Shapes of Distributions
The shape of a distribution describes how data is spread across its values. A symmetric distribution has matching halves, a positively skewed one has a long tail to the right, a negatively skewed one has a long tail to the left, and a bimodal one has two peaks. In NSW Year 10 Advanced maths you describe shape from dot plots, histograms, stem-and-leaf plots and box plots (NESA outcome MA5‑DAT‑C‑01).
You will describe the shape of data sets, pick a skewed distribution from its five-number summary, and explain why the mean is pulled towards a long tail. Data include ferry delays, Perth house prices and cafe customers. You will see how adding bonus marks changes the mean, median and IQR, and find scores that fit given statistics. It is part of Year 10 Advanced Single Variable and Bivariant Statistics.
Every question with a fully worked solution.
- Shapes of Distributions - Video - What is Skewness? | Statistics Watch
Theory
The shape of a distribution describes how the scores are spread: symmetrical, skewed with a tail to one side, or with two peaks (bimodal). This Year 10 Advanced maths page shows how to read the shape from any display, including box plots, how to confirm it with the mean and median, and what happens to the statistics when every score is changed.
A distribution is symmetrical if its left and right halves are mirror images. It is positively skewed if most scores are low and a tail stretches to the right (large values), and negatively skewed if most are high with a tail to the left.
A bimodal distribution has two peaks, often because two groups are mixed (for example breakfast and lunch customers).
In a box plot, skew shows as a median off-centre in the box and one whisker longer than the other. Median close to \(Q_1\) with a long upper whisker means positive skew.
The mean is pulled towards a tail or an outlier; the median is not. So for skewed data the median is usually the better measure of a typical value.
| Evidence | Shape |
|---|---|
| mean \(>\) median; median near \(Q_1\); long upper whisker | positively skewed |
| mean \(\approx\) median; median central; equal whiskers | symmetrical |
| mean \(<\) median; median near \(Q_3\); long lower whisker | negatively skewed |
These are strong guides for real data, not laws. A small data set can break them, so give reasons from the display as well.
Effect of changing every score \(x\):
| Change | Mean, median | Range, IQR | Shape |
|---|---|---|---|
| \(x + c\) | add \(c\) | unchanged | unchanged |
| \(kx\) (\(k > 0\)) | multiply by \(k\) | multiply by \(k\) | unchanged |
| \(c - x\) | \(c -\) old value | unchanged | skew reverses |
Describing the shape
- Find the peak (tallest column, longest row of leaves, or where the box is) and the tail (the side where the data thin out).
- Name the skew after the tail: tail to the right (high values) is positive; tail to the left is negative. Even spread is symmetrical; two peaks is bimodal.
- Back it up with numbers: compare the mean and median, or the two halves of the box and the two whiskers.
- Interpret: say what the shape means in context and which measure of centre is more typical.
Questions with unknown scores
- Total from the mean: total \(=\) mean \(\times\) number of scores.
- Median: for an odd number of scores, the middle one in order; it may not depend on an unknown at all.
- Mode and range: the mode must appear more often than any other score; range \(=\) largest \(-\) smallest.
- Symmetry: a frequency table is symmetrical when frequencies match at equal distances either side of the middle score.
- Test a claim with a small counterexample, such as five scores whose mean exceeds the median but whose upper whisker is short.
| Number of goals | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) |
|---|---|---|---|---|---|---|---|
| Frequency | \(1\) | \(2\) | \(4\) | \(6\) | \(9\) | \(5\) | \(3\) |
(i) Find the mean, correct to \(1\) decimal place, the median and the mode.
(ii) Describe the shape and justify it.
(i) Multiply each score by its frequency and add.
| \(\sum fx\) | \(=\) | \(0 + 2 + 8 + 18 + 36 + 25 + 18 = 107\) |
| \(\bar{x}\) | \(=\) | \(\dfrac{107}{30} = 3.566\ldots \approx \mathbf{3.6}\) |
The median is the average of the \(15\)th and \(16\)th scores. Running totals: \(1, 3, 7, 13, 22\). Both are in the column for \(4\), so the median is \(\mathbf{4}\). The mode (highest frequency, \(9\)) is also \(\mathbf{4}\).
(ii) The frequencies build up to a peak at \(4\) and the low scores trail off to the left. The mean \((3.6)\) is less than the median \((4)\): negatively skewed.
Write the running totals (cumulative frequencies): \(2,\ 7,\ 16,\ 31,\ 38,\ 40\).
(i) The median is the average of the \(20\)th and \(21\)st times. \(16 < 20\) and \(21 \le 31\), so both lie in \(\mathbf{30}\)–\(\mathbf{40}\) minutes.
(ii) \(Q_1\) is the middle of the lower \(20\): the \(10\)th and \(11\)th, both in \(\mathbf{20}\)–\(\mathbf{30}\). \(Q_3\) is the middle of the upper \(20\): the \(30\)th and \(31\)st, both in \(\mathbf{30}\)–\(\mathbf{40}\).
(iii) The tallest column is towards the right and the shorter columns stretch out to the left: negatively skewed. Most students took \(20\) to \(50\) minutes; a few were very quick.
(i) A new booking system adds \(4\) minutes to every wait. Find the new mean, median and IQR.
(ii) Instead, every time is converted to seconds. Find the new IQR.
(iii) Instead, each patient records \(60 -\) their waiting time. Describe the new shape and find the new median.
(i) Adding the same amount moves every score, and so every measure of centre, up by \(4\). The gaps between scores do not change.
| \(\text{mean}\) | \(=\) | \(23 + 4 = \mathbf{27}~\text{min}\) |
| \(\text{median}\) | \(=\) | \(19 + 4 = \mathbf{23}~\text{min}\) |
| \(\text{IQR}\) | \(=\) | \(\mathbf{8}~\text{min}\) |
(ii) Multiplying every score by \(60\) multiplies every gap by \(60\).
| \(\text{IQR}\) | \(=\) | \(8 \times 60 = \mathbf{480}~\text{s}\) |
(iii) \(60 - t\) reverses the order of the scores, so the long tail of big waits becomes a long tail of small values: negatively skewed. The middle score stays in the middle.
| \(\text{median}\) | \(=\) | \(60 - 19 = \mathbf{41}~\text{min}\) |
(i) Find the scores.
(ii) Describe the shape.
(i) Write the scores in order as \(a, b, c, d, e\).
- The median is the \(3\)rd score: \(c = 6\).
- The mode \(3\) must appear at least twice, and it is less than \(6\): \(a = b = 3\).
- Range \(9\): \(e = 3 + 9 = 12\).
- Mean \(6.4\): the total is \(5 \times 6.4 = 32\).
| \(3 + 3 + 6 + d + 12\) | \(=\) | \(32\) |
| \(d\) | \(=\) | \(8\) |
The scores are \(\mathbf{3,\ 3,\ 6,\ 8,\ 12}\). Check: \(3\) is the only repeated score, so the mode is \(3\ \checkmark\)
(ii) The mean \((6.4)\) is greater than the median \((6)\) and the scores spread further above the median (up to \(12\)) than below it (down to \(3\)): positively skewed.
Common pitfalls
Frequently asked questions
How do you tell if a distribution is positively or negatively skewed?
Look for the tail, the side where the data thin out. A tail to the right, towards larger values, is positive skew. A tail to the left is negative skew.
How do you see skewness in a box plot?
If the median is close to the lower quartile and the upper whisker is long, the data are positively skewed. If the median is close to the upper quartile and the lower whisker is long, they are negatively skewed.
Why is the mean bigger than the median in positively skewed data?
The mean uses the size of every score, so the few large values in the tail pull it up. The median is just the middle score and is not affected by how large the extreme values are.
What happens to the mean and IQR if you add a number to every score?
The mean, median and quartiles all increase by that number, but the gaps between scores do not change, so the range and IQR stay the same.
What does a bimodal distribution mean?
It has two peaks. This often means the data come from two different groups, so it can be better to describe each group separately.