IGCSE / O Level Mathematics · Topic 9 Statistics · Chapter 37 · pp. 532–540

Cumulative frequency

With a big set of grouped data you cannot list every value to find the median. Instead you add up the frequencies as you go, draw a cumulative frequency curve, and read the median, quartiles and any percentile straight off it. The same numbers then make a box-and-whisker plot, which is a fast way to compare two sets of data.

Learning objectives

Build a cumulative frequency table by adding up the frequencies as you go along.
Plot a cumulative frequency diagram, with each point at the upper boundary of its class.
Read the median from the curve and say what it tells us.
Read quartiles and percentiles, and calculate the interquartile range.
Draw box-and-whisker plots and use them, with the IQR, to compare two sets of data.

Book exercises: 37.1 and 37.2 and Student assessment 1 (pp. 532–540).

Cumulative frequency tables

🔑 Key ideas

Cumulative frequency = the running total of the frequencies: how many values are up to and including this class.
The last cumulative frequency is the total number of values.
It is most useful for large, grouped or continuous data, and for judging how consistent the results are.

From a frequency table to a median in five steps.

📘 Example 1 · Test scores of 20 students: build the cumulative frequency table

Score x	Frequency	Cumulative frequency
0 ≤ x < 10	2	2
10 ≤ x < 20	6	2 + 6 = 8
20 ≤ x < 30	9	8 + 9 = 17
30 ≤ x < 40	3	17 + 3 = 20

The first cumulative frequency is just the first frequency: 2.
Each next one is the previous total plus this class's frequency: 2 + 6 = 8, then 8 + 9 = 17, then 17 + 3 = 20.
Check: the last value, 20, is the number of students. ✔

The cumulative frequency diagram

🔑 Key ideas

Plot each point at the upper boundary of the class, not the middle: (5, 3), (10, 8), …
Start the curve at the lower bound of the first class with cumulative frequency 0, here (0, 0).
The median is at half way up the vertical axis: read across from N ÷ 2, then down to the horizontal axis.

Type A batteries (50 tested). Points are at the upper bounds 5, 10, … 35. Read across from 25 and down: median about 19 h.

📘 Example 2 · Type A batteries from the book (page 532): find the median

Duration t (h)	0–5	5–10	10–15	15–20	20–25	25–30	30–35
Frequency	3	5	8	10	12	7	5
Cumulative	3	8	16	26	38	45	50

Add up as you go: 3, 3 + 5 = 8, 8 + 8 = 16, 16 + 10 = 26, 26 + 12 = 38, 38 + 7 = 45, 45 + 5 = 50.
Plot (5, 3), (10, 8), (15, 16), (20, 26), (25, 38), (30, 45), (35, 50), starting from (0, 0). Join them in a smooth curve.
The total is 50, so the median is at 50 ÷ 2 = 25 on the vertical axis.
Read across from 25 to the curve, then straight down: about 19 h (the book's reading).
Check: 25 lies between 16 (at 15 h) and 26 (at 20 h), so the median is just under 20 h. ✔

💡 What the median tells us: half the batteries lasted less than about 19 h and half lasted longer.

📘 Example 3 · Check with the numbers (interpolation)

25 is between cumulative 16 (at 15 h) and 26 (at 20 h). The class 15–20 has 10 batteries.
We need 25 − 16 = 9 of those 10, so we go 9/10 of the way across the 5 h class.
Median ≈ 15 + 9/10 × 5 = 15 + 4.5 = 19.5 h.
This is the same as reading a straight-line version of the graph. Reading a smooth curve by eye gives about 19, so both are fine for an estimate.

✏️ Try it

Frequencies 6, 10, 14, 10 in order. Write the cumulative frequencies.
The last cumulative frequency is 40. At what height do you read the median?
Class 30 ≤ x < 40 has cumulative frequency 28 and the class before it has 19. What is its frequency?

Answers

1. 6, 16, 30, 40 · 2. 40 ÷ 2 = 20 · 3. 28 − 19 = 9

Next: quartiles and the interquartile range →

Percentiles, quartiles and the interquartile range

🔑 Key ideas

A percentile scale cuts the vertical axis into hundredths. The top of the curve is the 100th percentile.
Lower quartile (Q1) = 25th percentile, at ¼ of the total. Median (Q2) = 50th percentile, at ½. Upper quartile (Q3) = 75th percentile, at ¾.
Interquartile range (IQR) = Q3 − Q1: the spread of the middle half of the data.
The range (highest − lowest) is often not useful because one extreme value changes it. The IQR is not affected by extremes.

The cumulative frequency axis as percentiles. The IQR covers the middle 50% of the values.

Value	Percentile	Read at height (N values)
Lower quartile	25th	N ÷ 4 (for N = 50: 12.5)
Median	50th	N ÷ 2 (25)
Upper quartile	75th	3N ÷ 4 (37.5)
Any percentile p	p-th	p% of N

Type A batteries: read across at 12.5 and 37.5, then down. Q1 ≈ 13 h and Q3 ≈ 25 h, so IQR ≈ 12 h.

📘 Example 4 · Batteries A and B (book page 535): quartiles and IQR

Both brands have N = 50, so read across at 12.5 (lower quartile) and 37.5 (upper quartile).
Type A: Q1 ≈ 13 h and Q3 ≈ 25 h. IQR = 25 − 13 = 12 h.
Type B: Q1 ≈ 15 h and Q3 ≈ 21 h. IQR = 21 − 15 = 6 h.
Compare: B's IQR is half of A's, so the middle half of B batteries are much closer together. B is more consistent.
Advertising: A is on average the longer-lasting battery (higher median). B is the more reliable battery (smaller IQR).

📘 Example 5 · A percentile: the top 20% of 200 students go to the final

Mark	0–20	20–40	40–60	60–80	80–100
Frequency	10	30	60	70	30
Cumulative	10	40	100	170	200

The top 20% start above the 80th percentile.
80% of 200 = 160, so read across at 160.
160 is between 100 (at 60) and 170 (at 80). The class has 70 students, and we need 160 − 100 = 60 of them.
Mark ≈ 60 + 60/70 × 20 = 60 + 17.1 = 77 (about).
Check: 170 students scored under 80, so 77 is a little under 80. ✔

✏️ Try it

A graph for 120 values: at what heights do you read Q1, the median and Q3?
Q1 = 42 and Q3 = 67. Find the IQR.
A graph for 300 values: at what height do you read the 90th percentile?

Answers

1. 30, 60, 90 · 2. 67 − 42 = 25 · 3. 90% of 300 = 270

Next: box-and-whisker plots →

Box-and-whisker plots

🔑 Key ideas

A box-and-whisker plot shows five numbers: minimum, Q1, median, Q3, maximum.
The box runs from Q1 to Q3 with a line at the median. Its length is the IQR.
The whiskers run from the box out to the minimum and maximum.
For grouped data we do not know where values lie in the end classes. Take the minimum as the lower bound of the first class and the maximum as the upper bound of the last class.
Draw two plots on the same scale to compare.

The anatomy of a box-and-whisker plot.

Batteries A and B on one scale. Same overall range, but B's box is much narrower.

Type	Min	Q1	Median	Q3	Max	IQR
A	0	13	19	25	35	12
B	0	15	18	21	35	6

📘 Example 6 · Draw a box plot for the 20 test scores of Example 1

N = 20 and cumulative frequencies 2, 8, 17, 20 at the upper bounds 10, 20, 30, 40.
Median at 10: between 8 (at 20) and 17 (at 30): 20 + 2/9 × 10 ≈ 22.2.
Q1 at 5: between 2 (at 10) and 8 (at 20): 10 + 3/6 × 10 = 15.
Q3 at 15: between 8 (at 20) and 17 (at 30): 20 + 7/9 × 10 ≈ 27.8.
Minimum = lower bound of the first class = 0. Maximum = upper bound of the last class = 40.
Draw a scale 0 to 40. Box from 15 to 27.8 with a line at 22.2, whiskers to 0 and 40. IQR ≈ 27.8 − 15 = 12.8.

📘 Example 7 · Compare the batteries (book page 536)

Both types have the same overall range: 0 to 35 h.
A's median (about 19 h) is slightly bigger than B's (about 18 h).
A's box is much longer, IQR 12 h against 6 h.
Conclusion: the middle 50% of B batteries are more consistent, while A is slightly better on average.

✏️ Try it

Which five numbers do you need to draw a box-and-whisker plot?
Minimum 5, Q1 12, median 20, Q3 26, maximum 40. Find the IQR and say what fraction of the data lies in the box.
Plots P and Q have the same median. P's box is shorter. Which set is more consistent?

Answers

1. Minimum, lower quartile, median, upper quartile, maximum · 2. IQR = 26 − 12 = 14, and half (50%) of the data lies in the box · 3. P, because its middle 50% is squeezed into a shorter range

Next: the plotter →

Cumulative frequency plotter

Type the class boundaries and the frequencies. The plotter builds the table, draws the curve, marks the quartiles and any percentile, draws the box plot, and shows each step.

🧪 Try it: cumulative frequency plotter

Class boundaries (separate with commas)

Frequencies (one fewer than the boundaries)

Percentile to find %

The curve with dashed guide lines for Q1, the median, Q3 and the chosen percentile, and the box-and-whisker plot underneath on the same scale.

Working

📘 Example 8 · Compare two data sets with the plotter

Press Battery A. Note the median 19.5 h, Q1 12.8 h, Q3 24.8 h and IQR 12.0 h.
Press Battery B. The median is 17.8 h and the IQR is 6.3 h.
The medians are close, so the two types last about the same time on average.
B's IQR is about half of A's, so B is the more consistent brand.
The plotter joins points with straight lines, so its values differ slightly from the book's smooth-curve readings (19 h and 12 h for A).

✏️ Try it

Enter boundaries 0, 10, 20, 30, 40 and frequencies 5, 15, 20, 10. Predict the median, then check.
For the same data, find the 90th percentile.
Why does the curve always start at height 0 and end at the total?

Answers

1. N = 50, half is 25. Cumulative 5, 20, 40, 50. In the class 20–30: 20 + (25 − 20)/20 × 10 = 22.5 · 2. 90% of 50 = 45, in the class 30–40: 30 + (45 − 40)/10 × 10 = 35 · 3. At the lower bound of the first class no values are below it, so the cumulative frequency is 0. At the upper bound of the last class every value is included, so it equals N.

Next: summary and quiz →

Chapter summary

Cumulative frequency is a running total. The last value equals the number of items.
Plot at the upper boundary of each class, starting at (lower bound, 0).
Median: read at N ÷ 2. Lower quartile: N ÷ 4. Upper quartile: 3N ÷ 4. Percentile p: p% of N.
IQR = Q3 − Q1. A smaller IQR means more consistent data. The median says which is higher on average.
Box-and-whisker plot: minimum, Q1, median, Q3, maximum. For grouped data use the end bounds of the first and last classes.
Compare on the same scale: say something about the median (average) and the IQR (consistency).

📄 Practice questions

The book's Exercises 37.1 and 37.2 and Student assessment 1 are in the printable assignment for this chapter.

Mini-quiz

1. Class frequencies are 4, 7, 9, 5 in order. What is the last cumulative frequency?

2. For the class 10 ≤ t < 15, where is its point plotted on a cumulative frequency graph?

3. A cumulative frequency curve ends at 80. At what height do you read the median?

4. For 60 values, at what cumulative frequency do you read the lower quartile?

5. For 200 values, at what cumulative frequency do you read the upper quartile?

6. The lower quartile is 13 h and the upper quartile is 25 h. What is the interquartile range?

7. Two data sets have the same median. Set P has IQR 6 and set Q has IQR 12. Which is more consistent?

8. Which five values are needed to draw a box-and-whisker plot?

9. In a box-and-whisker plot, what does the length of the box show?

10. A top 20% of students go to the final. Which percentile gives the qualifying mark?

11. Frequency table: first class is 0 ≤ t < 5. What minimum value do we use on the box plot?

12. Why is the interquartile range often better than the range?

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Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 37 (Cumulative frequency). Explanations, the Try it answers, the plotter and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.