What Cumulative Frequency Questions Test
Cumulative frequency questions are a staple of IGCSE Maths statistics. They appear on both Core and Extended papers and typically carry five to eight marks. The questions test your ability to construct a cumulative frequency table, draw a smooth curve, and read off key statistical measures such as the median, quartiles, and interquartile range.
These questions are among the most accessible on the paper because they follow a rigid structure. If you learn the method, you can collect every mark.
The Problem
The table below shows the heights of 120 students:
| Height (h cm) | Frequency |
|---|---|
| 140 < h ≤ 150 | 8 |
| 150 < h ≤ 155 | 15 |
| 155 < h ≤ 160 | 24 |
| 160 < h ≤ 165 | 32 |
| 165 < h ≤ 170 | 25 |
| 170 < h ≤ 175 | 12 |
| 175 < h ≤ 185 | 4 |
(a) Complete the cumulative frequency table. (b) Draw the cumulative frequency curve. (c) Find the median height. (d) Find the interquartile range. (e) Find the number of students taller than 172 cm.
Step 1: Build the Cumulative Frequency Table
Cumulative frequency means a running total. You add each frequency to the total of all previous frequencies:
| Height (h cm) | Frequency | Cumulative Frequency |
|---|---|---|
| h ≤ 150 | 8 | 8 |
| h ≤ 155 | 15 | 23 |
| h ≤ 160 | 24 | 47 |
| h ≤ 165 | 32 | 79 |
| h ≤ 170 | 25 | 104 |
| h ≤ 175 | 12 | 116 |
| h ≤ 185 | 4 | 120 |
The last cumulative frequency should equal the total number of students. Check: 8 + 15 + 24 + 32 + 25 + 12 + 4 = 120. Correct.
Important: The cumulative frequency is plotted against the upper class boundary, not the midpoint. So the point (150, 8) means 8 students have height ≤ 150 cm.
Step 2: Plot the Cumulative Frequency Curve
Plot these points on a graph:
- (150, 8)
- (155, 23)
- (160, 47)
- (165, 79)
- (170, 104)
- (175, 116)
- (185, 120)
You should also plot the starting point (140, 0) because no students have a height of 140 cm or less.
Join the points with a smooth S-shaped curve (an ogive), not straight lines. The curve should be smooth and continuous. On the exam, use a sharp pencil and take your time — marks are awarded for the quality of the curve.
Step 3: Find the Median
The median is the value at the halfway point. With 120 students, the median corresponds to the 60th student (120 ÷ 2 = 60).
On your cumulative frequency curve:
- Go across from 60 on the y-axis (cumulative frequency)
- Draw a horizontal line until it meets the curve
- Drop a vertical line down to the x-axis
- Read off the value
From our data, the 60th student falls in the 160-165 range. Reading from the curve, the median is approximately 162 cm.
Step 4: Find the Quartiles and Interquartile Range
The lower quartile (Q1) is at the ¼ position: 120 ÷ 4 = 30th student.
The upper quartile (Q3) is at the ¾ position: 3 × 120 ÷ 4 = 90th student.
Using the same method as above:
- Q1: Go across from 30 on the y-axis. Read off approximately 157 cm.
- Q3: Go across from 90 on the y-axis. Read off approximately 168 cm.
The interquartile range (IQR) is:
- IQR = Q3 − Q1 = 168 − 157 = 11 cm
Step 5: Find the Number Taller than 172 cm
Go to 172 on the x-axis. Draw a vertical line up to the curve, then across to the y-axis. The cumulative frequency at 172 cm is approximately 110.
This means 110 students have heights of 172 cm or less. Therefore, the number taller than 172 cm is:
- 120 − 110 = 10 students
Understanding What Each Part Tests
Let us break down the marks:
- Part (a) tests whether you can add a running total correctly. This is typically worth 1-2 marks and is essentially free if you are careful with arithmetic.
- Part (b) tests your graph-drawing skills. You earn marks for correct plotting and for drawing a smooth curve (not connecting dots with straight lines).
- Parts (c) and (d) test your ability to read from the curve. Examiners allow a tolerance of about 1 unit either side of the correct answer.
- Part (e) tests whether you understand that cumulative frequency gives “less than or equal to” values, so you need to subtract from the total to find “greater than” values.
Common Mistakes to Avoid
- Plotting at the midpoint instead of the upper class boundary. This is the most common error and costs you all the marks for the graph and subsequent readings.
- Drawing straight lines instead of a smooth curve. The examiner will specifically look for a smooth S-shaped curve. Practise freehand curve drawing before the exam.
- Reading the wrong axis. When finding the median, start from the cumulative frequency axis (y-axis) and read across to the curve, then down to the value axis (x-axis). Going the wrong way gives a meaningless answer.
- Using n/2 + 1 for the median position. At IGCSE level, use n/2 for cumulative frequency curves. The “+1” rule applies to discrete listed data, not grouped data.
- Forgetting to subtract from the total. When the question asks “how many students scored more than X,” you must subtract the cumulative frequency at X from the total.
Extending to Box Plots
Some IGCSE questions ask you to draw a box-and-whisker plot from your cumulative frequency data. You need five values:
- Minimum value (lower bound of the first class)
- Q1 (from the curve)
- Median (from the curve)
- Q3 (from the curve)
- Maximum value (upper bound of the last class)
Draw a number line, mark these five values, draw a box from Q1 to Q3, draw a vertical line inside the box at the median, and extend whiskers to the minimum and maximum. This is a straightforward extension once you have the cumulative frequency readings.
Practice Suggestion
The best practice for cumulative frequency is to work through past paper questions with a ruler and sharp pencil. Time yourself — you should complete the table, graph, and all readings within ten minutes. Pay attention to the scale given on the exam paper and use it accurately.
If you find that your readings are consistently off by more than 1 or 2 units, your curve is probably not smooth enough. Ask a tutor to review your graph-drawing technique.
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