Semi-interquartile range
The semi-interquartile range is half of the difference between the upper quartile and the lower quartile.
In the previous example, the quartiles were \(Q_1 = 4\) and \(Q_3 = 11\).
The semi-interquartile range is:
\(\frac{1}{2}\,(Q_3 - Q_1)\)
\(=\frac{1}{2}\,\times\,(11-4) \)
\(\frac{1}{2}\,\times \,(11-4)\)
\(=\frac{1}{2}\,\times \,7=3.5\)
Example
A survey was carried out to find the number of pets owned by each child in a class. The results are given below:
| Number of pets | Frequency |
| 0 | 3 |
| 1 | 5 |
| 2 | 2 |
| 3 | 7 |
| 4 | 10 |
| 5 | 3 |
| 6 | 1 |
| Number of pets | 0 |
|---|---|
| Frequency | 3 |
| Number of pets | 1 |
|---|---|
| Frequency | 5 |
| Number of pets | 2 |
|---|---|
| Frequency | 2 |
| Number of pets | 3 |
|---|---|
| Frequency | 7 |
| Number of pets | 4 |
|---|---|
| Frequency | 10 |
| Number of pets | 5 |
|---|---|
| Frequency | 3 |
| Number of pets | 6 |
|---|---|
| Frequency | 1 |
Find the semi-interquartile range.
Answer
Adding a cumulative frequency column is helpful for finding median and quartiles.
| Number of pets | Frequency | Cumulative Frequency |
| 0 | 3 | 3 |
| 1 | 5 | 8 |
| 2 | 2 | 10 |
| 3 | 7 | 17 |
| 4 | 10 | 27 |
| 5 | 3 | 30 |
| 6 | 1 | 31 |
| Number of pets | 0 |
|---|---|
| Frequency | 3 |
| Cumulative Frequency | 3 |
| Number of pets | 1 |
|---|---|
| Frequency | 5 |
| Cumulative Frequency | 8 |
| Number of pets | 2 |
|---|---|
| Frequency | 2 |
| Cumulative Frequency | 10 |
| Number of pets | 3 |
|---|---|
| Frequency | 7 |
| Cumulative Frequency | 17 |
| Number of pets | 4 |
|---|---|
| Frequency | 10 |
| Cumulative Frequency | 27 |
| Number of pets | 5 |
|---|---|
| Frequency | 3 |
| Cumulative Frequency | 30 |
| Number of pets | 6 |
|---|---|
| Frequency | 1 |
| Cumulative Frequency | 31 |
There are 31 children in the class.
The lower quartile is the \((31+1)\div 4\) value which is the \(8^{th}\) value, which is \(1\).
The upper quartile is the \((31+1)\div 4\,\times 3=24^{th}\) value, which is \(4\).
Therefore the semi-interquartile range is \(\frac{1}{2}\times (4-1)=1.5\)
(Sometimes you are asked for the interquartile range. If so, then you do not half the answer to \(Q_3 - Q_1\). In the above example the interquartile range is \(3\)).
