Cumulative Frequency
Cumulative frequency is the running total of frequencies built up through a grouped frequency table, and a cumulative frequency graph plots these running totals against the upper boundary of each class to give a smooth curve.
Before you start
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Method
- Add a cumulative frequency column to the grouped frequency table by adding each class's frequency to the running total of all classes before it.
- Plot each cumulative frequency against the upper class boundary of its class interval, not the midpoint.
- Join the plotted points with a smooth curve or straight lines, starting from a cumulative frequency of 0 at the lower boundary of the first class.
- To estimate the median, find the value halfway up the cumulative frequency axis, draw across to the curve, then down to the horizontal axis.
- To estimate the lower or upper quartile, use one quarter or three quarters of the way up the cumulative frequency axis in the same way.
- To estimate how many items are above or below a given value, draw a line up from that value to the curve, then across to the cumulative frequency axis and read off the value.
Worked example
The table shows the cumulative frequency for the times, in minutes, taken by 60 students to complete a puzzle: under 10 minutes: 8 students, under 20 minutes: 26 students, under 30 minutes: 46 students, under 40 minutes: 56 students, under 50 minutes: 60 students. Use the cumulative frequency graph to estimate the median time.
- There are 60 students in total, so the median is found at the point halfway up the cumulative frequency axis, which is 60/2 = 30.
- Draw a line across from a cumulative frequency of 30 to meet the cumulative frequency curve.
- The curve passes from 26 (at 20 minutes) to 46 (at 30 minutes), so a cumulative frequency of 30 falls a small way into this interval.
- Reading down from that point on the curve to the horizontal axis gives an estimated time of approximately 22 minutes.
- Final answer: the estimated median time is approximately 22 minutes.
Practice questions
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Q1On a cumulative frequency graph, what do you plot each cumulative frequency against?Show answer
Answer: The upper class boundary of its class interval.
Q2A frequency table has class frequencies 5, 9, 14, 7 for four consecutive classes. Find the cumulative frequency of the third class.Show answer
Answer: 28 (5 + 9 + 14 = 28).
Q3A cumulative frequency graph for 80 data values is used to estimate the median. At what cumulative frequency value do you read across to the curve?Show answer
Answer: 40 (half of 80).
Q4For 80 data values, at what cumulative frequency do you read across to find the lower quartile?Show answer
Answer: 20 (one quarter of 80).
Q5A cumulative frequency graph shows that a cumulative frequency of 45 corresponds to a value of 60 on the horizontal axis, out of 90 total data values. How many data values are above 60?Show answer
Answer: 45 (90 - 45 = 45 values are above 60).
Q6Explain why a value read from a cumulative frequency graph, such as the median, is described as an estimate rather than an exact value.Show answer
Answer: The original data was grouped into class intervals, so the exact individual values within each class are unknown; the graph only shows running totals at class boundaries.
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
120 apples are weighed and grouped by mass. The cumulative frequency table shows: under 100g: 10 apples, under 120g: 40 apples, under 140g: 85 apples, under 160g: 110 apples, under 180g: 120 apples. Use the cumulative frequency graph to estimate the median mass.
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Using the apple data in the previous question, estimate the interquartile range.
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Explain why the cumulative frequency curve should always be non-decreasing (it never goes down) as you move from left to right.
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the cumulative frequency worksheet pack - 16 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 21 of GCSE Statistics Foundation Workbook 1 and chapter 25 of GCSE Statistics Higher Workbook 1, the whole course as one free printable PDF.
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