Measures of Spread
Every question with a fully worked solution.
- Measures of Spread - Video - Median, Quartiles and Interquartile Range Watch
Theory
The range measures how spread out a set of data is, and an outlier is a score far from the rest. This Year 9 maths page finds the range of any data, including negatives and decimals, shows how one outlier changes the mean, median and range, helps you choose the best measure of centre, and finds unknown scores and the effect of changing every score. Quartiles and the interquartile range come in Year 10.
The range is the highest score minus the lowest score. It is a measure of spread: a small range means the scores are close together and consistent, and a large range means they are spread out. The range is one number, not two.
An outlier is a score a long way from the rest of the data. It may be a mistake in recording, or a real but unusual case, such as one very long ride in a set of short ones. Look for a reason before deciding what to do with it.
An outlier affects the measures differently:
- the range changes a lot, because it uses the highest or lowest score;
- the mean is dragged towards the outlier, because it uses every score;
- the median barely moves, because it only depends on the middle;
- the mode is usually unchanged.
So for data with an outlier, the median describes a typical score better. For data that are close together with no outliers, the mean works well because it uses every score.
Changing every score. Adding the same number to every score moves the whole data set along, so the centre moves but the spread does not. Multiplying every score stretches the data, so every measure is multiplied.
| Every score is | Mean and median | Range |
|---|---|---|
| increased by \(k\) | increase by \(k\) | unchanged |
| decreased by \(k\) | decrease by \(k\) | unchanged |
| multiplied by \(k\) | multiplied by \(k\) | multiplied by \(k\) |
For example, \(3\), \(7\), \(8\), \(10\), \(12\) has a mean of \(8\) and a range of \(9\). Add \(4\) to each score and the mean is \(12\), the range still \(9\). Double each score instead and the mean is \(16\), the range \(18\).
How to deal with spread and outliers
- Sort the data, or read the lowest and highest values from the display.
- Range: highest minus lowest. Keep the signs and the units.
- Look for outliers: a score well away from the rest, often after a gap. Suggest a reason for it.
- Compare the mean, median and range with and without the outlier, if asked.
- Choose the median for data with outliers or a long tail, and the mean for data that are close together.
- Changed scores: adding or subtracting moves the mean and median only; multiplying changes all three.
The highest temperature is \(4.8\) and the lowest is \(-7.2\), the most negative.
| \(\text{range}\) | \(=\) | \(4.8 - (-7.2)\) |
| \(\) | \(=\) | \(4.8 + 7.2\) |
| \(\) | \(=\) | \(\mathbf{12.0^\circ\text{C}}\) |
Subtracting a negative is the same as adding it.
(i) Find the mean, median and range.
(ii) Find them again without the outlier.
(iii) Which measure of centre describes a typical time better?
| \(\text{mean}\) | \(=\) | \(\dfrac{260}{8} = 32.5\) |
| \(\text{median}\) | \(=\) | \(\dfrac{27 + 28}{2} = 27.5\) |
| \(\text{range}\) | \(=\) | \(74 - 21 = 53\) |
(ii) Without \(74\) there are \(7\) scores with a total of \(260 - 74 = 186\).
| \(\text{mean}\) | \(=\) | \(\dfrac{186}{7} \approx 26.6\) |
| \(\text{median}\) | \(=\) | \(27 \ (\text{the 4th score})\) |
| \(\text{range}\) | \(=\) | \(31 - 21 = 10\) |
(iii) The outlier moved the range by \(43\) and the mean by about \(6\), but the median by only \(0.5\). Only one of the eight students took longer than the mean of \(32.5\) seconds, so the median is the better summary.
Use the mean to get the total, then use the range.
| \(\text{total}\) | \(=\) | \(11 \times 5 = 55\) |
| \(x + y\) | \(=\) | \(55 - 30 = 25\) |
| \(y - x\) | \(=\) | \(13\) |
Two numbers that add to \(25\) and differ by \(13\): adding the two lines gives \(2y = 38\).
| \(y\) | \(=\) | \(\mathbf{19}\) |
| \(x\) | \(=\) | \(25 - 19 = \mathbf{6}\) |
Check: \(6\), \(8\), \(10\), \(12\), \(19\) is in order, the total is \(55\) and the range is \(19 - 6 = 13\).
Multiplying by \(3\) multiplies all three measures by \(3\). Subtracting \(2\) then lowers the mean and median by \(2\) but leaves the range alone.
| \(\text{mean}\) | \(=\) | \(15 \times 3 - 2 = \mathbf{43}\) |
| \(\text{median}\) | \(=\) | \(14 \times 3 - 2 = \mathbf{40}\) |
| \(\text{range}\) | \(=\) | \(8 \times 3 = \mathbf{24}\) |
The order matters: do the steps in the order they are given.
Common pitfalls
Frequently asked questions
How do you find the range?
Subtract the lowest score from the highest score. For \(4\), \(9\), \(2\), \(15\) the range is \(15 - 2 = 13\).
What is an outlier?
A score that is a long way from the rest of the data. It may be a recording mistake or a real but unusual result.
How does an outlier affect the mean, median and range?
It changes the range a lot and drags the mean towards it, but it barely moves the median. The mode is usually unchanged.
Should I use the mean or the median when there is an outlier?
Use the median. The mean is pulled towards the outlier, so it is not typical of most of the scores.
What happens to the range if you add the same number to every score?
Nothing. Every score moves by the same amount, so the highest and lowest are still the same distance apart. The mean and median increase by that number.