Cumulative Frequency
A cumulative frequency is a running total of frequencies as you work through a grouped frequency table. Plotting cumulative frequency against the upper boundary of each group produces a cumulative frequency graph, used to estimate the median and quartiles of grouped data.
Method
- Add a cumulative frequency column to the grouped frequency table by keeping a running total of the frequencies.
- Plot each cumulative frequency against the upper class boundary of its group, then join the points with a smooth curve or straight lines.
- To estimate the median, find n/2 (half the total frequency) on the vertical axis, read across to the curve, then down to the horizontal axis.
- To estimate the lower quartile, use n/4; for the upper quartile, use 3n/4, following the same read-across-then-down method.
- The interquartile range (IQR) is the upper quartile minus the lower quartile, and measures the spread of the middle 50% of the data.
Worked example
A cumulative frequency table for the times, in minutes, of 80 runners finishing a race gives a total frequency of 80. Explain how to estimate the median finishing time from the graph.
- Find half of the total frequency: 80 divided by 2 = 40.
- On the vertical (cumulative frequency) axis, find the value 40.
- Draw a horizontal line across from 40 until it meets the cumulative frequency curve.
- From that point on the curve, draw a vertical line down to the horizontal (time) axis.
- The value where this line meets the horizontal axis is the estimated median finishing time.
Practice questions
Try each question, then tap to reveal the answer.
Exam-style questions
Written in the style of a GCSE exam paper, with a full mark scheme.
120 students sat a maths test. Using a cumulative frequency graph, the lower quartile score was 38 and the upper quartile score was 62. Work out the interquartile range and explain what it tells you about the spread of the middle 50% of scores.
A cumulative frequency graph for the masses of 150 parcels shows a median mass of 4.2 kg. Estimate the number of parcels with a mass greater than the median.
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