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Forty students finish a school fun run. You have a table showing how many finished in each ten-minute interval. How can you tell how many finished within 30 minutes? And how can you estimate the median finish time?
Start with a running total. Once you plot those totals, the same graph can answer both questions. Drawing cumulative frequency diagrams and estimating the median, quartiles and interquartile range are part of Cambridge IGCSE Mathematics (0580) Extended. Cambridge IGCSE Mathematics syllabus, Section E9.6
Step 1. Turn the frequencies into a running total
In the table, (t) is the time taken to finish the run, in minutes.
| Finish time | Students in this interval | Cumulative frequency |
|---|---|---|
| (0 < t \leq 10) | 4 | 4 |
| (10 < t \leq 20) | 6 | 10 |
| (20 < t \leq 30) | 10 | 20 |
| (30 < t \leq 40) | 10 | 30 |
| (40 < t \leq 50) | 7 | 37 |
| (50 < t \leq 60) | 3 | 40 |
The second row shows that 6 students finished after 10 minutes and by 20 minutes. Add the 4 students who finished in the first interval, and you get 10 students who finished by 20 minutes.
Keep adding each group to the previous total. The last cumulative frequency should equal the total number of students. Here, it’s 40.
Step 2. Plot the cumulative frequency graph
Put finish time in minutes on the horizontal axis and cumulative frequency on the vertical axis.
For each row, plot the cumulative frequency against the upper end of the interval. For example, plot (20, 10) because 10 students finished in 20 minutes or less.
Plot the points (0, 0), (10, 4), (20, 10), (30, 20), (40, 30), (50, 37) and (60, 40). Mark each point clearly and join them with a smooth curve.
Step 3. Read the median and quartiles
There are 40 students, so the median is at a cumulative frequency of 20. Start at 20 on the vertical axis, move horizontally to the curve, then drop straight down to the time axis. You’ll read 30 minutes.
Use the same approach for the lower and upper quartiles.
| What you want | Start at this cumulative frequency | Read this finish time |
|---|---|---|
| Lower quartile | 40 ÷ 4 = 10 | 20 minutes |
| Median | 40 ÷ 2 = 20 | 30 minutes |
| Upper quartile | 3 × 40 ÷ 4 = 30 | 40 minutes |
Interquartile range = upper quartile − lower quartile = 40 − 20 = 20 minutes.
In this example, the middle half of finish times spans from 20 to 40 minutes. Here the quartile readings land on plotted points. When they fall between plotted points, read the estimate from your curve.
Which axis should you start from?
Look at what the question gives you.
“How many students finished within 50 minutes?” Start at 50 on the time axis, go up to the curve and then across. The answer is 37 students.
“What is the median finish time?” Start at 20 on the cumulative frequency axis, go across to the curve and then down. The answer is 30 minutes.
Try it yourself
How many students took longer than 40 minutes?
At 40 minutes, the cumulative frequency is 30. That means 30 students finished in 40 minutes or less. Subtract them from the total to find how many took longer. 40 − 30 = 10 students.
For a quick revision check, ask someone to give you one question that starts with a time and one that starts with a cumulative frequency. Point to the axis you’d use before drawing any guide lines.
Frequently asked questions
Why do I plot the upper end of the interval?
Each running total counts everyone who finished by that time. In the 10-to-20-minute interval, the total of 10 belongs at 20 minutes because it includes everyone in that interval and the one before it.
Can a cumulative frequency graph go down?
The running total can stay flat when an interval contains no observations. It can only stay the same or increase as you include more of the data.
For a broader study plan, read TheTuitionE’s IGCSE Maths revision guide for UAE students.
If your child can fill in the table but gets stuck reading the curve, bring a recent graph question to a free TheTuitionE demo class.
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