ACARA v9 CONTENT DESCRIPTION “create different types of numerical data displays including stem-and-leaf plots using software where appropriate; describe and compare the distribution of data, commenting on the shape, centre and spread including outliers and determining the range, median, mean and mode”
Once data has been collected, the next task is to make sense of it. A long list of numbers tells you very little until you can see its overall pattern, its centre and its spread. This year you learn to display numerical data and to describe and compare distributions using their shape, their range and a measure of centre, turning raw figures into a clear summary you can reason about.
Describing a distribution well means answering three questions: what shape is the data, where is its centre, and how spread out is it. Together these give a faithful picture of the whole data set, far more useful than any single number on its own.
The shape of the data
The shape of a distribution is the overall pattern you see when the data is plotted. A simple dot plot, with one dot for each value, reveals this immediately. The data might pile up in the middle and tail off at both ends, forming a symmetric mound, or it might lean to one side, or spread out evenly. Reading the shape is the first step, because it shows where the data clusters and where it is sparse.
The shape of a distribution
A dot plot reveals whether data clusters, spreads out or leans to one side.
A distribution is the pattern of how data is spread. Plotting each value as a dot shows the shape at a glance: here the data forms a symmetric mound, clustered around a central value with fewer points at the extremes.
Different displays suit different purposes. Dot plots show every value and suit small data sets, while stem-and-leaf plots organise larger sets while keeping the individual figures visible. The right display makes the shape of the distribution easy to see, which is why choosing and creating a suitable display is part of the skill.
Centre and spread
Beyond shape, two summaries capture a distribution: a measure of centre and a measure of spread. The centre, such as the mean or the median, gives a single typical value that represents the data. For the values clustered around 6 in the example, a centre of about 6 tells you what is typical. The mean balances the data, while the median sits at its middle when the values are ordered.
Centre and spread
Summarise data by where it centres and how widely it spreads.
Two numbers summarise a distribution well: a measure of centre and a measure of spread. The centre, such as the mean of about 6 here, gives the typical value; the spread, like the range from 3 to 9, shows how varied the data is.
The spread tells you how varied the data is, and the simplest measure is the range, the largest value minus the smallest. A small range means the data is tightly bunched; a large range means it is widely scattered. This is what makes spread essential for comparison: two data sets can share the same centre yet behave very differently. If two classes have the same mean test score but one has a far larger range, that class has much more varied results. Describing data by its shape, centre and spread, and comparing distributions on all three, is how statistics turns a column of numbers into genuine understanding.
Teaching tip: collect a small real data set with the student, such as the number of letters in each family member name, and plot it as a dot plot together. Asking what is typical and how spread out the values are introduces centre and spread naturally, grounded in data they helped create.
Emphasise that centre and spread answer different questions. A common habit is to report only an average and stop. Prompt the student to also ask how varied the data is, so that describing a distribution always covers both the typical value and the variation around it.
Builds on: Range, median, mean and mode (AC9M7ST01). That unit collected and organised data; this unit describes the shape, centre and spread of a distribution.
Quick self-check
1. What does the shape of a data distribution show you?
2. A measure of centre, such as the mean or median, tells you
3. For the data 3, 5, 6, 6, 9, what is the range?
4. Two classes have the same mean score, but one has a much larger range. This tells you
5. Why describe data with both a centre and a spread?