Summary Statistics
Every question with a fully worked solution.
- Summary Statistics - Video - The 5 Number Summary Watch
Theory
Quartiles split ordered data into four equal parts. This Year 10A maths page shows how to find \(Q_1\), the median and \(Q_3\) for odd and even numbers of scores, write the five-number summary, find the range and interquartile range, see what an extreme score does to each, and use the IQR to test for outliers.
When data is in order, the median \(Q_2\) splits it into a lower half and an upper half. The lower quartile \(Q_1\) is the median of the lower half, and the upper quartile \(Q_3\) is the median of the upper half.
With an odd number of scores, the median is a single score: leave it out of both halves. With an even number, the data splits exactly in two.
The five-number summary is: minimum, \(Q_1\), median, \(Q_3\), maximum. These cut the data into four quarters, each holding about \(25\%\) of the scores.
Two measures of spread: the range (maximum \(-\) minimum) uses only the two extremes, while the interquartile range (IQR \(= Q_3 - Q_1\)) is the spread of the middle \(50\%\). One extreme score can change the range a lot but hardly moves the IQR.
An outlier is a score far from the rest. A common test: a score is an outlier if it is more than \(1.5 \times \text{IQR}\) below \(Q_1\) or above \(Q_3\).
Outlier limits:
| Number of scores \(n\) | Median | Halves |
|---|---|---|
| odd, e.g. \(13\) | the 7th score | 1st–6th and 8th–13th |
| even, e.g. \(12\) | halfway between 6th and 7th | 1st–6th and 7th–12th |
How to find the quartiles
- Order the scores from smallest to largest (or use running totals for a frequency table, dot plot or stem-and-leaf plot).
- Find the median. With an odd number of scores, mark it and leave it out of both halves.
- Find the middle of the lower half (\(Q_1\)) and of the upper half (\(Q_3\)). If a half has an even number of scores, average the middle two.
- Subtract: IQR \(= Q_3 - Q_1\).
How to test for outliers
- Find \(1.5 \times \text{IQR}\).
- Subtract it from \(Q_1\) and add it to \(Q_3\) to get the limits.
- Check both ends of the data: any score outside the limits is an outlier.
Order the scores first:
| \(\) | \(\) | \(8,\ 9,\ 11,\ 12,\ 13,\ 14,\ \boldsymbol{15},\) \(16,\ 17,\ 19,\ 21,\ 23,\ 25\) |
\(13\) scores: the median is the 7th, \(15\). Leave it out and split the rest into two halves of \(6\).
| \(Q_1\) | \(=\) | \(\dfrac{11 + 12}{2} = 11.5\ \text{(middle of } 8 \ldots 14)\) |
| \(Q_3\) | \(=\) | \(\dfrac{19 + 21}{2} = 20\ \text{(middle of } 16 \ldots 25)\) |
Five-number summary: \(\mathbf{8,\ 11.5,\ 15,\ 20,\ 25}\).
| \(\text{range}\) | \(=\) | \(25 - 8 = \mathbf{17}\) |
| \(\text{IQR}\) | \(=\) | \(20 - 11.5 = \mathbf{8.5}\) |
| Hours | Frequency | Positions |
|---|---|---|
| 2 | 2 | 1st–2nd |
| 3 | 3 | 3rd–5th |
| 4 | 6 | 6th–11th |
| 5 | 4 | 12th–15th |
| 6 | 3 | 16th–18th |
| 7 | 2 | 19th–20th |
Use running totals to see which score sits in each position (last column).
\(20\) scores: the median is halfway between the 10th and 11th. The lower half is the 1st to 10th, so \(Q_1\) is halfway between the 5th and 6th. The upper half is the 11th to 20th, so \(Q_3\) is halfway between the 15th and 16th.
| \(\text{median}\) | \(=\) | \(\dfrac{4 + 4}{2} = \mathbf{4}\) |
| \(Q_1\) | \(=\) | \(\dfrac{3 + 4}{2} = \mathbf{3.5}\) |
| \(Q_3\) | \(=\) | \(\dfrac{5 + 6}{2} = \mathbf{5.5}\) |
| \(\text{IQR}\) | \(=\) | \(5.5 - 3.5 = \mathbf{2}\) |
Halves of \(5\): \(31, 44, \mathbf{46}, 47, 49\) and \(50, 52, \mathbf{53}, 55, 72\).
| \(Q_1\) | \(=\) | \(46,\quad Q_3 = 53\) |
| \(\text{IQR}\) | \(=\) | \(53 - 46 = 7\) |
| \(1.5 \times \text{IQR}\) | \(=\) | \(10.5\) |
| \(\text{lower limit}\) | \(=\) | \(46 - 10.5 = 35.5\) |
| \(\text{upper limit}\) | \(=\) | \(53 + 10.5 = 63.5\) |
\(31 < 35.5\) and \(72 > 63.5\), so the outliers are 31 and 72.
(i) About how many took between \(18\) and \(27\) minutes?
(ii) About how many took longer than \(27\) minutes?
(iii) Write the IQR as a percentage of the range, to \(1\) decimal place.
Each quarter holds \(25\%\) of \(40 = 10\) students.
| \(\text{(i) } Q_1 \text{ to } Q_3\) | \(=\) | \(50\% \text{ of } 40 = \mathbf{20}\) |
| \(\text{(ii) above } Q_3\) | \(=\) | \(25\% \text{ of } 40 = \mathbf{10}\) |
| \(\text{(iii) } \dfrac{\text{IQR}}{\text{range}} \times 100\%\) | \(=\) | \(\dfrac{27 - 18}{41 - 12} \times 100\%\) |
| \(\) | \(=\) | \(\dfrac{9}{29} \times 100\% \approx \mathbf{31.0\%}\) |
Common pitfalls
Frequently asked questions
How do you find the lower and upper quartiles?
Put the data in order and find the median. The lower quartile is the median of the lower half and the upper quartile is the median of the upper half. With an odd number of scores, leave the median out of both halves.
What is the interquartile range?
It is the upper quartile minus the lower quartile, \(Q_3 - Q_1\). It measures the spread of the middle \(50\%\) of the data.
Why is the IQR better than the range?
The range depends only on the two most extreme scores, so one unusual value can change it a lot. The IQR ignores the top and bottom quarters, so it is hardly affected by outliers.
How do you work out if a number is an outlier?
Find the IQR and multiply it by \(1.5\). Subtract that from \(Q_1\) and add it to \(Q_3\). Any score below the lower limit or above the upper limit is an outlier.
What is a five-number summary?
The minimum, lower quartile, median, upper quartile and maximum. It is all you need to draw a box plot.
What happens to the IQR if every score increases by 10%?
It also increases by \(10\%\), because every quartile is multiplied by \(1.1\). Adding a fixed amount to every score leaves the IQR unchanged.