The Shape of a Distribution
The shape of a distribution tells you how the data are spread out: symmetrical, positively skewed (a long tail of high values), negatively skewed (a long tail of low values), uniform or bimodal. This NSW Year 9 Core maths subtopic is Stage 5 Core content under NESA outcome MA5‑DAT‑C‑01.
You will name the shape of histograms and dot plots, explain which way a skew points, and use the mean and median to work out the shape without a picture: a mean of \(38\) with a median of \(44\) means a negative skew. You will see how outliers change the shape and the mean, choose the median for strongly skewed data such as weekly incomes, compare the shapes of two sets, and decide what a stated skew does and does not tell you. It is part of Year 9 Core Investigating Data.
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Theory
The shape of a distribution describes how data are spread across a histogram or dot plot. This Year 9 maths page shows how to name a shape as symmetrical, positively skewed or negatively skewed, spot uniform and bimodal data, use the mean and median to tell the skew without a picture, and choose the right measure of centre.
The shape of a distribution is the outline its histogram or dot plot makes. Look for where most of the data sit and which way the tail (the thin end) runs.
A symmetrical distribution has two halves that match when folded down the middle.
A positively skewed distribution has most scores at the low end and a long tail running to the right, towards the larger numbers. A negatively skewed distribution has most scores at the high end and a tail running to the left. The skew is named after the tail, not the hump.
Uniform data have every column the same height, so there is no mode. Unimodal data have one peak and bimodal data have two.
A cluster is a group of scores close together. A gap is a stretch with no scores. An outlier is a score far from the rest. A few outliers on one side can turn a symmetrical set into a skewed one.
The tail drags the mean towards it, but the median barely moves. So comparing them tells you the shape:
| Shape | Where most scores sit | Mean and median |
|---|---|---|
| symmetrical | evenly about the centre | \(\text{mean} = \text{median}\) |
| positively skewed | at the low end, tail to the right | \(\text{mean} > \text{median}\) |
| negatively skewed | at the high end, tail to the left | \(\text{mean} < \text{median}\) |
| uniform | spread evenly, no peak | no mode |
How to describe the shape of a distribution
- Draw or read the display. From raw data, sort and count each score, then make a dot plot.
- Find where most of the scores sit, and which way the thin tail runs.
- Name the shape: tail right is positive, tail left is negative, matching halves is symmetrical.
- Count the peaks: none (uniform), one (unimodal) or two (bimodal).
- Note any clusters, gaps or outliers.
- Without a picture, compare the mean with the median.
Find where the bulk of the data sits, then which way the tail runs.
| \(\text{tall bars}\) | \(:\) | \(\text{at the HIGH scores}\) |
| \(\text{tail}\) | \(:\) | \(\text{runs LEFT}\) |
A tail to the left is a negative skew. The distribution is negatively skewed.
Compare the two measures of centre. The tail drags the mean towards it, while the median barely moves.
| \(\text{mean}\) | \(=\) | \(46\) |
| \(\text{median}\) | \(=\) | \(39\) |
| \(46\) | \(>\) | \(39\) |
The mean sits above the median, so a tail of high scores has pulled it up. The distribution is positively skewed.
Fold the plot about the middle column. The two halves, \(7, 2\) and \(2, 7\), match, so it is symmetrical.
Now count the peaks. Two columns tie at \(7\), at opposite ends, so there are two peaks.
It is symmetrical and bimodal. A full description gives both the shape and the number of peaks.
Seven times are close together and one sits far above them, making a tail to the right. The data are positively skewed.
| \(\text{mean}\) | \(=\) | \(\dfrac{269}{8}\) |
| \(\) | \(\approx\) | \(33.6 \text{ min}\) |
| \(\text{median}\) | \(=\) | \(\dfrac{25 + 27}{2}\) |
| \(\) | \(=\) | \(26 \text{ min}\) |
Seven of the eight deliveries were faster than the mean. The median, \(26\) minutes, describes a typical delivery better.
Common pitfalls
Frequently asked questions
What does positively skewed mean?
Most of the scores are at the low end and a long tail runs to the right, towards the larger numbers. The mean is greater than the median.
How do you tell if data are positively or negatively skewed?
Look at which way the tail runs. A tail to the right is positive and a tail to the left is negative. Without a picture, compare the mean and median: mean above median is positive, mean below median is negative.
What is a symmetrical distribution?
One whose two halves match when the display is folded down the middle. Its mean and median are equal.
What is the difference between uniform and bimodal?
Uniform data have every column the same height, so there is no peak. Bimodal data have two separate peaks.
Should I use the mean or median for skewed data?
Use the median. The tail drags the mean towards it, so the mean is not typical of most scores. The median stays with the bulk of the data.