Investigating and comparing data distributions

Statistics begins with a question and a set of observations. Each characteristic we record about the people, objects or events in a study is called a variable, and before we can choose a graph or a summary statistic we must decide what type of variable we are dealing with. Getting the type wrong leads to meaningless results, such as calculating an 'average postcode'.

Categorical variables place each observation into a group or category. They are nominal when the categories have no natural order (eye colour, favourite sport, type of car, postcode) and ordinal when the categories can be ranked (shirt size S/M/L/XL, survey response 'strongly disagree' to 'strongly agree', year level). Numbers can still be categorical: a jersey number or a phone number is a label, not a quantity, because adding or averaging them makes no sense.

Numerical variables record a quantity that can be counted or measured. They are discrete when they take only separate, countable values (number of siblings, goals scored, cars in a car park) and continuous when they can take any value within an interval, limited only by the precision of the measuring instrument (height, time to run 100 m, mass, temperature). Numerical data can also be described by its measurement scale.

On an interval scale differences are meaningful but zero is arbitrary, so ratios are not; temperature in °C is the classic example, since 20 °C is not 'twice as hot' as 10 °C. On a ratio scale there is a true zero, so both differences and ratios are meaningful; a 4 kg bag really is twice as heavy as a 2 kg bag.

Categorical data are usually organised in a frequency table, listing each category with its count (frequency) and its percentage frequency, found by dividing the count by the total and multiplying by 100. The display of choice is a bar chart (or a segmented bar chart to compare groups), with gaps between the bars because the categories are separate. The only measure of centre that makes sense for nominal data is the mode, the most frequent category.

When describing categorical data, name the modal category and support statements with percentages rather than raw counts, especially when group sizes differ.

When comparing a categorical variable across two groups (for example Year 7 students versus Year 11 students), convert each group to percentages first. A percentage segmented bar chart, where each bar has a total height of 100%, makes the comparison easy to see. A good comparison statement quotes the relevant percentages for both groups, such as 'a higher percentage of Year 11 students drove themselves (20%) compared with Year 7 students (0%)'.