The Shape of a Distribution
A distribution can be symmetrical, positively skewed with a tail to the right, or negatively skewed with a tail to the left. This NSW Year 9 Advanced maths subtopic describes shape and links it to the mean and median (NESA outcome MA5‑DAT‑C‑01).
You will describe the shape, clusters and outliers of dot plots and histograms, recognise bimodal data, and use the mean and median to predict the shape. You will see how one large score moves the mean but not the median, match histograms to their statistics, explain the shape of the heights of a mixed group, and choose a measure of centre for skewed incomes. It is part of Year 9 Advanced Investigating Data.
Every question with a fully worked solution.
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Theory
The shape of a data set tells you more than any single number. Is it symmetrical, or does it have a tail on one side? One peak or two? This Year 9 Advanced maths page shows how to describe the shape from a graph, how the mean and median give it away, and which of them to trust.
A distribution is symmetrical when its two halves are mirror images about the centre.
It is positively skewed when most scores are at the low end and a tail runs to the right, towards the larger numbers. It is negatively skewed when most scores are high and the tail runs to the left. The name comes from the tail, not the peak.
The number of peaks is a separate description. Unimodal means one peak, bimodal two peaks, and uniform means every value occurs about equally often. Two peaks often mean the data mix two different groups.
An outlier is a score well apart from the rest. A cluster is where most of the scores bunch together.
The mean is pulled towards a tail or an outlier. The median is the middle score, so it hardly moves.
| Compare | Shape | Best measure of centre |
|---|---|---|
| mean \(>\) median | positively skewed (tail right) | median |
| mean \(=\) median | symmetrical | mean or median |
| mean \(<\) median | negatively skewed (tail left) | median |
These are good guides for real data, not exact laws: a set with mean equal to median can still be lopsided. Always check the graph as well.
Describing a distribution
- Draw or read the graph. From raw data, count each value and stack the dots or draw the bars.
- Shape: symmetrical, or which side has the tail? Name the skew after the tail.
- Peaks: one, two or none (uniform). If there are two, ask what two groups are mixed.
- Outliers and clusters: point out any score far from the rest and where most scores bunch.
- Check with numbers: compare the mean and the median. They should agree with what the graph shows.
Choosing a measure of centre
- Symmetrical, no outliers: the mean uses every score and is fine.
- Skewed or with outliers: use the median, which is not dragged by the tail.
- Bimodal: neither describes the data well; split the groups and summarise each.
- Spot a misleading 'average', such as a mean salary raised by one large income.
(i) Count each score: \(3\) once, \(6\) once, \(7\) twice, \(8\) three times, \(9\) four times, \(10\) three times. Stack one dot per student.
(ii) Most scores are high, with a tail running left to the low scores: the shape is negatively skewed.
(iii) Yes: \(\mathbf{3}\) sits apart, with no scores at \(4\) or \(5\).
(iv) The tallest column is at \(9\): the data cluster around \(\mathbf{8}\) to \(\mathbf{10}\).
(i) Multiply each value by its frequency.
| \(\sum fx\) | \(=\) | \(14 + 18 + 15 + 12 + 10 + 6 = 75\) |
| \(\bar{x}\) | \(=\) | \(\dfrac{75}{34} = 2.2058\ldots \approx \mathbf{2.21}\) |
The median is halfway between the \(17\)th and \(18\)th. Running totals: \(14\), \(23\). Both are in the bar for \(2\).
| \(\text{median}\) | \(=\) | \(\mathbf{2}\) |
(ii) Mean \(>\) median, so the data are positively skewed. The graph agrees: the peak is at \(1\) and the tail runs right.
(iii) The median. The few full cars drag the mean up; the median stays with the bulk of the cars.
(i) Find the mean and median age of all \(12\) people.
(ii) Find the mean and median of the teenagers only.
(iii) Describe the effect of the parent on the shape and on each measure.
(i) Teenagers' total: \(182\). Add \(52\).
| \(\bar{x}\) | \(=\) | \(\dfrac{182 + 52}{12} = \dfrac{234}{12} = \mathbf{19.5}\) |
| \(\text{median}\) | \(=\) | \(\dfrac{6\text{th} + 7\text{th}}{2} = \dfrac{17 + 17}{2} = \mathbf{17}\) |
(ii) Eleven ages: the median is the \(6\)th.
| \(\bar{x}\) | \(=\) | \(\dfrac{182}{11} = 16.545\ldots \approx \mathbf{16.55}\) |
| \(\text{median}\) | \(=\) | \(\mathbf{17}\) |
(iii) The teenagers' ages are close to symmetrical. The single age of \(52\) is an outlier that makes a long tail to the right, so the data become positively skewed. It drags the mean up by about \(3\) years, to an age that nobody at the party is, while the median does not move.
(i) There are two clear peaks, at \(25\)–\(29\) and \(45\)–\(49\) minutes, with a gap between them: the distribution is bimodal.
(ii) The data mix two groups. The runners make the faster peak and the walkers make the slower one.
(iii) Class centre \(\times\) frequency:
| \(\sum fx\) | \(=\) | \(3(22) + 7(27) + 4(32) + 1(37)\) |
| \(\) | \(\) | \(+\ 2(42) + 6(47) + 4(52) = 994\) |
| \(\bar{x}\) | \(\approx\) | \(\dfrac{994}{27} = 36.81\ldots \approx \mathbf{36.8}~\text{min}\) |
The mean lands in the \(35\)–\(39\) class, the valley between the peaks, where only one person finished. It describes neither group: summarise the runners and the walkers separately.
Common pitfalls
Frequently asked questions
What does positively skewed mean?
Most of the scores are at the low end and a tail of a few larger scores runs to the right. The mean is usually bigger than the median.
How can you tell the shape from the mean and median?
If the mean is greater than the median the data are usually positively skewed; if it is less, negatively skewed; if they are equal, roughly symmetrical. Check the graph too.
Which measure of centre should I use for skewed data?
The median. The tail pulls the mean away from where most of the data are, but the median stays in the middle of the ordered scores.
What does bimodal mean?
A bimodal distribution has two peaks. It often means two different groups have been measured together, such as runners and walkers.
What is an outlier?
An outlier is a score that lies well away from the rest of the data, with a gap before it. It pulls the mean towards it but barely changes the median.
What is a uniform distribution?
Every value occurs about equally often, so the graph is flat with no peak.