Shapes of Distributions
Shapes of Distributions in NSW Year 10 Core maths describes how data is spread out when it is graphed (NESA outcome MA5‑DAT‑C‑01). A distribution can be symmetrical, positively skewed with a long tail to the right, negatively skewed with a tail to the left, or bimodal with two peaks.
You will describe the shape of data shown in histograms, dot plots, stem-and-leaf plots and box plots, and decide whether statements about skewness are true. You will also compare the mean and median in skewed data, such as bus delays with one very late bus, and use a five-number summary to describe the shape of test marks and fun-run times. It is part of Year 10 Core Statistics.
Every question with a fully worked solution.
- Shapes of Distributions - Video - What is Skewness? | Statistics Watch
Theory
The shape of a data set tells you where most scores sit and where the tail goes. This Year 10 maths page covers symmetrical, positively skewed and negatively skewed distributions, unimodal and bimodal data, how to read the shape from any graph or a five-number summary, and what skew does to the mean and median.
A distribution is symmetrical if its left and right halves are mirror images. The mean and median are then about equal (and, with one peak, so is the mode).
A distribution is skewed if the scores bunch up on one side and trail off in a long tail on the other. It is named after the side of the tail: tail to the right (towards high values) is positively skewed; tail to the left (towards low values) is negatively skewed.
The tail pulls the mean towards it, but hardly moves the median. So in positive skew the mean is usually greater than the median, and in negative skew it is usually less. For skewed data the median is often the better measure of centre.
The number of peaks matters too: one peak is unimodal, two is bimodal, more is multimodal. Two peaks often mean the data mixes two groups.
An outlier is a score far from the rest. It can create a tail on its own.
| Shape | Graph | Box plot | Mean vs median |
|---|---|---|---|
| Symmetrical | mirror halves | median in the middle of the box, whiskers equal | about equal |
| Positive skew | peak on the left, tail to the right | right part of box and right whisker longer | mean \(>\) median |
| Negative skew | peak on the right, tail to the left | left part of box and left whisker longer | mean \(<\) median |
From a five-number summary, compare the two sides:
How to describe the shape of a data set
- Draw or read a graph: histogram, dot plot, or a stem-and-leaf plot turned on its side.
- Find where most scores are (the peak or peaks).
- Find the tail. Tail right: positively skewed. Tail left: negatively skewed. No tail, mirror halves: symmetrical.
- Count the peaks: unimodal, bimodal or multimodal.
- Link to the centre: the mean moves towards the tail; the median stays with the bulk.
From a box plot or five-number summary
- Measure each part: left whisker, left part of the box, right part of the box, right whisker.
- Compare: the longer side is the side of the tail.
(i) Describe the shape.
(ii) Is the mean likely to be more or less than the median?
(i) The columns are tallest at \(40\)–\(50\) hours and get shorter towards the low values. The tail is on the left, so the distribution is negatively skewed. It has one peak, so it is also unimodal.
(ii) The few short battery lives in the tail pull the mean down. The mean is likely to be less than the median.
| \(\text{tail on left}\) | \(\Rightarrow\) | \(\text{negative skew}\) |
| \(\) | \(\Rightarrow\) | \(\text{mean} < \text{median}\) |
(i) Find the mean, median and mode.
(ii) Describe the shape.
(iii) Which measure best describes a typical worker?
(i) The scores are already in order.
| \(\text{mean}\) | \(=\) | \(\dfrac{130}{10} = \mathbf{13}\) |
| \(\text{median}\) | \(=\) | \(\dfrac{11 + 12}{2} = \mathbf{11.5}\) |
| \(\text{mode}\) | \(=\) | \(\mathbf{12}\) |
(ii) The \(33\) sits far above the rest and makes a tail to the right: positively skewed. The mean is pulled above the median.
(iii) Only one worker did more than \(13\) hours, so the mean overstates a typical week. The median (\(11.5\) h) is the better summary. Without the \(33\), the mean would drop to about \(10.8\) h, while the median would only move to \(11\) h.
Compare the lengths on each side of the median.
| \(\text{median} - Q_1\) | \(=\) | \(41 - 30 = 11\) |
| \(Q_3 - \text{median}\) | \(=\) | \(45 - 41 = 4\) |
| \(Q_1 - \min\) | \(=\) | \(30 - 12 = 18\) |
| \(\max - Q_3\) | \(=\) | \(48 - 45 = 3\) |
The left side is much longer: the marks are bunched at the top and spread out towards the low marks. The distribution is negatively skewed.
| Stem | Leaf |
|---|---|
| 1 | 2 5 7 |
| 2 | 0 3 4 6 8 9 |
| 3 | 1 5 |
| 4 | 2 4 |
| 5 | 0 1 3 6 7 8 |
| 6 | 2 4 |
Key: \(2 \,|\, 3 = 23\) minutes
Turn the plot on its side: each row of leaves is like a column of a histogram.
| \(\text{row lengths}\) | \(:\) | \(3,\ 6,\ 2,\ 2,\ 6,\ 2\) |
There are two separate peaks, in the \(20\)s and the \(50\)s, with a dip between them. The distribution is bimodal. It might come from two groups, such as dine-in and takeaway customers.
Common pitfalls
Frequently asked questions
What is the difference between positive and negative skew?
Positive skew has the tail on the right, towards the high values, with most scores bunched on the left. Negative skew has the tail on the left, towards the low values, with most scores bunched on the right.
Is the mean bigger than the median in positive skew?
Usually, yes. The few high scores in the right-hand tail pull the mean up, while the median stays with the middle of the data.
How do you tell the shape from a box plot?
Compare the two sides of the median. If the right part of the box and the right whisker are longer, the data is positively skewed. If the left side is longer, it is negatively skewed. If they match, it is symmetrical.
What does bimodal mean in statistics?
A bimodal distribution has two separate peaks. It often happens when the data comes from two different groups mixed together.
Which is better for skewed data, the mean or the median?
The median, because an extreme score or a long tail pulls the mean away from where most of the data lies, but barely moves the median.