Cumulative frequency

Cumulative frequency is a running total of the frequencies in grouped data. Plot each total at the upper end of its class and join the points with a smooth curve; then read off the median, the quartiles and how many values lie above or below any value.

Key facts

  • Cumulative frequency is the running total of the frequencies.
  • Plot each cumulative frequency at the upper end of its class and join the points.
  • Median ≈ the value at n/2 on the cumulative frequency axis. Q1 is at n/4 and Q3 at (3n)/4.
  • Interquartile range = Q3 − Q1: the spread of the middle half of the data.
  • Estimated mean of grouped data = Σ(f × midpoint) ÷ Σf.

Worked examples

Example 1

The times 40 people spent in a shop were grouped: 0–10 min: 4, 10–20: 10, 20–30: 14, 30–40: 8, 40–50: 4. Estimate the median time.

  1. Cumulative frequencies: 4, 14, 28, 36, 40.
  2. Half of 40 is 20. Go across from 20 to the line, then down. median ≈ 24 minutes

Answer: About 24.3 minutes

Example 2

Estimate the mean time for the shop data in the first example.

  1. Midpoints: 5, 15, 25, 35, 45.
  2. Multiply by the frequencies and add. Σfx = 4×5 + 10×15 + 14×25 + 8×35 + 4×45 = 980
  3. Divide by the total frequency. 980 ÷ 40 = 24.5

Answer: About 24.5 minutes

Example 3

A cumulative frequency graph shows 80 values. Where do you read the median?

  1. The median is the middle value.
  2. Half of 80 is 40: read across from 40 to the curve, then down. 80 ÷ 2 = 40

Answer: At a cumulative frequency of 40

Example 4

40 people were timed. The cumulative frequency at 30 minutes is 26. How many people took longer than 30 minutes?

  1. 26 people took 30 minutes or less.
  2. Subtract from the total. 40 − 26 = 14

Answer: 14