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Cumulative frequency: tables, graphs, median and quartiles

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Cumulative frequency is a running total of frequencies. Plotting it as a graph lets you estimate the median, the quartiles and how many values lie above or below any point, even when you only have grouped data. It's a higher-tier GCSE topic and part of IGCSE Extended, and it leads straight into box plots.

Key takeaways

  • Add up the frequencies as you go to get the running total.
  • Plot each total at the upper end of its class.
  • Median at n ÷ 2, lower quartile at n ÷ 4, upper quartile at 3n ÷ 4.
  • Interquartile range = upper quartile − lower quartile.

Step 1: the table

The time 50 students took to finish a task:

Time, tt (minutes)FrequencyCumulative frequency
0<t≤100 < t \le 1044
10<t≤2010 < t \le 201014
20<t≤3020 < t \le 301832
30<t≤4030 < t \le 401244
40<t≤5040 < t \le 50650

The last cumulative frequency must equal the total: 50.

Step 2: the graph

Plot each cumulative frequency at the upper class boundary: (10,4)(10, 4), (20,14)(20, 14), (30,32)(30, 32), (40,44)(40, 44), (50,50)(50, 50). Also plot (0,0)(0, 0), because no one took less than 0 minutes. Join the points with a smooth curve (or straight lines if the question says so). The curve is usually an S shape.

Plotting at the midpoint is the most common mistake: by the end of each class, all of its values have been counted.

Step 3: median and quartiles

With n=50n = 50:

MeasurePositionRead across fromEstimate
Lower quartile50÷4=12.550 \div 4 = 12.512.5about 18.5 minutes
Median50÷2=2550 \div 2 = 2525about 26 minutes
Upper quartile3×50÷4=37.53 \times 50 \div 4 = 37.537.5about 34.5 minutes

Draw a horizontal line from each position to the curve, then down to the time axis. The interquartile range is about 34.5−18.5=1634.5 - 18.5 = 16 minutes. The exact readings depend on how the curve is drawn, so mark schemes accept a small range of answers.

Step 4: values above or below a point

To estimate how many students took more than 35 minutes, draw up from 35 on the time axis to the curve and read across: about 38. So about 50−38=1250 - 38 = 12 students took longer than 35 minutes.

To estimate a percentage, divide by the total: 12÷50=24%12 \div 50 = 24\%.

Why the interquartile range matters

The interquartile range (IQR) measures the spread of the middle half of the data. Unlike the range, it isn't affected by one very high or very low value, so it's better for comparing groups. When comparing, mention both an average (the median) and the spread (the IQR), in context: "The median time was higher for group B, so they were slower on average; group A had a smaller IQR, so their times were more consistent."

Common mistakes

  • Plotting at the midpoint or the start of each class instead of the upper bound.
  • Forgetting the starting point at zero.
  • Reading the median at the middle of the time axis instead of at n÷2n \div 2 on the frequency axis.
  • Answering "more than" questions with the reading itself, instead of subtracting it from the total.

Practice questions

  1. Frequencies are 6, 15, 20, 9. Write the cumulative frequencies.
  2. For 80 values, at what cumulative frequencies do you read the quartiles and median?
  3. The quartiles are 42 and 57. Find the IQR.

Answers: 1. 6, 21, 41, 50. 2. Lower quartile 20, median 40, upper quartile 60. 3. 15.

More practice in our cumulative frequency questions, or see histograms for another way to show grouped data.

Frequently asked questions

How do you work out cumulative frequency?

Add each frequency to the total of the ones before it. The final cumulative frequency equals the total number of values.

Where do you plot points on a cumulative frequency graph?

At the upper class boundary of each class, because by then all the values in the class have been counted.

How do you find the median from a cumulative frequency graph?

Divide the total frequency by 2, read across from that value to the curve, then read down to the horizontal axis.

Sources

  1. GCSE mathematics: subject content and assessment objectives. Department for Education.
  2. Cambridge IGCSE Mathematics 0580 syllabus. Cambridge International.

Keep reading

Cumulative frequency questions and answers

8 GCSE and IGCSE questions with worked answers, checked as you type.

Box plots: how to draw, read and compare them

The five values, finding quartiles, drawing the plot, outliers and comparing two groups.

Histograms: frequency density and unequal class widths

Frequency density, drawing and reading histograms, and estimating from part of a class.

Mean, median, mode and range explained (with examples)

How to find each measure, which average to use, and how to compare data sets in exam answers.