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 are interested in learning more about statistics, and in particular finding out about other statistical tests, you may like to make use of one or more of the following resources:

- The Beginner’s Guide to Statistical Analysis
- Statistics by Jim
- Online Statistics Education: A Multimedia Course of Study

The following is a glossary of statistical terms used throughout this module. For more information on any of the terms please 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 always states the opposite of the null hypothesis; i.e. it states that there *is* a difference or relationship between variables in a population.

See *Hypothesis testing*, *Null hypothesis*, *Variable*.

Categorical data

Data which 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 and ANOVA.

See *Effect size*, *\(t\) test*, *One-way ANOVA*.

Confidence interval

A range of values that a population statistic (e.g. the population mean) is expected to lie between with a given level of certainty (known as the confidence coefficient). The confidence coefficient is typically \(95\%\), in which case it is referred to as 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 which is measured on a continuous numerical scale and which can take on 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 the independent variable are listed in the rows and categories of the dependent variable are listed in the columns, with each cell containing the frequency (number) of subjects that fall into that combination of categories. Percentages are often also included, along with totals.

See *Categorical data*, *Dependent variable*, *Independent variable*.

Data

Observations and measurements which have been collected in some way, often through research. Quantitative data measures quantities and is recorded as numbers, while qualitative data records qualities in terms of different categories or in terms of 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. This is commonly reported as part of the results of various hypothesis tests.

See *Hypothesis testing*.

Dependent variable (outcome variable)

When testing for a relationship between pairs of variables, the dependent variable is the one that is potentially influenced, affected or predicted by the other 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 is used to provide evidence of whether it is meaningful in real life (i.e. has practical significance), 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 appropriate in situations where the median is used as the measure of central tendency. It is calculating 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 exactly how small the \(p\) value can be before the null hypothesis is rejected. It is typically \(5\%\) (\(.05\))

See *Hypothesis testing*, *Null hypothesis*, *\(p\) value*

Mean

The mean is the arithmetic average of a data set, calculated by adding all of the data together and dividing through by the total number of values. It is the most commonly used measure of central tendency. The sample standard deviation 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 which summarises a continuous variable by finding the average, central or typical member. Examples of measures of central tendency are the mean, median and mode.

See *Continuous data*, *Descriptive statistic*, *Mean*, *Median*, *Mode*.

Measure of dispersion

A descriptive statistic which summarises a continuous variable by finding out how widely it is spread or dispersed. Examples of measures of dispersion are 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 calculated by finding the middle value (or average of two middle values) when the data set is sorted from smallest to largest.

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 doesn’t require the variable(s) to be normally distributed, and doesn’t require continuous 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 always states that there is *no* difference or relationship between variables in a population.

See *Alternative hypothesis*, *Hypothesis testing*, *Variable*.

Odds ratio

A measure of effect size used when testing for association between an exposure and an outcome (e.g. using a Chi-square test), an odds ratio compares the odds of exposure in the group with the outcome to the odds of exposure in the group without the outcome. An odds ratio of \(1\) indicates no difference between the two groups, while an odds ratio greater than \(1\) indicates that the group with the outcome are more likely to have had the exposure, and an odds ratio less than \(1\) indicates that the group with the outcome are less likely to have had the exposure.

See *Chi-square test of independence*.

One sample \(t\) test

A parametric inferential statistical test used to determine whether there is a statistically significant difference between the mean of a continuous variable and 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 any data point that lies well above or below the other data; in particular, over \(1.5\) interquartile ranges below the first quartile or \(1.5\) interquartile ranges above the third quartile.

See *Interquartile range*, *Quartiles*.

\(p\) value

The \(p\) value for a hypothesis test is the probability of obtaining a given test statistic if the null hypothesis is true. A small \(p\) value indicates a low probability, and in particular if the \(p\) value is less than the level of significance it provides evidence to reject the null hypothesis (and hence of statistical significance).

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 continuous variables for two related groups.

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 \(r\), is used to determine whether there is a linear correlation (straight line relationship) between two continuous variables. It can range from \(-1\) to \(1\), with values close to \(-1\) indicating strong negative correlation, values close to \(1\) indicating strong positive correlation, and values close to \(0\) indicating no correlation.

See *Continuous data*.

Percentiles

A measure of dispersion that measures position from the beginning of an ordered data set, and can be used to measure the relative standing of a particular data point.

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, that an incorrect null hypothesis will in fact be rejected.

See *Hypothesis test*, *Null hypothesis*.

Practical significance

Practical significance refers to whether or not a difference or relationship between variables is meaningful in a practical sense (i.e. in real life). It is determined by calculating an effect size.

See *Effect size*, *Variable*.

Quartiles

A specific type of percentiles which divide the data set into quarters. In particular, 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 the most commonly used measure of dispersion, appropriate in situations where the mean is used as the measure of central tendency. It is the square root of the variance, and measures how much deviation there is from the mean. Sample standard deviation is denoted by \(s\), while 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 or not a difference or relationship between variables observed in a sample could have occurred due to random chance alone. 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 that you are observing, measuring and recording data for, e.g height, weight, eye colour, dog breed, etc.

See *Data*.

Variance

A measure of dispersion that measures how much deviation there is from the mean, the square root is usually taken in order to find the standard deviation.

See *Mean*, *Measure of dispersion*, *Standard deviation*.