9.6 Cumulative Frequency Diagrams

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Draw and interpret cumulative frequency tables and diagrams.
  2. Estimate and interpret the median, percentiles, quartiles, and interquartile range from cumulative frequency diagrams.
  3. Plot points clearly (using small crosses ×) and join them with a smooth curve.

Part 1: What Is Cumulative Frequency?

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Frequency simply means how many times something occurs. For example, if 9 customers bought between 0 and 10 litres of petrol, the frequency for that group is 9.

Cumulative frequency means a running total — you keep adding up the frequencies as you go from one group to the next. Think of it like counting how many people have crossed a finish line so far, not just in one minute, but all the way up to that point.

Example — Building a Cumulative Frequency Table

A garage records how much petrol (in litres) 120 customers bought. The results are shown below:

Petrol (litres)Number of customers (frequency)
0 < k ≤ 109
10 < k ≤ 2013
20 < k ≤ 3036
30 < k ≤ 4030
40 < k ≤ 5016
50 < k ≤ 609
60 < k ≤ 705
70 < k ≤ 802

To build the cumulative frequency table, you add up all the frequencies as you go along:

Petrol (k litres)Cumulative FrequencyHow we got it
k ≤ 1099
k ≤ 20229 + 13 = 22
k ≤ 305822 + 36 = 58
k ≤ 408858 + 30 = 88
k ≤ 5010488 + 16 = 104
k ≤ 60113104 + 9 = 113
k ≤ 70118113 + 5 = 118
k ≤ 80120118 + 2 = 120

Notice: The last cumulative frequency value always equals the total number of data values (120 in this case). This is a great way to check your work!

Part 2: Drawing a Cumulative Frequency Diagram

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A cumulative frequency diagram (also called a cumulative frequency curve) is a graph that shows the running total of data. It always produces an S-shaped curve — it rises steeply in the middle and flattens out at both ends.

Step-by-Step Guide to Drawing the Diagram

Step 1 — Set up your axes.

  • The horizontal axis (x-axis) shows the variable being measured (e.g., litres of petrol, time in seconds).
  • The vertical axis (y-axis) shows the cumulative frequency (the running total).
  • Make sure your axes are evenly scaled and clearly labelled with units.

Step 2 — Identify your plotting points.

  • Each point is plotted at the upper boundary (the end value) of each class interval and its corresponding cumulative frequency.
  • For the petrol example, your points would be: (10, 9), (20, 22), (30, 58), (40, 88), (50, 104), (60, 113), (70, 118), (80, 120).

Step 3 — Mark each point clearly with a small cross (×).

  • The syllabus specifically states that plotted points must be clearly marked — small crosses (×) are the standard and expected way to do this.

Step 4 — Join the points with a smooth curve.

  • Do not join the points with straight lines — you must draw one flowing, smooth S-shaped curve through all the crosses.
  • The curve should never go back downwards (because a running total can only stay the same or increase — it never decreases).

Step 5 — Add a starting point at zero.

  • At the lower boundary of the very first class interval, plot a point with a cumulative frequency of 0. This is where the curve begins.
  • For the petrol data, this starting point is (0, 0).

What the Completed Diagram Looks Like

Imagine a graph where:

  • The x-axis goes from 0 to 80 (litres).
  • The y-axis goes from 0 to 120 (cumulative frequency).
  • A smooth S-shaped curve starts at (0, 0), climbs steeply through the middle (near 30–40 litres), then flattens as it approaches (80, 120).

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