Grouped data and interpolation

The idea

With grouped data you only know how many values sit in each class, so the mean uses midpoints and the median and quartiles are estimated by linear interpolation inside a class.

Estimated mean: $\frac{\sum fx}{\sum f}$ with $x$ the class midpoint.
For the median, find the class holding the $\frac{n}{2}$ th value.
Interpolate: lower boundary $+ \frac{\text{position} - \text{cumulative before}}{\text{class frequency}} \times \text{width}$ .

Watch out Interpolated values are estimates: the data inside a class is assumed to be spread evenly.

See it worked

Worked example

Classes $0$ – $10$ , $10$ – $20$ , $20$ – $30$ have frequencies $8$ , $14$ , $18$ . Estimate the median.

$n = 40$ , so the median is the $20$ th value.
Cumulative: $8$ , $22$ , $40$ , so it is in $10$ – $20$ , which starts after $8$ values.
$10 + \frac{20 - 8}{14} \times 10 = 18.6$ (3 s.f.).

Answer $18.6$

Your turn

Question 1Non-calculator3 marks

For a grouped data set, $\sum f = 40$ and $\sum fx = 1240$ . Estimate the mean.

Question 2Non-calculator3 marks

Classes $0$ – $20$ , $20$ – $40$ , $40$ – $60$ have frequencies $12$ , $20$ , $8$ . Use interpolation to estimate the median.

Question 3Calculator allowed3 marks

For the same data ( $12$ , $20$ , $8$ in classes $0$ – $20$ , $20$ – $40$ , $40$ – $60$ ), estimate the lower quartile.

In past papers

21 official questions in 17 papers since 2018. The latest:

All A-level Maths past papers

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