Cumulative Frequency Graphs

Turning grouped data into a running total graph

A cumulative frequency graph shows how many values are at or below each class boundary. GCSE Higher and IGCSE students use it to estimate the median, quartiles, percentiles and interquartile range. It develops earlier work with frequency tables and grouped data.

Key knowledge and step-by-step method

Add the frequencies successively to form cumulative totals. Plot each total against the upper class boundary, begin at the lower boundary with cumulative frequency zero, then join the points with a smooth increasing curve. For a total frequency of n, read the median at n/2, the lower quartile at n/4 and the upper quartile at 3n/4. The interquartile range is upper quartile minus lower quartile.

Worked examples

Example 1: Frequencies 4, 7, 9 and 5 give cumulative frequencies 4, 11, 20 and 25. Example 2: For 40 values, locate the median at cumulative frequency 20. Example 3: If Q1 is 18 and Q3 is 31, the interquartile range is 31 - 18 = 13. The middle 50% of values spans 13 units.

Common mistakes and exam advice

Do not plot ordinary frequencies, use class midpoints, or allow the curve to decrease. Draw guide lines clearly and read the scale carefully. Estimates from a graph should be reported sensibly rather than with unrealistic precision.

Practice exercises and answers

Exercise 1: Frequencies are 3, 5, 8, 4. Find cumulative frequencies. Answer: 3, 8, 16, 20. Exercise 2: There are 60 values. At which cumulative frequency is the median read? Answer: 30. Exercise 3: A graph gives Q1 = 24 and Q3 = 39. Find the interquartile range. Answer: 39 - 24 = 15.

Related learning across the website

Build a wider statistics pathway through Statistics Lessons, the glossary entries for statistics, median, range, histogram, mean, mode, bar charts, scatter graphs and probability lessons

Benefits of one-to-one maths tutoring

One-to-one lessons with maths tutor Ryan Harvey can improve cumulative totals, plotting accuracy, scale reading and comparison language through personalised exam-style practice.

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Students and parents can enquire about one-to-one maths lessons for GCSE or IGCSE support.