Cumulative frequency, explained with examples

By The STEM-Tutor team · Updated October 2026

A cumulative frequency graph shows a running total of grouped data. It lets you estimate the median, quartiles and percentages for data you cannot see individually, and it is a common Higher tier GCSE question.

What cumulative frequency means

Cumulative frequency is a running total. For each class, it counts how many values are less than or equal to the upper end of that class. It always increases (or stays the same), and the last cumulative frequency equals the total number of values.

Building the table

Example 1: from frequencies to cumulative frequencies

Journey times for 69 people: 0–10 minutes, 10 people; 10–15, 15; 15–20, 20; 20–30, 16; 30–50, 8. Find the cumulative frequencies.

• Keep a running total: 10, then 10 + 15 = 25, then 45, then 61, then 69.
• So by 10 minutes, 10 people have arrived; by 15 minutes, 25; by 20, 45; by 30, 61; by 50, all 69.
• The last value must equal the total, which is a quick check.

Plotting the graph

1. Plot each cumulative frequency against the upper boundary of its class: (10, 10), (15, 25), (20, 45), (30, 61), (50, 69).
2. Plot a starting point at the lower boundary of the first class with cumulative frequency 0: here (0, 0).
3. Join the points with a smooth curve or with straight lines, as your teacher prefers.

Plotting at the upper boundary matters: by the end of each class, everyone in it has been counted. Plotting at midpoints is the most common mistake in this topic.

Checking your graph

Before reading values, check three things. The curve should never go down, because a running total cannot decrease. The first point should be at zero, at the start of the first class. And the final point should be at the total frequency. If any of these fail, recheck the table before using the graph.

Reading the median and quartiles

For a total of n:

• Median: read across from n/2, then down to the horizontal axis.
• Lower quartile, Q1: read across from n/4.
• Upper quartile, Q3: read across from 3n/4.
• Interquartile range: Q3 − Q1.

Draw the reading lines on the graph in pencil; examiners often give a method mark for them.

Example 2: median and quartiles

Using the graph for the 69 journeys, estimate the median, the quartiles and the IQR.

• Median: n/2 = 34.5, which reads to about 17.4 minutes.
• Q1: n/4 ≈ 17.25, which reads to about 12.4 minutes. Q3: 3n/4 ≈ 51.75, about 24.2 minutes.
• IQR ≈ 24.2 − 12.4 = 11.8 minutes. Answers read from a hand-drawn curve may differ slightly; examiners accept a small range.

Example 3: how many below a value

Estimate how many people took less than 25 minutes.

• Read up from 25 minutes to the curve, then across: about 53 people.
• With straight lines, the value is 45 + (5/10) × 16 = 53.

Example 4: how many above a value

Estimate how many people took more than 12 minutes.

• Read up from 12: the cumulative frequency is about 16 (10 + (2/5) × 15 = 16).
• More than 12 minutes: 69 − 16 = 53 people. Questions about “more than” always need this subtraction.

Example 5: a percentage

What percentage of people took more than 30 minutes?

• By 30 minutes, 61 people had arrived, so 69 − 61 = 8 took longer.
• 8/69 ≈ 11.6%.

From cumulative frequency to a box plot

A common follow-on question asks you to draw a box plot from the graph. Use the minimum and maximum from the table (here 0 and 50), and the median and quartiles you read off (12.4, 17.4 and 24.2). The box plot then makes it easy to compare this group with another, using the medians and IQRs, as described in the box plots guide.

Percentiles

The same reading method finds any percentile. The 90th percentile is the value below which 90% of the data lies: read across from 0.9 × 69 ≈ 62 and down, which gives just over 30 minutes here. Percentiles are used for things like children's growth charts and exam grade boundaries, and a question might ask for the time that only the slowest 10% exceeded.

Comparing two groups

When two cumulative frequency curves are drawn on the same axes, the curve that is further to the right has larger values overall. For example, if the curve for bus journeys lies to the right of the curve for car journeys, bus journeys were generally longer. For a full comparison, read off both medians and both IQRs, then make two comparisons in context: one about the typical value and one about the spread.

Why estimates can differ slightly

The histogram guide estimated the median of the same journey data as 17.5 minutes, using the 35th value, while this graph gives about 17.4 using n/2 = 34.5. Both are estimates from grouped data, which assume values are spread evenly within each class. For large data sets the difference is tiny, and mark schemes accept a range of answers, so use whichever convention your exam board and teacher expect.

Reading the shape

A steep section of the graph means many values in a short interval; a flat section means few. Here the curve is steepest between 15 and 20 minutes, where most journeys lie, and very flat after 30 minutes, showing a small number of long journeys: the distribution is positively skewed.

Common mistakes

• Plotting at class midpoints instead of upper boundaries.
• Forgetting the starting point at cumulative frequency 0.
• Reading the median from the vertical axis instead of the horizontal one.
• Forgetting to subtract from the total for “more than” questions.
• Using n + 1 instead of n for grouped data; for large grouped data, n/2 is standard.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Is cumulative frequency on the Foundation paper?

No. Cumulative frequency graphs are a Higher tier GCSE topic.

Should I use a curve or straight lines?

GCSE mark schemes accept a smooth curve or straight line segments through the plotted points. Use the method your teacher has taught, and draw it neatly through the points.

Key takeaways

• Cumulative frequency is a running total.
• Plot at upper class boundaries, starting from zero.
• Read the median at n/2 and the quartiles at n/4 and 3n/4.
• For “more than”, subtract the reading from the total.
• Steep sections show where most values lie.

Related guides

• Box plots
• Histograms
• GCSE Maths topics