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Inter-quartile range, cumulative frequency, box and whisker plots - Higher

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# Interquartile range

We know that the median divides the data into two halves. We also know that for a set of n ordered numbers the median is the (n + 1) ÷ 2 th value.

Similarly, the lower quartile divides the bottom half of the data into two halves, and the upper quartile also divides the upper half of the data into two halves.

Lower quartile is the (n + 1) ÷ 4 th value.

Upper quartile is the 3 (n + 1) ÷ 4 th value.

Question

Find the median, lower quartile and upper quartile for the following data:

11, 4, 6, 8, 3, 10, 8, 10, 4, 12 and 31.

Ordering the data, we get 3, 4, 4, 6, 8, 8,10, 10, 11, 12 and 31.

The median is the (11 + 1) ÷ 2 = 6th value.

The lower quartile is the (11 + 1) ÷ 4 = 3rd value.

The upper quartile is the 3 (11 + 1) ÷ 4 = 9th value.

Therefore, the median is 8, the lower quartile is 4, and the upper quartile is 11.

3, 4, 4, 6, 8, 8, 10, 10, 11, 12, 31

The interquartile range is the difference between the upper quartile and lower quartile.

In this example, the interquartile range is 11 - 4 = 7.

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 petsFrequency
03
15
22
37
410
53
61
>60

Find the interquartile range.

3.

Remember that there is a total of 31 children in the class.

• The lower quartile is the 8th value, which is 1.
• The upper quartile is the 24th value, which is 4.
• Therefore, the interquartile range is 4 - 1 = 3.

Note that the interquartile range ignores extreme values. The range includes extreme values.

Look at this set of data:

• 1, 5, 7, 8, 9, 12, 13, 15, 17, 18, 35.
• The interquartile range is 17 - 7 = 10.
• The range is 35 - 1 = 34.

In cases such as these, it is often preferable to use the interquartile range when comparing the data.

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