Cumulative Frequency

Cumulative frequency is a running total. Plot it against the upper class boundaries and join the points to get an S-shaped curve. Read across at $\tfrac n2$ for the median, and at $\tfrac n4$ and $\tfrac{3n}4$ for the quartiles. The interquartile range is the spread of the middle half.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Build a cumulative frequency table from a frequency table.
Plot cumulative frequency against upper class boundaries.
Estimate the median, quartiles, interquartile range and percentiles.
Find how many values lie above or below a given value.

Pause and think

$200$ runners finish a race. You want the time that splits the fastest quarter from the rest. Which runner's time do you look at?

Section 2 of 14

Core idea

Start hereWhere is the middle?

With 80 plant heights in a grouped table, you can't see the middle plant. But a running total graph climbs from 0 to 80; reading across at 40 finds the median.

The big idea

Plot running totals at upper class boundaries; join with a smooth curve. Median at $\frac n2$ ; quartiles at $\frac n4$ , $\frac{3n}4$ ; IQR $=Q_3-Q_1$ .

1. Reading the graph

Heights of 80 plants

The S-shaped curve rises steeply where most values are.

Worked example80 plants

Find the median, IQR, and the number above a value $h$ .

Goal: Readings.
Strategy: Read across, then down.

$n/2=40$
Read across at 40 to the curve, down to the median.
$20,\ 60$
Quartile positions; IQR $=Q_3-Q_1$ .
$80-\text{cf}(h)$
Read cf at $h$ ; subtract from total.

AnswerMedian, IQR and "more than" from the curve.

✓ CheckThe curve ends at 80 ✓.

Common misconceptionPlotting at midpoints Use the upper boundary: only then has the whole class been counted.

Pause and think

Why is IQR better than range here?

Summary

Upper boundaries; S-curve.
Median $n/2$ ; IQR from $n/4$ , $3n/4$ .

Cumulative frequency: Running total.
Interquartile range: $Q_3-Q_1$ .

HistoryThe ogive. Francis Galton, a Victorian scientist, popularised the cumulative frequency curve in the 1870s and called it an "ogive", after the curved arches in Gothic cathedrals. He also introduced the words "median" and "quartile" in their statistical sense.

Fun factGrowth charts. When a doctor says a baby is "on the

75

th percentile" for weight, it means

75\%

of babies of the same age weigh less. Those growth charts are built from cumulative frequencies of thousands of measurements.

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