What is cumulative frequency?
Cumulative frequency is a running total of the frequencies as you work through a frequency table. Plotting cumulative frequency against the upper boundary of each class gives an S-shaped curve, which can be used to estimate the median, quartiles and interquartile range of a set of grouped data.
Key facts:
- Cumulative frequency points are plotted at the upper class boundary of each interval, not the midpoint
- The final cumulative frequency always equals the total frequency, n
- The median is estimated at cumulative frequency n/2
- The lower quartile (Q1) is estimated at n/4, and the upper quartile (Q3) at 3n/4
- The interquartile range (IQR) = Q3 − Q1 measures the spread of the middle 50% of the data
Building a cumulative frequency table
Worked Example 1
The table shows the time taken by 80 runners to complete a fun run. Complete the cumulative frequency table.
| Time (minutes) | 20–30 | 30–40 | 40–50 | 50–60 | 60–70 |
|---|---|---|---|---|---|
| Frequency | 8 | 16 | 24 | 20 | 12 |
| Cumulative frequency | 8 | 24 | 48 | 68 | 80 |
1
Running total: 8, then 8 + 16 = 24, then 24 + 24 = 48, then 48 + 20 = 68, then 68 + 12 = 80
2
The final cumulative frequency (80) matches the total number of runners — a useful check
Answer8, 24, 48, 68, 80
Plot cumulative frequency against the upper class boundary. Draw a horizontal line at n/4, n/2 and 3n/4, then read down to the axis to estimate Q1, the median and Q3.
Reading the median from the graph
Worked Example 2
Using the cumulative frequency graph above (n = 80), estimate the median time.
1
The median is at cumulative frequency n/2 = 80 ÷ 2 = 40
2
Draw a horizontal line across at cumulative frequency 40 until it meets the curve, then drop straight down to the time axis
3
This gives a median of approximately 47 minutes
Answer≈ 47 minutes
Quartiles and the interquartile range
Worked Example 3
Using the same graph, estimate the lower quartile (Q1) and upper quartile (Q3), then find the interquartile range.
1
Q1 is at cumulative frequency n/4 = 80 ÷ 4 = 20, giving Q1 ≈ 37.5 minutes
2
Q3 is at cumulative frequency 3n/4 = 60, giving Q3 ≈ 56 minutes
3
IQR = Q3 − Q1 = 56 − 37.5 = 18.5 minutes
AnswerQ1 ≈ 37.5, Q3 ≈ 56, IQR ≈ 18.5 minutes
Comparing distributions
Worked Example 4 — Comparing two groups
A different group of 80 runners (Group B) has a median time of 52 minutes and an IQR of 10 minutes. Compare Group B's performance and consistency with the runners in Worked Example 2 and 3 (Group A: median ≈ 47 minutes, IQR ≈ 18.5 minutes).
1
Compare the medians: Group A's median (47 minutes) is lower than Group B's (52 minutes), so Group A tended to run faster on average
2
Compare the IQRs: Group A's IQR (18.5) is larger than Group B's (10), so Group A's times were more spread out — Group B was more consistent
AnswerGroup A was faster on average but less consistent than Group B
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1A cumulative frequency table has cumulative frequencies 5, 12, 20, 27, 30 for five class intervals. What is the total number of data values?Foundation
Q2In a cumulative frequency table, the cumulative frequency up to 10 is 5. The frequency for the next interval, 10–20, is 8. Find the cumulative frequency up to 20.Foundation
Q3A cumulative frequency graph is plotted for 60 data values. At what cumulative frequency should you draw a horizontal line to estimate the median?Foundation
Q4A cumulative frequency graph is plotted for 100 data values. At what cumulative frequency should you draw a horizontal line to find the upper quartile, Q3?Foundation
Q5True or false: cumulative frequency points are plotted at the midpoint of each class interval. Explain your answer.Foundation
Higher (Grade 5–7)
Q6
The table shows waiting times for 40 patients at a clinic. Find the cumulative frequency for waiting times up to 30 minutes.
Higher| Time (min) | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
| Frequency | 5 | 9 | 14 | 8 | 4 |
Q7Using the data from Q6, use linear interpolation to estimate the median waiting time.Higher
Q8Using the data from Q6, use linear interpolation to estimate the lower quartile, Q1.Higher
Q9Using the data from Q6, use linear interpolation to estimate the upper quartile, Q3.Higher
Q10Using your answers to Q8 and Q9, find the interquartile range for the clinic waiting times.Higher
Higher — Hard (Grade 8–9)
Q11A cumulative frequency graph for 200 students' test scores shows Q1 = 42, median = 58, Q3 = 71. Find the interquartile range, and explain what a small IQR would mean compared to a large one.Grade 8–9
Q12Two classes sit the same test. Class A: median = 65, IQR = 12. Class B: median = 61, IQR = 24. Compare the two classes' performance and consistency.Grade 8–9
Q13A cumulative frequency graph is drawn for 150 values. State the cumulative frequency needed to find: (a) the lower quartile, (b) the median, (c) the upper quartile.Grade 8–9
Q14Explain why estimates read from a cumulative frequency graph, such as the median or IQR, are described as "estimates" rather than exact values.Grade 8–9
Q15A cumulative frequency graph for 60 runners' race times rises steeply between 40 and 50 minutes, but is almost flat between 50 and 60 minutes. Explain what this tells you about the distribution of race times in these two intervals.Grade 8–9
Answers
Foundation (Q1–Q5)
Q130(the final cumulative frequency = the total)
Q213(5 + 8)
Q330(n/2 = 60 ÷ 2)
Q475(3n/4 = 3 × 100 ÷ 4)
Q5False(points are plotted at the upper class boundary, not the midpoint)
Higher (Q6–Q10)
Q628(5 + 9 + 14)
Q7≈ 24.3 minutes(n/2 = 20, falls in 20–30 class: 20 + (20−14)/(28−14) × 10)
Q8≈ 15.6 minutes(n/4 = 10, falls in 10–20 class: 10 + (10−5)/(14−5) × 10)
Q932.5 minutes(3n/4 = 30, falls in 30–40 class: 30 + (30−28)/(36−28) × 10)
Q10≈ 16.9 minutes(32.5 − 15.6)
Higher — Hard (Q11–Q15)
Q11IQR = 29(71 − 42). A small IQR means the middle 50% of scores are close together (consistent); a large IQR means they are more spread out
Q12Class A performed better on average (higher median, 65 vs 61) and was more consistent (smaller IQR, 12 vs 24) than Class B
Q13(a) 37.5 (b) 75 (c) 112.5(150/4, 150/2, 3×150/4)
Q14Because the graph is drawn from grouped data — the exact individual values within each class interval are unknown, and the graph assumes values increase steadily across each class, which may not exactly match the true distribution
Q15A steep section of the curve means many runners finished within that time range (high frequency), while an almost flat section means very few runners finished in that range (low frequency) — most runners finished between 40 and 50 minutes rather than 50 and 60
Common mistakes
Common Mistake 1
Plotting at the midpoint instead of the upper class boundary
Unlike an estimated mean (which uses midpoints), a cumulative frequency graph is plotted using the upper class boundary of each interval. For the class 20–30, plot the cumulative frequency at 30, not 25.
Common Mistake 2
Using n instead of n/2, n/4 or 3n/4
A common error is drawing the horizontal line at the total frequency, n, when finding the median. Remember: median uses n/2, the lower quartile uses n/4, and the upper quartile uses 3n/4.
Common Mistake 3
Forgetting to include the starting point at zero
The cumulative frequency curve should start at (lower boundary of the first class, 0). Omitting this point means the curve doesn't correctly represent that no data lies below the first class.
Common Mistake 4
Confusing IQR with range
The range uses the maximum and minimum values (highly affected by outliers). The interquartile range uses Q3 − Q1, focusing on the middle 50% of the data and ignoring extreme values. Don't mix the two up.
Exam tips
💡 Exam Tip 1
Draw your lines clearly on the graph
Always draw the horizontal and vertical construction lines used to read off the median or quartiles — examiners award method marks for showing where your estimate came from, even if the final reading is slightly off.
💡 Exam Tip 2
Join points with a smooth curve, not straight lines
A cumulative frequency graph is usually drawn as a smooth S-shaped curve through the points, rather than joining them with straight line segments — check your exam board's expectation, but a curve is the standard convention.
💡 Exam Tip 3
Use "estimate" in your final answer
Since values are read from a graph built on grouped data, always write "estimated median" or "estimated IQR" rather than stating the value as if it were exact.
💡 Exam Tip 4
Comment on both average and spread when comparing
A "compare the distributions" question almost always wants two comments: one about the median (which group did better/was higher on average) and one about the IQR (which group was more consistent). Giving only one loses marks.
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