 23.3\).

Next, subtract the mean from each value:

\[-4.3, 1.7, 4.7, 4.7, -0.3, -8.3, 4.7, -1.3, 0.7, -2.3\]

Then square each result:

\[18.49, 2.89, 22.09, 22.09, 0.09, 68.89, 22.09, 1.69, 0.49, 5.29\]

Add these values to get \(164.1\), divide by \(9 \) (that is, \(10 - 1\)) to get \(18.23\), and then take the square root to get a standard deviation of \(4.27\)

If you would like to practise these calculations and to practise deciding which measures to use, have a go at the following activities.

H5P interactive activity

Comparing means

Measures of central tendency and dispersion can also be used to compare a continuous variable across different categories of a categorical variable. This is often done by comparing the mean values of different groups.

For example, if the first five people in our sample are female and the second five people are male, we can compare the mean age of the two groups:

\[\begin{aligned} \textrm{Ages of females} &= 19, 25, 28, 28, 23 & \textrm{Mean age of females} = & 24.6 \\\ \textrm{Ages of males} &= 15, 28, 22, 24, 21 & \textrm{Mean age of males} = & 22 \end{aligned}\]

This tells us that the females in our sample are, on average, \(2.6\) years older than the males.

If you would like to practise comparing means, have a go at the following activity.

H5P interactive activity

Displaying data for two categorical variables

Cross-tabulations (also called contingency tables) are used to examine the relationship between two categorical variables in a sample. A cross-tabulation contains rows and columns that show the categories of the two variables being compared. Each cell in the table shows the frequency, or number of people, that fall into a particular combination of categories, and the table also includes subtotals for each row and column together with an overall total in the bottom-right corner.

For example, the following cross-tabulation displays data for two variables (gender and feelings towards statistics), for a fictional sample of 300 people:

Cross-tabulations often also include row percentages, column percentages and/or total percentages, depending on the purpose of the table. To learn more about these percentages, click the plus signs in the diagram below:

H5P interactive activity

Data for two categorical variables can also be displayed using a clustered bar chart or a stacked bar chart. A clustered bar chart is useful for comparing the categories of one variable across the different categories of the other variable, while a stacked bar chart is more useful for comparing the overall totals for the categories of one variable.

Descriptive statistics for two categorical variables

Data in cross-tabulations can be analysed using the frequencies and percentages shown previously. In addition, expected frequencies can also be used to analyse the association between the variables in the sample. These frequencies are calculated based on the assumption that there is no association between the variables in the sample.

The expected frequencies can then be compared to the actual frequencies to test for association. If the actual frequencies are similar to the expected frequencies, this suggests that there may be little or no association between the variables. If the actual frequencies differ substantially from the expected frequencies, this suggests that an association may exist between the variables.

For some examples of how expected frequencies are calculated, and how they can be interpreted, click on the plus signs in the diagram below:

H5P interactive activity

If you would like to practise interpreting cross-tabulations, have a go at one or both of the following activities.

H5P interactive activity

Displaying data for two continuous variables

The relationship (or lack of a relationship) between two continuous variables in a sample can be displayed using a scatterplot. This plots the values of the independent variable on the x-axis and the values of the dependent variable on the y-axis.

For example, the scatterplot below shows the ages and weights of a sample of 80 people. The data points do not closely follow a straight-line pattern, which suggests that there is a weak or no linear relationship between the two variables. It is important to note that two variables may still have a non-linear relationship, but non-linear relationships are not covered in this module.

When investigating linear relationships, a scatterplot can help you identify:

the strength of the relationship, by showing how closely the data points follow a straight-line pattern, and
the direction of the relationship, by showing whether the pattern of points slopes upwards (a positive relationship) or downwards (a negative relationship).

H5P interactive activity

Descriptive statistics for two continuous variables

Finally, the strength and direction of a linear relationship (or lack of a relationship) between two continuous variables can be measured using Pearson’s correlation coefficient (\(r\)).

The correlation coefficient is a number between \(-1\) and \(1\). Positive values indicate a positive linear relationship, while negative values indicate a negative linear relationship.

In general:

\(r = 0\) indicates no linear relationship between the variables.
Values of \(r\) between \(0\) and \(0.3\) (or between \(0\) and \(-0.3\)) indicate a weak positive (or negative) linear relationship.
Values of \(r\) between \(0.3\) and \(0.7\) (or between \(-0.3\) and \(-0.7\)) indicate a moderate positive (or negative) linear relationship.
Values of \(r\) between \(0.7\) and \(1\) (or between \(-0.7\) and -\(1\)) indicate a strong positive (or negative) linear relationship.
\(r = 1\) (or \(-1\)) indicates a perfect positive (or negative) linear relationship.

For example, suppose two variables contain data on age and scores on a memory test. If the correlation coefficient (\(r\)) is \(-.72\), this indicates a strong negative linear relationship between the variables.

If you would like to practise interpreting a correlation coefficient together with a scatterplot, have a go at the following activity.

H5P interactive activity
The normal distribution

Many of the statistical tests covered later in this module assume that continuous data follows, or approximately follows, a normal distribution, or that the sample size is large enough for the sampling distribution of the mean to be approximately normal. But what is a normal distribution? What is a sampling distribution? How large does a sample need to be? And how can you tell whether a continuous variable is approximately normally distributed?

This page answers these questions and provides important background knowledge for the inferential statistics topics covered in the following pages.

In brief, it covers the following:

The definition of the normal distribution
The definition of a sampling distribution
How to test for normality
Options for transforming variables if they are not normally distributed

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The normal distribution

The normal distribution is a special type of distribution that many naturally occurring continuous variables approximately follow. Because of this, many inferential statistical tests, known as parametric tests, are based on the properties of the normal distribution. These properties are:

The mean, median and mode are equal.
Fixed proportions of the data lie within certain numbers of standard deviations from the mean. In particular, approximately 68% of the data lies within one standard deviation of the mean, 95% lies within two standard deviations, and 99.7% lies within three standard deviations.

These properties give the normal distribution its characteristic bell-shaped curve, shown below. Note that the percentages displayed in the graph are given to two decimal places, while the percentages above are rounded values. Also note that the symbols \(\mu\) and \(\sigma\) represent the population mean and population standard deviation respectively.

If you would like to practise interpreting data from a normal distribution, have a go at the following activity.

H5P interactive activity

Sampling distributions

The descriptive statistics covered in the previous page of this module are used to analyse the data you have access to, which in most cases is a random sample of a larger population. It is important to remember though that any random sample is only one of many possible samples that could be selected from a population, and that sample statistics such as the sample mean will likely vary from sample to sample. For example, we could measure the heights of a random sample of 100 Curtin students and calculate the mean height, then we could select a different random sample of 100 Curtin students and calculate another mean height, which would likely be different, and so on. If we continued this process, we would obtain a mean for each possible sample of the same size.

All of these sample statistics (such as the mean heights in our example) form a distribution of their own, known as a sampling distribution. If we had access to all of these values then we could display the sampling distribution using a histogram, just as we would for any other set of data. In practice, however, this is not possible because it would require collecting data from every possible sample. Instead, inferential statistics use data from a single random sample to estimate properties of the sampling distribution.

Sampling distributions are therefore an important concept in statistics. In particular, it is useful to understand the following properties of the sampling distribution of the mean:

As the sample size increases, the shape of the sampling distribution of the mean becomes increasingly similar to a normal distribution. This important result is known as the Central Limit Theorem. Note that a sample size of 30 or more is often considered large enough for the sampling distribution to be approximately normal, and therefore to fulfil any normality requirements, although a larger sample size may be needed if the distribution is highly skewed.
The mean of the sampling distribution is equal to the population mean.
The standard deviation of the sampling distribution is equal to the population standard deviation divided by the square root of the sample size. This value is known as the standard error of the mean.

Testing for normality

If the inferential statistical test you want to use requires a normally distributed continuous variable (or variables) and your sample size is not sufficiently large (typically 30 or above), then you will need to test for normality. Unfortunately, there is no single test that can determine with certainty whether data are normally distributed. Instead, several statistics and graphs can be used to assess whether the data are approximately normal. Data are rarely perfectly normally distributed, and some deviation from a normal distribution is often acceptable, so examining multiple indicators provides a more complete understanding of the data and helps you make a more informed judgement about normality.

The following statistics and graphs can be used to assess normality:

Mean, median and mode
For a perfectly normal distribution, the mean, median and mode are all equal. Comparing these values is therefore a simple way to begin assessing normality. In practice, small differences are common, particularly for the mode, and this is usually less concerning than differences between the mean and median.
Skewness
Skewness measures the symmetry of a distribution. A skewness value of \(0\) indicates a perfectly symmetrical distribution, while negative values indicate negative skew (a longer tail on the left) and positive values indicate positive skew (a longer tail on the right). If the data is approximately normally distributed then the skewness should be close to \(0\), and as a general guide, values between \(-1\) and \(1\) are usually considered acceptable. You can also calculate a z-score for skewness by dividing it by its standard error and this z-score should fall between \(-1.96\) and \(1.96\)
Kurtosis
Kurtosis measures the shape of a distribution’s tails. A normal distribution has a kurtosis of \(3\), although many researchers and software programs (such as SPSS) report excess kurtosis instead. Excess kurtosis is calculated by subtracting \(3\) from the kurtosis, giving the normal distribution a value of \(0\). Using excess kurtosis, positive values indicate heavier tails and negative values indicate lighter tails, but again a value between \(−1\) and \(1\) is generally considered acceptable. You can also calculate a z-score for excess kurtosis by dividing it by its standard error. This z-score should fall between \(-1.96\) and \(1.96\)
Normality test (e.g. Shapiro-Wilk)
The Shapiro-Wilk test is a statistical test of normality, which tests the null hypothesis that the data comes from a normally distributed population. Another commonly used normality test is the Kolmogorov-Smirnov test. A significance (\(p\)) value greater than \(0.05\) suggests that there is insufficient evidence to conclude that the data is not normally distributed, which provides support for assuming normality (more information about hypothesis testing is provided in the Inferential statistics page of this module). For larger samples, normality tests can become very sensitive and may identify minor departures from normality that are not practically important. For this reason, they are usually interpreted alongside the other statistics and graphs described on this page.
Histogram
If the data is approximately normally distributed the histogram should be roughly symmetrical and centred around the mean (Figure 1). If there is a longer tail on the left the distribution is negatively skewed (Figure 2), and if there is a longer tail on the right the distribution is positively skewed (Figure 3).

Stem and leaf plot
A stem and leaf plot displays the frequency of values in a data set using ‘stems’ and ‘leaves’. For example, Figure 4 below shows that there is one value of \(63\), two values of \(65\), six values of either \(66\) or \(67\), and so on. A stem and leaf plot can be interpreted in much the same way as a histogram, only rotated on its side.

Normal Q-Q plot
If the data is approximately normally distributed, the points on a normal Q-Q plot will lie close to the straight diagonal line (Figure 7). If the data is not normally distributed, the points will tend to deviate from this line (Figures 8 and 9). Another version of this graph, called a detrended Q-Q plot, is sometimes also used. In a detrended Q-Q plot, there should be a roughly equal number of points above and below the horizontal line, with no obvious pattern.

Boxplot

If the data is approximately normally distributed, the median should be positioned near the centre of the box and the whiskers should be of similar length. Ideally, there should be few or no outliers (Figure 10). Alternatively, the boxplot may show negative skew (Figure 11) or positive skew (Figure 12).

After examining these indicators of normality, you should draw an overall conclusion based on the evidence as a whole. For example, you might conclude that the data is approximately normally distributed, or that it shows positive or negative skew.

For a worked example of assessing normality, you make like to view the Introduction to SPSS or Introduction to Stata modules. You can also practise assessing whether data approximates a normal distribution by completing the following activity.

H5P interactive activity

Transforming variables

If the tests for normality indicate that a variable is not approximately normally distributed, and the sample size is not sufficiently large, it may be possible to transform the variable so that its distribution more closely resembles a normal distribution. For example, if the data is negatively skewed, taking the square of the values may help reduce the skewness. If the data is positively skewed, applying a natural logarithm (\(ln\)), square root or reciprocal transformation may be useful.

After the data has been transformed, it should be tested for normality again to see whether the transformation has improved the distribution. If the transformed data is now approximately normally distributed, any further statistical analysis should be carried out using the transformed data. If the distribution is still not approximately normal, a non-parametric test should be used instead.

Inferential statistics

So far, we have used descriptive statistics to describe a sample of data by calculating sample statistics such as \(\bar{x}\) (the sample mean) and \(s\) (the sample standard deviation).

However, research is often conducted with the aim of using these sample statistics to estimate and compare true values of a population. These true values are called population parameters and are represented by Greek letters such as \(\mu\) (the population mean) and \(\sigma\) (the population standard deviation). Inferential statistics allow us to make statements about unknown population parameters using sample statistics from a random sample of the population. There are two main types of inferential statistics, estimation and hypothesis testing, and both are covered on this page.

In brief, it covers the following:

Using point estimates and confidence intervals for estimation
How to write hypotheses to test for statistical significance
How to interpret \(p\) values to assess statistical significance
How to interpret practical significance
The difference between parametric and non-parametric tests

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First, though, you may like to test your understanding of inferential statistics by choosing the best answer to the following question:

H5P interactive activity

Estimation

When sample statistics are used to estimate population parameters, this is known as estimation. Estimation can be done using either a single value, called a point estimate, or a range of values, called a confidence interval. These two types of estimation are explained in the following sections.

Point estimates

Suppose a researcher is interested in cholesterol levels in a population. If they select a random sample from that population, they can use statistics calculated from the sample to estimate the corresponding population parameters. In this case:

\(\bar{x}\) (the sample mean cholesterol level) is a point estimate of \(\mu\) (the population mean cholesterol level)
\(s\) (the sample standard deviation for cholesterol level) is a point estimate of \(\sigma\) (the population standard deviation).

While point estimates are useful, it is often better to estimate a population parameter using a range of values. This helps account for the fact that sample statistics are unlikely to be exactly the same as the corresponding population parameters. This is where confidence intervals come in.

Confidence intervals

A confidence interval gives a range of values that can be used to estimate a population parameter. It also includes a confidence coefficient, which shows how certain we are that the interval contains the population parameter. This is usually set at \(95\%\). For example, a \(95\%\) confidence interval for the population mean blood glucose level of (\(4\) mmol/L, \(6\) mmol/L) indicates that we are \(95\%\) certain that the population mean blood glucose level lies between \(4\) mmol/L and \(6\) mmol/L.

For another way of interpreting confidence intervals, recall the concept of sampling distributions and imagine calculating a confidence interval for every possible sample of the same size. A \(95\%\) confidence interval means that you would expect 95 out of every 100 of these intervals to contain the population mean.

While confidence intervals can be calculated for many different statistics, a common example is a confidence interval for the population mean. This can be calculated based on the sampling distribution of the sample mean using the following formula:

\[\textrm{confidence interval for population mean} = \textrm{sample mean} \pm \textrm{a multiple of the standard error of the mean}\]

There are two important things to note about this formula. First, the multiple depends on the confidence coefficient. For example, a \(95\%\) confidence interval requires a multiple of \(1.96\). This relates to the normal distribution, where approximately \(95\%\) of the area under the normal curve lies within 1.96 standard deviations of the mean. Second, the standard error of the mean for a sample of size \(n\) with standard deviation \(s\) is equal to \(\frac{s}{\sqrt{n}}\)

Therefore, the formula for a \(95\%\) confidence interval for a population mean, based on a sample of size \(n\) with mean \(\bar{x}\) and standard deviation \(s\), is:

\[95\% \textrm{ confidence interval for population mean} = \bar{x} \pm 1.96 \times \frac{s}{\sqrt{n}}\]

Before calculating a confidence interval for a population mean, you should make sure the following assumptions are met:

Assumption 1: The sample is a random sample that is representative of the population.

Assumption 2: The observations are independent. This means that measurements for one participant do not affect the measurements of another participant.

Assumption 3: The variable is normally distributed, or the sample size is large enough that the sampling distribution of the mean approximates a normal distribution.

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Some important points about confidence intervals are:

The confidence interval for the population mean is centred on the sample mean.
The width of a confidence interval increases as the level of confidence increases.
Confidence intervals become narrower as the sample size increases.
Variables with larger standard deviations produce wider confidence intervals because greater variation leads to less precise estimates.

Finally, remember that a population parameter is fixed. It is the sample statistic and confidence interval that change from sample to sample. Once a confidence interval has been calculated, the population parameter is either inside or outside the interval. We can only state the level of certainty that we believe the interval contains the population parameter.

If you would like to practise calculating and interpreting confidence intervals, have a go at one or both of the following activities.

H5P interactive activity

Hypothesis testing

Hypothesis testing involves developing hypotheses about a population based on information observed in a sample. These hypotheses can then be tested to determine whether differences or relationships observed in the sample are statistically significant for the population, or whether they may have occurred due to random chance in the sample.

To do this, two complementary but contradictory hypotheses need to be developed. These are called the null hypothesis and the alternative hypothesis (or research hypothesis).

The hypotheses are formulated differently depending on whether you are conducting a two-tailed or one-tailed hypothesis test. A two-tailed test tests for an effect in either direction, while a one-tailed test only tests for an effect in one direction.

The choice between a one-tailed and two-tailed test will depend on your methodology. However, two-tailed tests are typically used unless there is a reason to use a one-tailed test, such as when the effect is only possible in one direction. For this reason, this module will focus on two-tailed hypothesis tests. These tests use the following hypotheses:

Null hypothesis (\(\textrm{H}_\textrm{0}\)): This hypothesis states that there is no difference or relationship between variables in a population. For example, no significant difference between two population means, no significant association between two categorical variables, no significant correlation between two continuous variables or no significant difference from the normal distribution (as in the Shapiro-Wilk test).

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Alternative hypothesis (\(\textrm{H}_\textrm{A}\)): Also known as the research hypothesis, this hypothesis states the opposite of the null hypothesis. It states that there is a difference or relationship between variables in a population. For example, that there is a significant difference between two population means, a significant association between two categorical variables, a significant correlation between two continuous variables or a significant difference from the normal distribution (as in the Shapiro-Wilk test).

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Both hypotheses can be written using words or symbols, and there are often several ways to write them. For example, suppose we want to test whether there is a significant change in the mean blood pressure of a population of patients after they have taken a new medication. We could write the null and alternative hypotheses in several different ways:

\(\textrm{H}_\textrm{0}\): there is no significant difference in mean blood pressure before and after the medication
\(\textrm{H}_\textrm{0}: \mu_{\textrm{bp before}} = \mu_{\textrm{bp after}}\)
\(\textrm{H}_\textrm{0}: \mu_{\textrm{bp before}} - \mu_{\textrm{bp after}} = 0\)

\(\textrm{H}_\textrm{A}\): there is a significant difference in mean blood pressure before and after the medication
\(\textrm{H}_\textrm{A}: \mu_{\textrm{bp before}} \neq \mu_{\textrm{bp after}}\)
\(\textrm{H}_\textrm{A}: \mu_{\textrm{bp before}} - \mu_{\textrm{bp after}} \neq 0\)

If you would like to practise writing hypotheses, have a go at formulating null and alternative hypotheses for the following activity.

You wish to test whether there is a significant difference between the mean before and after heart rates for the population of people who complete a fun run.

\(\textrm{H}_\textrm{0}\): There is no significant difference in mean heart rate before and after the fun run (\(\mu_{\textrm{hr before}} = \mu_{\textrm{hr after}}\), or \(\mu_{\textrm{hr before}} - \mu_{\textrm{hr after}} = 0\))

\(\textrm{H}_\textrm{A}\): There is a significant difference in mean heart rate before and after the fun run (\(\mu_{\textrm{hr before}} \neq \mu_{\textrm{hr after}}\), or \(\mu_{\textrm{hr before}} - \mu_{\textrm{hr after}} \neq 0\))

You wish to test whether there is a significant difference in mean overall grades between the populations of male and female students enrolled in a particular unit.

\(\textrm{H}_\textrm{0}\): There is no significant difference in mean grades for male and female students (\(\mu_{\textrm{male}} = \mu_{\textrm{female}}\), or \(\mu_{\textrm{male}} - \mu_{\textrm{female}} = 0\))

\(\textrm{H}_\textrm{A}\): There is a significant difference in mean grades for male and female students (\(\mu_{\textrm{male}} \neq \mu_{\textrm{female}}\), or \(\mu_{\textrm{male}} - \mu_{\textrm{female}} \neq 0\))

You wish to test whether there is a significant correlation between hours of study and exam marks in a population of university students.

\(\textrm{H}_\textrm{0}\): There is no significant correlation between hours of study and exam marks (\(r = 0\))

\(\textrm{H}_\textrm{A}\): There is a significant correlation between hours of study and exam marks (\(r \neq 0\))

Once the hypotheses have been formulated, they can be tested to evaluate statistical significance, as explained in the following section. It is also important to consider practical significance, which is explained in the section after that.

Statistical significance

To evaluate statistical significance, you need to conduct an appropriate statistical test. Common examples include one sample, paired samples and independent samples \(t\) tests, one-way ANOVA, the chi-square test of independence and Pearson’s correlation. All of these tests are covered on later pages of this module. You will typically conduct these tests using statistical software or a programming language, such as SPSS, Stata, SAS, R or Python. However, they can also be calculated manually, although this is not covered here.

Whichever test you use, it will calculate a test statistic. This compares the value of the sample statistic (for example, the sample mean change in blood pressure in our blood pressure example) with the value specified by the null hypothesis for the population statistic (for example, a mean change in blood pressure of zero). The name of the test statistic, and how it is calculated, will depend on the test you are conducting. For example, a \(t\) test calculates a \(t\) value, a one-way ANOVA calculates an \(F\) value and a chi-square test of independence calculates a chi-square value. In each case, a large test statistic indicates a large difference between the hypothesised value and the sample statistic.

The test will also calculate a \(p\) value (or probability value). In some cases, two \(p\) values may be provided: a one-sided \(p\) value and a two-sided \(p\) value. You should use the two-sided \(p\) value unless you have previously decided to conduct a one-sided hypothesis test. The \(p\) value gives the probability of obtaining the observed test statistic, or one more extreme, if the null hypothesis is true. A small \(p\) value means that a result this extreme would be unlikely if the null hypothesis were true. It is the \(p\) value, rather than the test statistic itself, that is interpreted when deciding whether to reject the null hypothesis.

But how small does the \(p\) value need to be? To decide when to reject the null hypothesis, we need to choose a level of significance. This tells us how small the \(p\) value needs to be before we reject the null hypothesis. The level of significance is denoted by \(\alpha\) and is typically \(.05\) (\(5\%\)), although other values can also be used. Note that if:

\(p\) value \(\leqslant \alpha\) : A result this extreme, or more extreme, would have a probability of less than or equal to \(\alpha\%\) if the null hypothesis were true. Therefore, we reject the null hypothesis in favour of the alternative hypothesis. This means that the difference or relationship we have hypothesised about is statistically significant.
\(p\) value \( > \alpha\): A result this extreme, or more extreme, would have a probability greater than \(\alpha\) if the null hypothesis were true. Therefore, we cannot reject the null hypothesis. This means that the difference or relationship we have hypothesised about is not statistically significant.

If you would like to practise interpreting \(p\) values, have a go at the following activity.

H5P interactive activity

At this point, it is important to note that confidence intervals can also be used to determine whether a difference or relationship is statistically significant. For example, using the sample from our blood pressure example, we can calculate a \(95\%\) confidence interval. This gives a range of plausible values for the difference in mean blood pressure in the population. If this confidence interval does not contain \(0\), then \(0\) is not considered a plausible value for the population difference. This indicates that the difference is statistically significant at the \(5\%\) level.

Confidence intervals are useful because they provide more information than statistical significance alone. They also provide information about the magnitude and direction of a difference (or relationship).

If you would like to test your understanding of this concept, have a go at this activity.

H5P interactive activity

Error types

Because hypothesis testing uses incomplete information from a sample to draw conclusions about a complete population, there is always a possibility of making an error when deciding whether to reject the null hypothesis. This can happen even if our calculations are correct. There are two possible types of error:

Type I error: This occurs when we reject a null hypothesis that is actually correct. The probability of making a Type I error is equal to the level of significance \(\alpha\). This is why we generally select a low value for \(\alpha\), such as \(0.05\).

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Type II error: This occurs when we do not reject a null hypothesis that is actually incorrect. The probability of making a Type II error is denoted by \(\beta\), and it is usually desirable for this to be \(0.2\) or below.

To minimise the risk of a Type II error, a power analysis is often used to determine an appropriate sample size. The power of a statistical test is the probability that the test will detect an effect if one actually exists. Power is therefore the opposite of the Type II error rate and can be expressed as \(1-\beta\). Therefore, to keep the Type II error rate \(\leq 0.2\), the power needs to be \(\geq 0.8\)

The power of a test depends on three factors:

the effect size (how large the effect is; more on this shortly)
how strict we are when deciding whether the effect is significant (i.e. our \(\alpha\) level)
the sample size.

You can use this information to calculate the power of a test using software, for example SPSS (Version 27 or above). Alternatively, and ideally, you can use software to determine an appropriate sample size needed to achieve a power of \(\geq 0.8\)

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If you would like to test your understanding of the different error types, have a go at the following activity.

H5P interactive activity

Practical significance

Statistical significance is influenced by sample size. This means that with a very large sample, very small differences may be statistically significant. With a very small sample, even large differences may not be statistically significant. For this reason, it is often useful to consider practical significance as well. This can be assessed by calculating an effect size.

The effect size provides information about whether a difference or relationship is meaningful in a practical sense (i.e. in real life). Effect size is calculated differently for different statistical tests. Details on how to calculate effect size for each test are provided on later pages of this module.

Parametric and non-parametric tests

Different inferential statistical tests are used depending on the type of hypothesis being tested and the data being analysed. The following pages explain some of the most common tests. First, however, it is important to understand two main types of statistical tests:

Parametric tests: These make assumptions about the distribution of the data. They usually involve at least one continuous variable and commonly make assumptions relating to the normal distribution.

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Non-parametric tests: These do not require continuous data to follow a particular distribution, such as the normal distribution. They can also be used with some types of data that are not continuous, such as ordinal data. Many non-parametric tests use the ranks or order of the data rather than the original values.

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As a general rule, if the assumptions of a parametric test are met, the parametric test should be used because it generally has greater statistical power. If these assumptions are not met, or if the type of data is not suitable for a parametric test, a non-parametric test should be used instead.

The following pages explain five of the most commonly used parametric tests, with reference to their non-parametric alternatives, as well as one commonly used non-parametric test.

Comparing means

One reason for conducting a hypothesis test is to determine whether there is a statistically significant difference between means. You might compare the mean of a single sample with a constant value, or compare the means of two or more independent or related samples. Different statistical tests are used depending on the type of comparison being made. This page explains some of the most common tests, along with non-parametric alternatives that compare mean ranks for ordinal data.

In brief, it covers the following:

When to conduct a one sample \(t\) test, and how to interpret the results
When to conduct a paired samples \(t\) test, and how to interpret the results
When to conduct an independent samples \(t\) test, and how to interpret the results
When to conduct a one-way ANOVA, and how to interpret the results

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One sample \(t\) test

A one sample \(t\) test is used to test whether the mean of a continuous variable in a population is significantly different from a particular value. This value, which is often obtained from previous research, is called the test value. For example, suppose you have a sample of student final marks and want to test whether the students come from a population with a mean final mark equal to the previous year’s mean of \(70\). In this case, the hypotheses for a two-tailed hypothesis test would be:

\(\textrm{H}_\textrm{0}\): The sample comes from a population with a mean final mark of \(70\) (\(\mu_{\textrm{final mark}} = 70\))
\(\textrm{H}_\textrm{A}\): The sample does not come from a population with a mean final mark of \(70\) (\(\mu_{\textrm{final mark}} \neq 70\))

Before conducting a one sample \(t\) test, you need to check that the following assumptions are met:

Assumption 1: The sample is a random sample that is representative of the population.

Assumption 2: The observations are independent. This means that the measurement for one subject does not affect the measurements for any other subject.

Assumption 3: The variable is normally distributed, or the sample size is large enough that the sampling distribution of the mean approximates a normal distribution.

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If the assumptions for the one sample \(t\) test are not met, the one sample Wilcoxon signed-rank test may be an appropriate non-parametric alternative. This test uses the ranks of the data.

If the assumptions for the one sample \(t\) test are met, you can conduct the test using statistical software, such as SPSS. The results should include the following statistics:

When analysing the results, first look at the descriptive statistics to get an idea of what is happening in the sample. In this example, the sample mean is \(73.40\) as shown in the first table. The test value is \(70\), giving a difference of \(3.40\). This is shown as the mean difference in the second table.

To determine whether this difference is statistically significant, we need to look at the \(p\) value and the confidence interval for the difference. The \(p\) value is shown as ‘Sig. (2-tailed)’ in the second table because we have conducted a two-tailed hypothesis test. If we are using a level of significance of \(.05\):

If \(p \leqslant .05\), we reject \(\textrm{H}_\textrm{0}\) and conclude that the population mean is significantly different from the test value.
If \(p > .05\) we do not reject \(\textrm{H}_\textrm{0}\) and conclude that the population mean is not significantly different from the test value.

In this case, our \(p\) value of \(.025\) indicates that the difference is statistically significant.

This is also shown by the confidence interval of (\(.45\), \(6.36\)) for the difference between the population mean and our test value of \(70\). Because this confidence interval does not contain zero, the difference is statistically significant. In fact, we are \(95\%\) confident that the true population mean is between \(.45\) and \(6.36\) points higher than our test value.

Note that the test statistic (\(t\)) and degrees of freedom (\(df\)) should generally be reported as part of your results. However, you do not need to interpret these when deciding whether the difference is statistically significant.

If you would like to practise interpreting the results of a one sample \(t\) test for statistical significance, have a go at one or both of the following activities.

H5P interactive activity

To evaluate practical significance when using a one sample \(t\) test, Cohen’s \(d\) is often used to measure the effect size. This tells us how many standard deviations the sample mean is from the test value. Cohen’s \(d\) can also usually be obtained using statistical software. For example, the Cohen’s \(d\) for our final mark data is included in the following output:

This tells us that our sample mean is \(0.37\) standard deviations from the test value, indicating a small to medium effect. As a general guide, a Cohen’s \(d\) with a magnitude of \(0.2\) is considered small, \(0.5\) medium and \(0.8\) or above large.

Paired samples \(t\) test

A paired samples \(t\) test is used to test whether there is a significant difference between the means of a continuous variable measured for two related groups or under two related conditions. For example, suppose you have a sample of individuals whose heart rate is recorded twice, before and after exercise. You could use a paired samples \(t\) test to determine whether there is a significant difference between the mean heart rate before and after exercise. In this case, the hypotheses for a two-tailed hypothesis test would be:

\(\textrm{H}_\textrm{0}\): There is no significant difference in mean heart rate before and after exercise
(\(\mu_{\textrm{HR before}} = \mu_{\textrm{HR after}}\), or \(\mu_{\textrm{HR before}} - \mu_{\textrm{HR after}} = 0\))

\(\textrm{H}_\textrm{A}\): There is a significant difference in mean heart rate before and after exercise
(\(\mu_{\textrm{HR before}} \neq \mu_{\textrm{HR after}}\), or \(\mu_{\textrm{HR before}} - \mu_{\textrm{HR after}} \neq 0\))

Before conducting a paired samples \(t\) test, you need to check that the following assumptions are met:

Assumption 1: The sample is a random sample that is representative of the population.

Assumption 2: The observations are independent. This means that the measurements for one subject do not affect the measurements for any other subject.

Assumption 3: The differences between each pair of measurements are normally distributed, or the sample size is large enough that the sampling distribution of the mean difference approximates a normal distribution.

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If the assumptions for the paired samples \(t\) test are not met, the Wilcoxon signed-rank test may be an appropriate non-parametric alternative. This test uses the ranks of the data.

If the assumptions for the paired samples \(t\) test are met, you can conduct the test using statistical software, such as SPSS. The results should include the following statistics:

When analysing the results, first look at the descriptive statistics to get an idea of what is happening in the sample. In this example, the difference between the before and after sample means is \(-48.70\). This value can be calculated from the means in the first table and is also shown in the second table. The negative value simply indicates that the mean heart rate after exercise is higher than the mean heart rate before exercise.

If \(p \leqslant .05\) we reject \(\textrm{H}_\textrm{0}\) and conclude that the population means for the two related groups are significantly different.
If \(p > .05\) we do not reject \(\textrm{H}_\textrm{0}\) and conclude that the population means for the two related groups are not significantly different.

In this case, our \(p\) value of \(< .001\) indicates that the difference between the means is statistically significant.

his is also shown by the confidence interval of (\(-52.65\), \(-44.75\)) for the difference between the means. Because this confidence interval does not contain zero, the difference is statistically significant. In fact, we are \(95\%\) confident that the population mean heart rate after exercise is between \(44.75\) and \(52.65\) bpm higher than the population mean heart rate before exercise.

If you would like to practise interpreting the results of a paired samples \(t\) test for statistical significance, have a go at one or both of the following activities.

H5P interactive activity

To evaluate practical significance when using a paired samples \(t\) test, Cohen’s \(d\) can again be used to measure the effect size. In this case, it measures the size of the mean difference in standard deviation units and can usually be obtained using statistical software. For example, the Cohen’s \(d\) for our heart rate data is included in the following output:

This tells us that the difference between the two sample means is \(3.95\) standard deviations, indicating a very large effect. As a general guide, a Cohen’s \(d\) with a magnitude of \(0.2\) is considered small, \(0.5\) medium and \(0.8\) or above large.

Independent samples \(t\) test

An independent samples \(t\) test is used to test whether there is a significant difference between the means of a continuous variable for two independent groups. For example, suppose you collect data on the number of hours males and females spend watching TV each week. You could use an independent samples \(t\) test to determine whether there is a significant difference between the mean hours for the two groups. In this case, the hypotheses for a two-tailed hypothesis test would be:

\(\textrm{H}_\textrm{0}\): There is no significant difference in mean TV hours per week for males and females
(\(\mu_{\textrm{males}} = \mu_{\textrm{females}}\), or \(\mu_{\textrm{males}} - \mu_{\textrm{females}} = 0\))

\(\textrm{H}_\textrm{A}\): There is a significant difference in mean TV hours per week for males and females
(\(\mu_{\textrm{males}} \neq \mu_{\textrm{females}}\), or \(\mu_{\textrm{males}} - \mu_{\textrm{females}} \neq 0\))

Before conducting an independent samples \(t\) test, you need to check that the following assumptions are met:

Assumption 1: The sample is a random sample that is representative of the population.

Assumption 2: The observations are independent. This means that the measurement for one subject does not affect the measurements for any other subject.

Assumption 3: The variable is normally distributed within both groups, or the sample sizes are large enough that the sampling distributions of the means approximate a normal distribution.

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If the assumptions of the independent samples \(t\) test are not met, the Mann-Whitney U test may be an appropriate non-parametric alternative. This test uses the ranks of the data.

If the assumptions for the independent samples \(t\) test are met, you can conduct the test using statistical software, such as SPSS. The results should include the following statistics:

When analysing the results, first look at the descriptive statistics to get an idea of what is happening in the sample. In this example, the difference between the sample means for males and females is \(1.45\). This value can be calculated from the means in the first table and is also shown in the second table.

Next, you need to determine which row of the second table to use. The top row is used when the variances of the two groups can be assumed to be equal. The bottom row is used when the variances cannot be assumed to be equal. Each row provides a one-sided and two-sided \(p\) value. Because we are conducting a two-tailed hypothesis test, we will use the two-sided \(p\) value.

To determine which row to use, first look at the \(p\) value for Levene’s Test for Equality of Variances, shown in the ‘Sig.’ column. The null hypothesis for this test is that the variances are equal. Therefore:

If \(p \leqslant .05\) we reject \(\textrm{H}_\textrm{0}\) and use the Equal variances not assumed row.
If \(p > .05\) we do not reject \(\textrm{H}_\textrm{0}\) and use the Equal variances assumed row.

In this example, the \(p\) value for Levene’s test is \(.257\). This means that we do not reject the null hypothesis that the variances are equal. We therefore use the two-sided \(p\) value and confidence interval in the Equal variances assumed row. The interpretation of this \(p\) value is:

If \(p \leqslant .05\) we reject \(\textrm{H}_\textrm{0}\) and conclude that the population means of the two groups are significantly different.
If \(p > .05\) we do not reject \(\textrm{H}_\textrm{0}\) and conclude that the population means of the two groups are not significantly different.

In this case, our \(p\) value of \(.682\) indicates that the difference between the means is not statistically significant.

This is also shown by the confidence interval of \(-5.64, 8.54\) for the difference between the means. Because this confidence interval contains zero, the difference is not statistically significant. In fact, we are \(95\%\) confident that the difference in mean hours spent watching TV each week for males and females is between \(-5.64 and 8.54\) hours.

If you would like to practise interpreting the results of an independent samples \(t\) test for statistical significance, have a go at one or both of the following activities.

H5P interactive activity

To evaluate practical significance when using an independent samples \(t\) test, Cohen’s \(d\) can again be used to measure the effect size. In this case, it measures the difference between the two means in standard deviation units and can usually be obtained using statistical software. For example, the Cohen’s \(d\) for our TV hours data is included in the following output:

This tells us that our two sample means are separated by \(0.13\) standard deviations, indicating a very small effect. As a general guide, a Cohen’s \(d\) with a magnitude of \(0.2\) is considered small, \(0.5\) medium and \(0.8\) or above large.

One-way ANOVA

One-way ANOVA is similar to the independent samples \(t\) test but is used to compare the means of three or more independent groups. One-way ANOVA is the only type of ANOVA covered here. However, it is part of a family of ANOVA tests that also includes:

Factorial ANOVA: used to test for differences in means between groups when there are two or more independent variables.
One-way repeated measures ANOVA: used to test for differences in means between three or more related samples.
ANCOVA (one-way analysis of covariance): used to test for differences between two or more independent samples after controlling for the effects of a third variable (covariate).
MANOVA (multivariate analysis of variance): used to test for differences between groups on multiple dependent variables at the same time.

Returning to the one-way ANOVA, suppose you have a data set containing the final marks of a sample of students studying one of three different units. You could use a one-way ANOVA to determine whether there is a significant difference between the mean marks for any of the units. In this case, the hypotheses would be:

\(\textrm{H}_\textrm{0}\): There is no significant difference in the mean final mark for any of the three units.
(\(\mu_{\textrm{unit 1}} = \mu_{\textrm{unit 2}} = \mu_{\textrm{unit 3}}\))

\(\textrm{H}_\textrm{A}\): The mean final mark of at least one of the units is significantly different to the others.

Before conducting a one-way ANOVA, you need to check that the following assumptions are met:

Assumption 1: The sample is a random sample that is representative of the population.

Assumption 2: The observations are independent. This means that the measurement for one subject does not affect the measurements for any other subject.

Assumption 3: The variable is normally distributed within each group, or the sample sizes are large enough that the sampling distributions of the means approximate a normal distribution.

Assumption 4: The populations being compared have equal variances.

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If the assumptions of the one-way ANOVA are not met, another test may be more appropriate. If the normality assumption is not met, the Kruskal-Wallis test may be an appropriate non-parametric alternative. This test uses the ranks of the data. If the equal variances assumption is not met, the Brown-Forsythe or Welch test can be used instead.

If the assumptions for the one-way ANOVA are met, you can conduct the test using statistical software, such as SPSS. The results should include the following statistics:

When analysing the results, first look at the descriptive statistics to get an idea of what is happening in the sample. In this example, the sample means are \(73.22\) for Unit 1, \(74.85\) for Unit 2 and \(72.15\) for Unit 3. To determine whether there is a statistically significant difference between these means, we need to look at the \(p\) value, which is shown as ‘Sig.’ in the second table. If we are using a level of significance of \(.05\):

If \(p \leqslant .05\) we reject \(\textrm{H}_\textrm{0}\) and conclude that the mean of at least one of the groups is significantly different from the others.
If \(p > .05\) we do not reject \(\textrm{H}_\textrm{0}\) and conclude that there are no statistically significant differences between the group means.

In this case, our \(p\) value of \(.764\) indicates that there are no statistically significant differences in mean final marks between the units.

Note that the test statistic (\(F\)) and degrees of freedom (\(df\)) should generally be reported as part of your results. However, you do not need to interpret these when deciding whether the difference is statistically significant.

If a one-way ANOVA shows that at least one of the means is significantly different, you will need to investigate further to determine which groups are different. This can be done using post hoc tests, such as Tukey’s HSD.

For now though, if you would like to practise interpreting the results of a one-way ANOVA for statistical significance, have a go at one or both of the following activities.

H5P interactive activity

To evaluate practical significance when using a one-way ANOVA, eta-squared can be used to measure the effect size. This indicates how much of the variability in the dependent variable can be attributed to the independent variable, expressed as a percentage. It can also usually be obtained using statistical software. For example, the eta-squared for our final mark data is included in the following output:

The eta-squared value of \(0.014\) tells us that \(1.4\%\) of the variability in final marks can be attributed to the unit of study. This indicates a small effect. As a general guide, an eta-squared of \(0.01\) is considered small, \(0.059\) medium, and \(0.138\) or above large.Assessing relationships

One reason for conducting a hypothesis test is to determine whether there is a statistically significant relationship between two or more variables. The appropriate test depends on the types of variables being analysed. This page explains two of the most common tests for assessing the relationship between two variables.

In brief, it covers the following:

When to conduct a Chi-square test of independence, and how to interpret the results
When to conduct a test of Pearson’s correlation, and how to interpret the results

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Chi-square test of independence

The chi-square test of independence is a non-parametric test used to determine whether there is a statistically significant association between two categorical variables. For example, it could be used to test whether there is a statistically significant association between employment status (full time or part time) and owning a pet. In this case, the hypotheses would be:

\(\textrm{H}_\textrm{0}\): There is no association between employment status and owning a pet
\(\textrm{H}_\textrm{A}\): There is an association between employment status and owning a pet

Before conducting a chi-square test of independence, you need to check that the following assumptions are met:

Assumption 1: The categories used for the variables are mutually exclusive. This means that each person or item can only fit into one category for each variable.

Assumption 2: The categories used for the variables are exhaustive. This means that there is an appropriate category for every person or item.

Assumption 3: The observations are independent. This means that each person or item only contributes to one cell in the cross-tabulation.

Assumption 4: No more than \(20\%\) of the expected frequencies are less than \(5\). If this assumption is not met, Fisher’s exact test may be an appropriate alternative.

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The first two assumptions require appropriate categories for both variables. Independence depends on how the data were collected. Information about the expected frequencies should be provided with the results of the test. In this example, the \(0.0\%\) shown below the second table indicates that none of the expected frequencies are less than \(5\).

If the assumptions for the chi-square test are met, you can conduct the test using statistical software, such as SPSS. The results should include the following statistics:

When analysing the results, first look at the descriptive statistics to get an idea of what is happening in the sample. In this example, the differences between the Counts and Expected Counts in the cross-tabulation indicate that there is some association between the variables in the sample.

To determine whether this association is statistically significant, we need to look at the \(p\) value for Pearson’s chi-square test. This is shown as ‘Asymptotic Significance (2-sided)’. If we are using a level of significance of \(.05\):

If \(p \leqslant .05\) we reject \(\textrm{H}_\textrm{0}\) and conclude that there is a statistically significant association between the variables.
If \(p > .05\) we do not reject \(\textrm{H}_\textrm{0}\) and conclude that there is not a statistically significant association between the variables.

In this case, our \(p\) value of \(.015\) indicates that the association between employment status and owning a pet is statistically significant.

Note that the chi-square value and degrees of freedom (\(df\)) should generally be reported as part of your results. However, you do not need to interpret these when deciding whether the association is statistically significant.

If you would like to practise interpreting the results of a chi-square test for statistical significance, have a go at the following activity.

H5P interactive activity

Several statistics can be used to calculate effect size, and therefore evaluate practical significance, when conducting a chi-square test of independence. The appropriate statistic depends on factors such as the number of categories for each variable, the type of variables (nominal or ordinal), and the nature of the study.

When there are only two categories for each variable, effect size can be measured using phi (\(\phi\)). An extension of phi for nominal variables with more than two categories is Cramer’s V, while other measures of effect size include relative risk and odds ratio. These statistics can also usually be obtained using statistical software. For example, phi for our employment status and pet ownership example is included in the following output:

The value of \(0.39\) indicates a medium to large effect. As a general guide, a phi with a magnitude of \(0.1\) is considered small, \(0.3\) medium, and \(0.5\) or above large.

Pearson’s correlation

A hypothesis test of Pearson’s correlation coefficient is used to determine whether there is a statistically significant linear correlation between two continuous variables. For example, it could be used to test whether there is a statistically significant linear correlation between heart rates before and after exercise. In this case, the hypotheses would be:

\(\textrm{H}_\textrm{0}\): There is no linear correlation between heart rates before and after exercise in the population
\(\textrm{H}_\textrm{A}\): There is a linear correlation between heart rates before and after exercise in the population

Before conducting a hypothesis test of Pearson’s correlation coefficient, you need to check that the following assumptions are met:

Assumption 1: The observations are independent. This means that the measurements for one subject do not affect the measurements for any other subject.

Assumption 2: Both variables are normally distributed, or the sample size is large enough that the sampling distribution approximates a normal distribution.

Assumption 3: There is a linear relationship between the variables, as shown by the scatter plot. Pearson’s correlation coefficient can still be calculated if the relationship is not linear, but it will not provide an appropriate measure of the relationship.

Assumption 4: There is a homoscedastic relationship between the variables. This means that the variability in one variable is similar across the values of the other variable. In a scatter plot, the dots should have a similar spread around the line of best fit along its length.

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If the assumptions for Pearson’s correlation are not met, or if the data are ordinal rather than continuous, Spearman’s rho or Kendall’s tau-\(b\) may be appropriate non-parametric alternatives.

If the assumptions for Pearson’s correlation are met, you can conduct the test using statistical software, such as SPSS. The results should include the following statistics:

When analysing the results, first look at Pearson’s correlation coefficient (\(r\)) to get an idea of what is happening in the sample. In this example, \(r=.316\). The positive value indicates a positive linear correlation, while the relatively small magnitude indicates that the correlation is fairly weak.

To determine whether this linear correlation is statistically significant, we need to look at the \(p\) value, which is shown as ‘Sig. (2-tailed)’ in the table. If we are using a level of significance of \(.05\):

If \(p \leqslant .05\) we reject \(\textrm{H}_\textrm{0}\) and conclude that there is a statistically significant linear correlation between the variables.
If \(p > .05\) we do not reject \(\textrm{H}_\textrm{0}\) and conclude that there is not a statistically significant linear correlation between the variables.

In this case, our \(p\) value of \(.047\) indicates that the linear correlation between the before and after heart rates is statistically significant.

If you would like to practise interpreting the results of a Pearson’s correlation hypothesis test for statistical significance, have a go at the following activity.

H5P interactive activity

Finally, note that the correlation coefficient itself can be used as a measure of effect size, so a separate measure does not need to be calculated to evaluate practical significance. As a general guide, an \(r\) with a magnitude of \(0.1\) is considered small, \(0.3\) medium, and \(0.5\) or above large.

The coefficient of determination, \(r^2\), can also be calculated. This tells us the percentage of variation in the dependent variable that can be accounted for by variation in the independent variable.

In our heart rate example, the correlation coefficient of \(.316\) indicates a medium effect. The coefficient of determination of \(.0999\) indicates that \(9.99\%\) of the variation in heart rate after exercise can be accounted for by variation in heart rate before exercise.

Regression analysis

Regression analysis is a statistical method used to model and explain the relationship between variables, and to make predictions about one variable based on one or more others. Different types of regression are required depending on the type of variables involved, and this page outlines two of the most common types.

In brief, it covers the following:

When to use linear regression, and how to interpret the results
When to use logistic regression, and how to interpret the results

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Linear regression

Linear regression is used to model the relationship between one or more independent variables, which can be continuous or categorical, and a continuous dependent variable. It builds on the idea of correlation because it also describes a linear relationship between variables, but it allows this relationship to be expressed as an equation that can be used to make predictions. Furthermore, while simple linear regression involves only one independent variable (as with correlation), most of the time there are actually multiple factors that may influence the outcome - and these can all be accounted for in multiple linear regression.

For example, multiple linear regression can be used to examine how factors such as heart rate before exercise, age and gender relate to heart rate after exercise. It provides information about how well these variables explain heart rate after exercise overall, as well as the effect of each variable individually while accounting for the other variables in the model.

Before conducting linear regression, you need to check that the following assumptions are met:

Assumption 1: The observations are independent. This means that measurements for one subject do not affect any other subject’s measurements.

Assumption 2: There is a linear relationship between each independent variable and the dependent variable. This can be assessed by examining scatter plots.

Assumption 3: There are no significant outliers that have too much influence on the results.

Assumption 4: The independent variables are not too highly correlated with each other. This is known as multicollinearity, and it can make it difficult to determine how each variable is influencing the dependent variable.

Assumption 5: The residuals (differences between the observed and predicted values) are approximately normally distributed, and their variability is similar across all values of the predicted variable. This is known as homoscedasticity.

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In addition, it is important to have a sufficiently large sample size relative to the number of independent variables included in the regression model. A commonly used rule of thumb is to have around \(10–15\) cases for each independent variable.

If the assumptions for linear regression are met and the analysis is conducted using statistical software, such as SPSS in this example, the results should include the following statistics:

The first output table is the Model Summary, which includes \(R^2\) and adjusted \(R^2\) values. Both provide an indication of how well the model explains variation in the dependent variable. However, adjusted \(R^2\) takes into account the number of independent variables in the model and provides a more accurate estimate of how well the model would perform in the population, so it is typically the value used. In this case, the adjusted \(R^2\) value of \(.131\) indicates that \(13.1\%\) of the variation in heart rate after exercise can be explained by the independent variables in the model (heart rate before exercise, age and gender).

The second output table is the ANOVA table, which includes the overall \(p\) value for the model. This is used to determine whether the model as a whole is statistically significant. In this case, the \(p\) value of \(.045\) indicates that the model is statistically significant at the \(.05\) level.

The final output table is the Coefficients table, which shows the effect of each independent variable on the dependent variable while accounting for the other variables in the model. It includes both unstandardised coefficients (\(B\)) and standardised coefficients (\(\beta\)), along with their associated \(p\) values. While both types of coefficient provide information about the relationship between the independent and dependent variables, the unstandardised coefficients are typically used for interpretation. They indicate the expected change in the dependent variable for a one-unit increase in the independent variable, while accounting for the other variables in the model.

In this example, heart rate before exercise has a positive coefficient (\(B = 0.76\)) and is statistically significant (\(p = .020\)). This indicates that, after accounting for age and gender, heart rate after exercise is expected to increase by \(0.76\) beats per minute for each additional beat per minute in heart rate before exercise.

Age also has a positive coefficient (\(B = 0.22\)) but is not statistically significant (\(p = .346\)). This indicates that, after accounting for the other variables, heart rate after exercise is expected to increase by \(0.22\) beats per minute for each additional year of age, although there is insufficient evidence that this relationship exists in the population.

Gender also has a positive coefficient (\(B = 7.24\)) but is not statistically significant (\(p = .066\)). Given that gender is coded as \(1\) for male and \(2\) for female, this indicates that, after accounting for the other variables, heart rate after exercise is expected to be \(7.24\) beats per minute higher for females than males. However, there is insufficient evidence that this difference exists in the population.

Overall, while the model is statistically significant, heart rate before exercise is the only independent variable with a statistically significant relationship with heart rate after exercise after accounting for age and gender.

Logistic regression

Logistic regression is used to model the relationship between one or more independent variables, which can be continuous or categorical, and a categorical dependent variable. Most commonly, the dependent variable has two categories, such as yes/no or pass/fail, although logistic regression can also be extended to outcomes with more than two categories. Like linear regression, logistic regression allows the effects of multiple independent variables on an outcome to be examined while accounting for the other variables in the model. However, rather than modelling a linear relationship, it models the probability of an outcome occurring.

For example, logistic regression can be used to examine how factors such as study time, attendance and gender relate to whether a student passes or fails a unit. It provides information about how well these variables explain pass status overall, as well as the effect of each variable individually while accounting for the other variables in the model.

Before conducting logistic regression, you need to check that the following assumptions are met:

Assumption 1: The observations are independent. This means that measurements for one subject do not affect any other subject’s measurements.

Assumption 2: There is a linear relationship between any continuous independent variables and the log odds of the dependent variable.

Assumption 3: There are no significant outliers that have too much influence on the results.

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In addition, it is important to have a sufficiently large sample size relative to the number of independent variables included in the regression model. A commonly used rule of thumb for logistic regression is to have at least \(10\) outcome events for each independent variable.

If the assumptions for logistic regression are met and the analysis is conducted using statistical software, such as SPSS in this example, the results should include the following statistics:

The first output table is the Omnibus Tests of Model Coefficients table, which includes the overall \(p\) value for the model in the ‘Model’ row. This is used to determine whether the model as a whole is statistically significant. In this case, the \(p\) value is less than \(.001\), indicating that the model is statistically significant at the \(.05\) level.

The second output table is the Model Summary table, which includes pseudo \(R^2\) values (Cox & Snell and Nagelkerke \(R^2\)). Both provide an indication of how well the model accounts for the outcome, however, Nagelkerke \(R^2\) is scaled to range from \(0\) to \(1\) which makes it easier to interpret. In this case, the Nagelkerke \(R^2\) value of \(.61\) indicates that the model accounts for a substantial amount of the variation in pass status, although some variation remains unexplained.

The final output table is the Variables in the Equation table, which shows the effect of each independent variable on the dependent variable while accounting for the other variables in the model. It includes the regression coefficients (\(B\)), their associated \(p\) values, and odds ratios (\(Exp(B)\)). While the regression coefficients are used to construct the model, the odds ratios are typically easier to interpret because they show how the odds of the outcome change for a one-unit increase in an independent variable.

In this example, study time has an odds ratio of \(2.13\), but is not statistically significant (\(p=.661\)). This indicates that, after accounting for attendance and gender, each additional hour of study time is associated with approximately twice the odds of passing the unit. However, there is insufficient evidence that this relationship exists in the population.

Attendance has an odds ratio of \(1.07\), but is not statistically significant (\(p=.895\)). This indicates that, after accounting for the other variables, each additional percentage point in attendance is associated with a small increase in the odds of passing. However, there is insufficient evidence that this relationship exists in the population.

Gender has an odds ratio of \(2.60\), but is also not statistically significant (\(p=.351\)). Given that gender is coded as \(1\) for male and \(2\) for female, this indicates that, after accounting for the other variables, females have approximately \(2.6\) times the odds of passing compared with males. However, there is insufficient evidence that this difference exists in the population.

Overall, while the model is statistically significant, there is insufficient evidence that any of the individual variables are significantly associated with pass status once the other variables are taken into account.
Further resources & glossary

Further Resources

Congratulations on making it to the end of the module! We hope you found it a useful introduction to the world of statistics. If you would like to learn more, particularly about other statistical tests, you may find one or more of the following resources helpful:

Glossary

The following glossary explains statistical terms used throughout this module. For more information about a term, click on the relevant link.

A C D E H I L M N O P Q R S T V

Alternative hypothesis
Also known as the research hypothesis and denoted by \(\textrm{H}_\textrm{A}\), the alternative hypothesis states the opposite of the null hypothesis. It states that there is a difference or relationship between variables in a population.
See Hypothesis testing, Null hypothesis, Variable.

Categorical data
Data that is grouped into categories, such as data for a ‘gender’ or ‘smoking status’ variable. Categorical data can be further classified as nominal or ordinal. See Data, Nominal data, Ordinal data.

Chi-square test of independence
A non-parametric test used to determine whether there is a statistically significant association between two categorical variables. The chi-square value is represented by \(\chi^2\)
See Categorical data, Non-parametric test.

Cohen’s \(d\)
A measure of effect size which determines how many standard deviations two means are separated by. It is commonly used to evaluate practical significance for \(t\) tests.
See Effect size, \(t\) test, One-way ANOVA.

Confidence interval
A range of values that a population statistic, such as the population mean, is expected to lie between with a given level of confidence. This level is known as the confidence coefficient and is typically \(95\%\), in which case it is called a \(95\%\) confidence interval.
See Population.

Confounding variable
A variable that may have an influence on the dependent (outcome) variable.
See Variable.

Continuous data
Data that is measured on a continuous numerical scale and can take a large number of possible values, such as data for a ‘weight’ or ‘distance’ variable. Continuous data can be further classified as interval or ratio. See Interval data, Ratio data, Variable.

Cross-tabulation (contingency table)
A table used to display information for two categorical variables. Categories of one variable are listed in the rows and categories of the other variable are listed in the columns. Each cell contains the frequency (number) of subjects that fall into that combination of categories. Percentages and totals are often also included. See Categorical data, Dependent variable, Independent variable.

Data
Observations and measurements that have been collected in some way, often through research. Quantitative data measures quantities and is recorded as numbers, while qualitative data records qualities as categories or as thoughts, feelings and opinions.

Discrete data
Data which measures counts or numbers of events, such as data for a ‘class attendance’ variable. It can be treated as either categorical or continuous, depending on how many values are possible.
See Data.

Degrees of freedom
The number of values that are free to vary when calculating an estimate. Degrees of freedom are commonly reported as part of the results of various hypothesis tests.
See Hypothesis testing.

Dependent variable (outcome variable)
When testing for a relationship between variables, the dependent variable is the one that is potentially influenced, affected or predicted by another variable.
See Variable.

Descriptive statistics
Statistics that are used to summarise and describe a variable or variables for a sample of data.
See Data, Sample, Variable.

Effect size
Effect size measures the magnitude of a difference or relationship between variables. It provides information about whether the difference or relationship is meaningful in a practical sense (i.e. in real life) and is calculated differently for different statistical tests.
See Cohen’s \(d\), Odds ratio, Practical significance, Variable.

Hypothesis testing
Hypothesis testing is used to determine whether a difference or relationship observed in a sample is statistically significant in terms of the population from which the sample was drawn. This can be assessed by interpreting the resulting \(p\) value.
See Alternative hypothesis, Null hypothesis, \(p\) value, Population, Sample, Statistical significance.

Independent samples \(t\) test
A parametric inferential statistical test used to determine whether there is a statistically significant difference between the mean of a continuous variable for two independent (unrelated) groups.
See Continuous data, Inferential statistics, Mean, Parametric test, Statistical significance.

Independent variable (predictor or exposure variable)
When testing for a relationship between pairs of variables, the independent variable is the one that potentially influences, affects or predicts the other variable.
See Variable.

Inferential statistics
Statistics that are used to draw inferences about the wider population from which a sample of data was drawn.
See Population, Sample.

Interquartile range
The interquartile range is a measure of dispersion that is appropriate when the median is used as the measure of central tendency. It is calculated by finding the difference between the first and third quartiles.
See Measure of central tendency, Measure of dispersion, Median, Quartiles.

Interval data
Continuous data that does not have an absolute zero, and where negative numbers also have meaning, such as for a ‘temperature in degrees Celsius variable’.
See Continuous data.

Level of significance
In a hypothesis test, the level of significance (denoted by \(\alpha\)) determines how small the \(p\) value needs to be before the null hypothesis is rejected. It is typically \(5\%\) (\(.05\))
See Hypothesis testing, Null hypothesis, \(p\) value

Linear regression
A statistical technique used to examine the relationship between one or more independent variables and a continuous dependent variable.
See Continuous data, Dependent variable, Independent variable.

Logistic regression
A statistical technique used to examine the relationship between one or more independent variables and a categorical dependent variable.
See Categorical data, Dependent variable, Independent variable.

Mean
The mean is the arithmetic average of a data set, calculated by adding all the values together and dividing by the total number of values. It is the most commonly used measure of central tendency. The sample mean is denoted by \(\bar{x}\), while the population mean is denoted by \(\mu\).
See Measure of central tendency, Population, Sample.

Measure of central tendency
A descriptive statistic that summarises a variable by identifying its average, central or typical value. Examples include the mean, median and mode.
See Continuous data, Descriptive statistic, Mean, Median, Mode.

Measure of dispersion
A descriptive statistic that summarises a variable by showing how widely the data is spread. Examples include the range, interquartile range, variance and standard deviation.
See Continuous data, Descriptive statistic, Interquartile range, Range, Standard deviation, Variance.

Median
The median is a more appropriate measure of central tendency than the mean when the data is affected by outliers or is skewed. It is the middle value when the data set is sorted from smallest to largest. If there are two middle values, the median is their average.
See Mean, Measure of central tendency, Outlier, Skewness.

Mode
The mode is the most frequently occurring value (or values) in the data set; it is a less commonly used measure of central tendency.
See Measure of central tendency.

Nominal data
Categorical data where the categories do not have an order, such as for a ‘marital status’ variable. If there are only two categories, then the terms binary and/or dichotomous are often also used.
See Categorical data.

Non-parametric test
An inferential statistical test that does not require continuous data to follow a particular distribution, such as the normal distribution. Non-parametric tests can also be used with some types of data that are not continuous, such as ordinal data.
See Continuous data, Inferential statistics, Normal distribution.

Normal distribution
A distribution (spread) of data that has two key properties:
1. The mean, median and mode are all equal.
2. Fixed proportions of the data lie within certain numbers of standard deviations from the mean (\(68\%\) within one standard deviation, \(95\%\) within two standard deviations and \(99.7\%\) within three standard deviations).
See Data, Mean, Median, Mode, Standard deviation.

Null hypothesis
Denoted by \(\textrm{H}_\textrm{0}\), the null hypothesis states that there is no difference or relationship between variables in a population.
See Alternative hypothesis, Hypothesis testing, Variable.

Odds ratio
A measure that compares the odds of an outcome between groups or shows how the odds of an outcome change as an independent variable changes. An odds ratio of \(1\) indicates no difference in the odds, a value greater than \(1\) indicates higher odds, and a value less than \(1\) indicates lower odds.
See Chi-square test of independence, Logistic regression.

One sample \(t\) test
A parametric inferential statistical test used to determine whether the mean of a continuous variable in a population is significantly different from a test value (some hypothesised value).
See Continuous data, Inferential statistics, Mean, Parametric test, Statistical significance.

One-way ANOVA (analysis of variance)
A parametric inferential statistical test used to determine whether there are any statistically significant differences between the means of a continuous variable for three or more independent (unrelated) groups.
See Continuous data, Inferential statistics, Mean, Parametric test, Statistical significance.

Ordinal data
Categorical data where the categories do have an order, such as for a ‘satisfaction level’ variable.
See Categorical data.

Outlier
An outlier is a data point that lies well above or below the other data. One way to identify possible outliers is to look for values more than \(1.5\) interquartile ranges below the first quartile or above the third quartile.
See Interquartile range, Quartiles.

\(p\) value
The \(p\) value for a hypothesis test is the probability of obtaining the observed test statistic, or one more extreme, if the null hypothesis is true. A small \(p\) value means that a result this extreme would be unlikely if the null hypothesis were true. If the \(p\) value is less than or equal to the level of significance, we reject the null hypothesis.
See Hypothesis test, Level of significance, Null hypothesis, Statistical significance, Test statistic.

Paired samples \(t\) test
A parametric inferential statistical test used to determine whether there is a statistically significant difference between the means of a continuous variable measured for two related groups or under two related conditions.
See Continuous data, Inferential statistics, Mean, Parametric test, Statistical significance.

Parametric test
An inferential statistical test that requires at least one continuous variable, and which requires continuous variables to be normally distributed (or the sample size to be large enough that the sampling distribution of the mean approximates a normal distribution).
See Continuous data, Inferential statistics, Normal distribution.

Pearson’s correlation coefficient
Pearson’s correlation coefficient, denoted by \(r\), measures the strength and direction of the linear relationship between two continuous variables. It can range from \(-1\) to \(1\). Values close to \(-1\) indicate a strong negative linear correlation, values close to \(1\) indicate a strong positive linear correlation, and values close to \(0\) indicate little or no linear correlation.
See Continuous data.

Percentiles
Values that show the position of a data point within an ordered data set. They can be used to show the relative standing of a particular value.
See Measure of dispersion.

Population
A population is every member of a group of interest. Normally it is not possible or feasible to collect data from the entire population, so a random sample is used instead to draw inferences about the population.
See Data, Inferential statistics, Sample.

Power
The power of a hypothesis test is the probability that the test will find an effect if one actually exists. In other words, it is the probability of correctly rejecting an incorrect null hypothesis.
See Hypothesis test, Null hypothesis.

Practical significance
Practical significance refers to whether a difference or relationship between variables is meaningful in a practical sense (i.e. in real life). It can be evaluated by calculating an effect size.
See Effect size, Variable.

Quartiles
Specific percentiles that divide an ordered data set into four parts. The\(25\)th percentile is known as the first or lower quartile, the \(50\)th percentile is known as the median, and the \(75\)th percentile is known as the third or upper quartile.
See Median, Percentiles

Range
The simplest measure of dispersion, the range is the difference between the smallest and largest value in a data set.
See Measure of dispersion.

Ratio data
Continuous data that does have an absolute zero, and where negative numbers do not have meaning, such as for a ‘height’ variable.
See Continuous data.

Sample
A sample is a subset of a population. It can be analysed using descriptive statistics, or used to draw inferences about the wider population using inferential statistics.
See Descriptive statistics, Inferential statistics, Population.

Standard deviation
Standard deviation is a commonly used measure of dispersion and is appropriate when the mean is used as the measure of central tendency. It is the square root of the variance and measures how much the values typically vary around the mean. The sample standard deviation is denoted by \(s\), while the population standard deviation is denoted by \(\sigma\).
See Mean, Measure of central tendency, Measure of dispersion, Population, Sample, Variance.

Statistical significance
Statistical significance refers to whether there is enough evidence from a sample to conclude that a difference or relationship exists in the population. It is determined by conducting a hypothesis test.
See Hypothesis test, Sample, Variable.

Test statistic
A value calculated as the result of a hypothesis test, the test statistic compares the value of the sample statistic (for example, the sample mean) with the value specified by the null hypothesis for the population statistic.
See Hypothesis testing, Mean, Null hypothesis, Population, Sample.

Variable
A characteristic or attribute for which you observe, measure or record data, such as height, weight, eye colour or dog breed.
See Data.

Variance
A measure of dispersion that shows how much the data varies around the mean. The square root of the variance is the standard deviation.
See Mean, Measure of dispersion, Standard deviation. Feedback

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