Statistics

Cumulative frequency

Connect grouped counts, medians and quartiles.

Predict, test and explain

Step 1 of 3

Predict

The four class frequencies are 5, 10, 5 and 5. At what cumulative count do you read the median?

N=40

Scroll the diagram sideways to see every label.

\text{target frequency}=20

Q_2\approx 20

Counts are plotted at upper class boundaries. Straight segments estimate an even spread within each class; the quartiles and median are estimates, not exact raw-data values.

Grouped frequencies
Class	Frequency	Cumulative count
0 ≤ x < 10	5	5
10 ≤ x < 20	15	20
20 ≤ x < 30	12	32
30 ≤ x < 40	8	40

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

A cumulative-frequency graph represents 60 values. At counts 15, 30 and 45 it reads 8, 12 and 20. Estimate the median and interquartile range.

$N=60$

Hint

The median is at count 30; subtract the lower quartile from the upper quartile.

Reveal answer

The estimated median is 12 and the estimated interquartile range is 12.

$\text{median}\approx12,\quad \mathrm{IQR}\approx20-8=12$

Build an explanation

Hint 1

Cumulative frequency is a running total. For grouped data, plot each total against its upper class boundary.

Hint 2

For N values, read the lower quartile at N/4, the median at N/2 and the upper quartile at 3N/4 on the cumulative-frequency axis.

Hint 3

Move across to the curve and then down to the value axis. The interquartile range is upper quartile minus lower quartile; readings from grouped data are estimates.

Worked example

This example uses fixed values, separate from the diagram controls.

A cumulative-frequency graph represents 40 values. At cumulative frequencies 10, 20 and 30, the curve gives values 12, 18 and 24. Estimate the median and interquartile range.

The quartile positions are 40/4 = 10, 40/2 = 20 and 3 × 40/4 = 30.
Read the corresponding values: lower quartile 12, median 18 and upper quartile 24.
Subtract the quartiles: 24 − 12 = 12.

Answer: Estimated median 18; estimated interquartile range 12.

Connect this idea

Watch out: The cumulative-frequency axis gives a position in the data. Read across and down to find the quartile value; do not report N/4 as the lower quartile itself.