Quartiles, box plots and comparing distributions

Quartiles split ordered data into four equal parts. The interquartile range measures spread without being upset by extreme values.

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Quartiles from a list

Order the data. With n values:
lower quartile (LQ) = the (n + 14)th value
median = the (n + 12)th value
upper quartile (UQ) = the 3(n + 1)4th value
Data: 12, 15, 17, 18, 20, 21, 23, 25, 26, 30, 41 (n = 11)
LQ = 17, median = 21, UQ = 26, so IQR = 26 − 17 = 9.

Box plots and outliers

A box plot shows the minimum, LQ, median, UQ and maximum on a scale.
An outlier is a value far from the rest. A common rule, given in the question when it is needed, is: more than 1.5 × IQR above the UQ or below the LQ.
Here 26 + 1.5 × 9 = 39.5, so 41 is an outlier.
min 12 LQ 17 median 21 UQ 26 30 outlier 41 39.5 10 15 20 25 30 35 40 45
Box plot with an outlier: 41 is more than 1.5 × IQR above the UQ.

Comparing two distributions

Always compare an average (median) and a spread (IQR or range), and say what each means in context.
'Group A's median is higher, so on average they scored more. Group B's IQR is smaller, so their scores were more consistent.'
The IQR is better than the range when there are outliers.
Worked example

Find the interquartile range of 12, 15, 17, 18, 20, 21, 23, 25, 26, 30, 41.

  1. n = 11: LQ is the 3rd value, UQ is the 9th value
  2. LQ = 17, UQ = 26
  3. IQR = 26 − 17 = 9

Answer: 9

Key idea

Order the data. LQ, median and UQ sit at (n+1)/4, (n+1)/2 and 3(n+1)/4. IQR = UQ − LQ. Compare an average and a spread, in context.

Key words
Quartiles
The values a quarter (lower quartile) and three quarters (upper quartile) of the way through ordered data.
Range
The largest value minus the smallest value. It measures spread.
Median
The middle value when the data are in order.
Box plot
A diagram showing the minimum, lower quartile, median, upper quartile and maximum of a data set.

All GCSE and International GCSE Mathematics key terms and definitions →

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Extra practice

2 more questions

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  1. Find the median of 7, 10, 12, 16, 19, 33, 34, 40, 42, 50, 55.

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    Answer: 33

    The 6th value is 33.

  2. Find the lower quartile of 6, 7, 15, 28, 43, 44, 48, 50, 51, 53, 58.

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    Answer: 15

    There are 11 values. The (n + 1)/4th = 3rd value is 15.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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