Cumulative Frequency Curves — An Examiner's Guide

Cumulative frequency curves come up on virtually every GCSE Higher paper, usually for six to nine marks across two or three parts. They're some of the easiest marks on the paper if you know the routine, and some of the most lost if you don't. This guide walks through the full process: building the table, plotting the curve, and reading off the median, quartiles and interquartile range.

What "cumulative" means

"Cumulative" means a running total. So a cumulative frequency tells you how many data points are at or below a given value. By the end of the table, the cumulative frequency equals the total number of data points.

Step 1 — Build the cumulative frequency table

Take the standard frequency table you're given and add a third column with the running total.

Step 2 — Plot the curve

Plot the cumulative frequency at the upper end of each class interval — not the middle, not the lower end.

• For 140 < h ≤ 150 → plot at h = 150, cf = 8
• For 150 < h ≤ 160 → plot at h = 160, cf = 30
• For 160 < h ≤ 170 → plot at h = 170, cf = 58
• And so on.

Join the points with a smooth curve (or a series of straight lines — the mark scheme usually accepts either at GCSE).

Step 3 — Read off the median, quartiles and IQR

For a sample of size n, find the cumulative frequency value of the position you want, then read across to the curve and down to the x-axis.

• Median — the n/2 value. For our 80 students, that's the 40th value.
• Lower quartile (Q1) — the n/4 value. Here, the 20th.
• Upper quartile (Q3) — the 3n/4 value. Here, the 60th.
• Interquartile range (IQR) — Q3 − Q1.

For the table above, reading off the curve roughly gives: median ≈ 164 cm, Q1 ≈ 156 cm, Q3 ≈ 172 cm, so IQR ≈ 16 cm.

Estimating values above or below a point

Questions sometimes ask "estimate how many students are taller than 175 cm". Read up from x = 175 to the curve, then across to the y-axis. Subtract from the total.

The mistakes that cost the most marks