4.5.1 Calculating The Range And Interquartile Range

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Find the quartiles of this data set: 6, 47, 49, 15, 43, 41, 7, 39, 43, 41, 36.

You first need to arrange the data points in increasing order. As you do so, you can give them a rank to indicate their position in the data set. Rank 1 is the data point with the smallest value, rank 2 is the data point with the second-lowest value, etc.

 Table 4.5.1.1 Rank of data points Table summary This table displays the results of Rank of data points. The information is grouped by Rank (appearing as row headers), Value (appearing as column headers).
Rank Value
1 6
2 7
3 15
4 36
5 39
6 41
7 41
8 43
9 43
10 47
11 49

Then you need to find the rank of the median to split the data set in two. As we have seen in the section on the median, if the number of data points is an uneven value, the rank of the median will be

(n + 1) ÷ 2 = (11 + 1) ÷ 2 = 6

The rank of the median is 6, which means there are five points on each side.

Then you need to split the lower half of the data in two again to find the lower quartile. The lower quartile will be the point of rank (5 + 1) ÷ 2 = 3. The result is Q1 = 15. The second half must also be split in two to find the value of the upper quartile. The rank of the upper quartile will be 6 + 3 = 9. So Q3 = 43.

Once you have the quartiles, you can easily measure the spread. The interquartile range will be Q3 - Q1, which gives 28 (43-15). The semi-interquartile range is 14 (28 ÷ 2) and the range is 43 (49-6).

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