Interquartile range
We know that for a set of ordered numbers, the median \({Q_2}\), is the middle number which divides the data into two halves.
Similarly, the lower quartile \({Q_2}\) divides the bottom half of the data into two halves, and the upper quartile \({Q_3}\) also divides the upper half of the data into two halves.
The The interquartile range is the difference between the upper quartile and the lower quartile. is the difference between the upper quartile and lower quartile.
To calculate the interquartile range (IQR):
\(IQR = {Q_3} - {Q_1}\)
Example
Find the median, lower quartile and upper quartile for the following data:
\(11,\,4,\,6,\,8,\,3,\,10,\,8,\,10,\,4,\,12\,and\,31\)
In order to find the median, we need to put the numbers in order first.
\(3,\,4,\,4,\,6,\,8,\,8,\,10,\,10,\,11,\,12,\,31\)
\(Median({Q_2}) = 8\)
\(Lower\,Quartile({Q_1}) = 4\)
\(Upper\,Quartile({Q_3}) = 11\)
Therefore the IQR = \({Q_3} - {Q_1} = 11 - 4 = 7\)
Look at this set of data:
\(1,\,5,\,7,\,8,\,9,\,12,\,13,\,15,\,17,\,18,\,35\)
The interquartile range is \(17 - 7 = 10\).
(Sometimes we are asked for the semi-interquartile range . This is half of the interquartile range and in this case would be 5)
Now try the example questions below.
Question
Find the median and interquartile range for the following data:
\(13,\,18,\,15,\,11,\,6,\,9,\,23,\,8,\,17,\,21,\,6\)
Arranging the numbers in order we get:
\(6,\,6,\,8,\,9,\,11,\,13,\,15,\,17,\,18,\,21,\,23\)
Median = 13
Q1 = 8
Q3 = 18
Interquartile range = 18 鈥 8 = 10.
Question
A survey was carried out to find the number of pets owned by each child in a class.
The results are shown in the table:
| 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 interquartile range.
There are 31 children in the class.
The median is the 16th value.
By using cumulative frequency this is in the 3 pets row. (3+5+2+7 = 17).
Median =3.
The lower quartile is the 8th value.
By using cumulative frequency this is in the 1 pet row (3+5+2=8).
Lower Quartile = 1.
The upper quartile is the 24th value.
By using cumulative frequency this is in the 4 pets row. (3+5+2+7+10= 27).
Upper Quartile = 4.
Therefore, the interquartile range is 4 - 1 = 3.
Question
In a different class, their interquartile range was 9. What does this tell you about the results from this class?
There was more variation in the number of pets owned by the children in this class.