GCSE Maths · Statistics

Cumulative Frequency GCSE Questions and Answers

Q: Why does a cumulative-frequency graph always rise?

The running total can stay the same or increase as more classes are included; it cannot decrease.

Q: Where do I plot each cumulative frequency?

At the upper boundary of its class interval.

Q: How do I find the median?

Use half the total frequency, move across to the curve and then down to the value axis.

Q: Are answers from a cumulative-frequency graph exact?

Usually not. They are estimates based on the drawn curve and should be given to sensible accuracy.

Cumulative frequency is a running total of frequencies up to each class boundary. Plot upper class boundaries against cumulative totals, then use N ÷ 4, N ÷ 2 and 3N ÷ 4 to estimate the quartiles and median from the curve.

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Count everything up to a boundary

What you need to know about cumulative frequency

Suppose 4 journeys took up to 10 minutes and another 8 took more than 10 but no more than 20 minutes. A frequency is the number of journeys in one group. Adding 4 + 8 = 12 tells us how many journeys took no more than 20 minutes altogether. This running total is called the cumulative frequency. A class boundary is an end of a group’s time interval: here, 10 or 20 minutes.

Running total → curve → estimates

Build the running total before drawing the graph

Start with the first frequency. For every later group, add its frequency to the total you already have.

Up to 104begin with the first frequency

Up to 204 + 8 = 12include both groups

Up to 3012 + 11 = 23keep accumulating

Journey times and cumulative-frequency plotting points
Time t (minutes)	Frequency	Cumulative frequency	Point
0 < t ≤ 10	4	4	(10, 4)
10 < t ≤ 20	8	12	(20, 12)
20 < t ≤ 30	11	23	(30, 23)
30 < t ≤ 40	9	32	(40, 32)
40 < t ≤ 50	8	40	(50, 40)

At the upper boundary 20, the cumulative frequency 12 means 12 journeys lasted no more than 20 minutes. That is why the point is (20, 12), not a midpoint.

The curve begins at the lower boundary with cumulative frequency zero and passes through every upper-boundary total from the table.

The final total is 40 journeys. The median is the middle journey time: half the journeys are at or below it. Start at 20 on the vertical count axis, go across to the curve and then down to the time axis. The result is a time in minutes, not the count 20.

The lower quartile, Q1, marks a quarter of the data; the upper quartile, Q3, marks three quarters. For these 40 journeys, start at counts 10 and 30 respectively. The interquartile range (IQR) is Q3 − Q1: the width of the middle half of the data.

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Complete cumulative totals

What the problem asks: A grouped frequency table has an empty cumulative-frequency column.

How to solve it: Add each new class frequency to the previous running total. Check the last total equals the sample size.

Recover class frequencies

What the problem asks: Cumulative totals are given but ordinary frequencies are missing.

How to solve it: Subtract consecutive cumulative totals. The first frequency equals the first cumulative total.

Plot a curve

What the problem asks: The question gives class intervals and cumulative totals.

How to solve it: Plot each cumulative total at the upper class boundary, include the lower starting boundary at zero, then join with a smooth increasing curve.

Estimate median and quartiles

What the problem asks: A curve and total frequency N are given.

How to solve it: Read Q1 at N/4, the median at N/2 and Q3 at 3N/4 using horizontal and vertical guide lines.

Estimate a frequency in an interval

What the problem asks: The question asks how many values lie between two boundaries.

How to solve it: Read both cumulative frequencies and subtract the lower cumulative total from the upper one.

A reliable routine

Method for quartiles from a cumulative-frequency curve

Use this when a cumulative-frequency graph asks for a median, quartile or IQR. Dividing the total into halves or quarters gives a count on the vertical axis; the curve links that count to a value on the horizontal axis.

Find the total frequency N from the final cumulative total.
Calculate N ÷ 4, N ÷ 2 or 3N ÷ 4 for the requested position.
Draw horizontally from that cumulative frequency to the curve, then vertically to the value axis.
For IQR, subtract the lower-quartile estimate from the upper-quartile estimate.

Check: A cumulative-frequency curve must never fall: a running total cannot lose values. Over equal-width intervals, a greater rise means more values. A flat section means none were added. Readings inside grouped intervals are estimates because the individual values are unknown.

Fully worked

Cumulative frequency GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Build a cumulative-frequency column

4 marks

Question

Complete the cumulative-frequency column in the table.

Journey times and cumulative-frequency plotting points
Time t (minutes)	Frequency	Cumulative frequency	Point
0 < t ≤ 10	4	4	(10, 4)
10 < t ≤ 20	8	?	?
20 < t ≤ 30	11	?	?
30 < t ≤ 40	9	?	?
40 < t ≤ 50	8	?	?

Complete each running total, then pair it with the upper end of its time interval.

Keep a running total.

4

4+8=12

12+11=23

23+9=32

32+8=40

\boxed{4,\ 12,\ 23,\ 32,\ 40}

Example 2

Recover ordinary frequencies

3 marks

Question

The cumulative frequencies are $4, 11, 19, 27$ . Find the four class frequencies.

The first frequency is $4$ .

11-4=7

19-11=8

27-19=8

\boxed{4,\ 7,\ 8,\ 8}

Example 3

Find quartile positions

3 marks

Question

A cumulative-frequency graph represents $60$ values. State the cumulative frequencies used to read $Q_1$ , the median and $Q_3$ .

Q_1:\ \frac{60}{4}=15

\text{Median}:\ \frac{60}{2}=30

Q_3:\ \frac{3\times60}{4}=45

\boxed{15,\ 30,\ 45}

Example 4

Estimate an interquartile range

4 marks

Question

From a cumulative-frequency curve, the lower quartile is estimated as $18$ and the upper quartile as $43$ . Estimate the IQR.

\text{IQR}=Q_3-Q_1

=43-18

\boxed{25}

Example 5

Estimate a frequency between two values

4 marks

Question

A curve gives cumulative frequency $18$ at $20$ and cumulative frequency $72$ at $45$ . Estimate how many values are greater than $20$ and no more than $45$ .

The total up to $45$ includes the total up to $20$ , so subtract it.

72-18=54

\boxed{54\text{ values}}

Example 6

Compare two distributions

4 marks

Question

The graph shows cumulative-frequency curves for Groups A and B, each containing $80$ values. At cumulative frequency $40$ , A has value $33$ and B has value $24$ . The estimated IQRs are $20$ for A and $31$ for B. Compare the distributions.

Compare the horizontal readings at the same cumulative-frequency levels.

Distribution A has the greater median, so its typical value is higher.

33>24

Distribution A also has the smaller IQR because $20<31$ , so its middle half is less spread out.

\boxed{\text{A typically has larger values and is more consistent than B.}}

Example 7

Plot a complete cumulative-frequency curve

5 marks

Question

Use the completed table to plot a cumulative-frequency curve. Include the starting point.

Journey times and cumulative-frequency plotting points
Time t (minutes)	Frequency	Cumulative frequency	Point
0 < t ≤ 10	4	4	(10, 4)
10 < t ≤ 20	8	12	(20, 12)
20 < t ≤ 30	11	23	(30, 23)
30 < t ≤ 40	9	32	(40, 32)
40 < t ≤ 50	8	40	(50, 40)

At the upper boundary 20, the cumulative frequency 12 means 12 journeys lasted no more than 20 minutes. That is why the point is (20, 12), not a midpoint.

The lower boundary is $0$ , so begin at $(0,0)$ . Plot the cumulative totals at upper class boundaries:

\boxed{(0,0),\ (10,4),\ (20,12),\ (30,23),\ (40,32),\ (50,40)}

Join the points with a smooth increasing curve rather than separate straight bars.

The curve begins at the lower boundary with cumulative frequency zero and passes through every upper-boundary total from the table.

Example 8

Convert quartile readings to a box plot

4 marks

Question

A cumulative-frequency curve gives minimum $4$ , lower quartile $18$ , median $29$ , upper quartile $43$ and maximum $58$ . Draw the corresponding box plot.

Use one value scale. Draw the box from $18$ to $43$ , the median line at $29$ , and whiskers to $4$ and $58$ .

Read each vertical line against the shared linear scale.

15 original questions · total 43 marks

Cumulative frequency GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 50 minutes · show graph guide lines and label estimated answers · answers start collapsed

Continue a running total

3 marks

The first four class frequencies are $6, 9, 4, 11$ . Find the cumulative frequencies.

Show worked answer

6,\quad6+9=15,\quad15+4=19,\quad19+11=30

\boxed{6,\ 15,\ 19,\ 30}

Check the sample size

1 mark

The final cumulative frequency in a table is $84$ . How many values are in the data set?

Show worked answer

The final running total counts every value.

\boxed{84}

Find frequencies by subtraction

3 marks

The cumulative totals are $3, 10, 24, 30$ . Find the ordinary frequencies.

Show worked answer

3

10-3=7

24-10=14

30-24=6

\boxed{3,\ 7,\ 14,\ 6}

Choose plotting coordinates

2 marks

For the class $20<x\le30$ , the cumulative frequency is $17$ . Which point should be plotted?

Show worked answer

Cumulative frequency is plotted against the upper class boundary.

\boxed{(30,17)}

Calculate quartile levels

3 marks

A graph represents $96$ values. Find the cumulative-frequency levels for $Q_1$ , the median and $Q_3$ .

Show worked answer

\text{Level for }Q_1=\frac{96}{4}=24

\text{Level for median}=\frac{96}{2}=48

\text{Level for }Q_3=\frac{3\times96}{4}=72

\boxed{24,\ 48,\ 72}

Use two graph readings

2 marks

A graph shows cumulative frequency $21$ at $30$ and $67$ at $55$ . Estimate the number of values in $30<x\le55$ .

Show worked answer

Subtract the number at or below $30$ from the number at or below $55$ .

67-21=\boxed{46}

Estimate a number above a value

2 marks

There are $90$ values. A curve shows cumulative frequency $58$ at $40$ . Estimate how many values are greater than $40$ .

Show worked answer

90-58=\boxed{32}

Find an IQR from graph estimates

2 marks

The graph estimates $Q_1=24.5$ and $Q_3=61.0$ . Estimate the IQR.

Show worked answer

\text{IQR}=61.0-24.5

\boxed{36.5}

Compare two cumulative-frequency curves

4 marks

The graph shows Groups A and B, each containing $80$ values. At cumulative frequency $40$ , A has value $33$ and B has value $24$ . Their estimated IQRs are $20$ and $31$ respectively. Compare them.

Compare the horizontal readings at the same cumulative-frequency levels.

Show worked answer

Group A has the higher median, so its typical value is greater.

Group A also has the smaller IQR, so its middle half is less spread out.

\boxed{\text{A typically has higher values and is more consistent.}}

Interpret a complete graph

4 marks

A cumulative-frequency graph for $120$ journey times gives $Q_1=18$ minutes, median $26$ minutes and $Q_3=41$ minutes. Estimate the IQR and the number of journeys lasting no more than the median.

Show worked answer

\text{IQR}=41-18=23\text{ minutes}

The median is at half the total frequency.

\frac{120}{2}=60

\boxed{\text{IQR}=23\text{ minutes; about }60\text{ journeys}}

Complete a table and plot the curve

5 marks

Complete the cumulative-frequency column, list the plotting coordinates and draw a smooth curve.

Journey times and cumulative-frequency plotting points
Time t (minutes)	Frequency	Cumulative frequency	Point
0 < t ≤ 10	4	4	(10, 4)
10 < t ≤ 20	8	?	?
20 < t ≤ 30	11	?	?
30 < t ≤ 40	9	?	?
40 < t ≤ 50	8	?	?

Complete each running total, then pair it with the upper end of its time interval.

Show worked answer

The cumulative totals are

4,\ 12,\ 23,\ 32,\ 40

Include the starting point at the lower boundary.

\boxed{(0,0),\ (10,4),\ (20,12),\ (30,23),\ (40,32),\ (50,40)}

The curve begins at the lower boundary with cumulative frequency zero and passes through every upper-boundary total from the table.

Explain the upper-boundary rule

2 marks

Why is cumulative frequency $12$ for $10<t\le20$ plotted at $(20,12)$ rather than $(15,12)$ ?

Show worked answer

The total $12$ counts all values no more than $20$ . That statement is known at the upper boundary, while the midpoint $15$ does not represent the end of the class.

\boxed{(20,12)}

Draw a box plot from graph readings

4 marks

A curve gives minimum $2$ , $Q_1=17$ , median $28$ , $Q_3=44$ and maximum $59$ . Draw the corresponding box plot.

Show worked answer

Draw the box from $17$ to $44$ , place the median at $28$ , and extend the whiskers to $2$ and $59$ .

Read each vertical line against the shared linear scale.

Compare median and spread

4 marks

Curve R gives median $35$ and IQR $16$ . Curve S gives median $31$ and IQR $24$ . Compare the distributions.

Show worked answer

R has the higher median, so its typical value is higher.

R has the smaller IQR, so its middle half is less spread out.

\boxed{\text{R typically has higher values and is more consistent than S.}}

Interpret steep and flat sections

2 marks

A cumulative-frequency curve is steep between $20$ and $30$ but almost flat between $50$ and $60$ . What does this show?

Show worked answer

The running total rises quickly from $20$ to $30$ , so many values lie there. It rises only slightly from $50$ to $60$ , so few values lie there.

\boxed{\text{many values from 20–30; few from 50–60}}

Examiner-style feedback

Common cumulative frequency mistakes

Plotting class midpoints

Cumulative totals are plotted at upper class boundaries, not at class midpoints.

Missing the zero starting point

Begin at the lower boundary of the first class with cumulative frequency 0 before plotting the upper-boundary totals.

Resetting each total

Every cumulative entry includes all earlier classes. It is a running total, not an ordinary frequency.

Using value-axis quarters

Quartile positions come from the total frequency on the vertical axis, not from quartering the horizontal scale.

Adding for an interval count

To count between two boundaries, subtract the lower cumulative total from the upper cumulative total.

30-SECOND RECAP

Choose, carry out, check

Translate the wording before you reach for a rule.

Add frequencies to create a running total.
Plot totals at upper class boundaries.
Use N/4, N/2 and 3N/4 for quartiles.
Subtract graph readings for interval frequencies or IQR.

Quick answers

Cumulative frequency FAQ

Why does a cumulative-frequency graph always rise?

The running total can stay the same or increase as more classes are included; it cannot decrease.

Where do I plot each cumulative frequency?

At the upper boundary of its class interval.

How do I find the median?

Use half the total frequency, move across to the curve and then down to the value axis.

Are answers from a cumulative-frequency graph exact?

Usually not. They are estimates based on the drawn curve and should be given to sensible accuracy.

Build connected skills

What to revise next

Display the summary

Box plots

Turn quartile estimates into a compact comparison diagram.

Revise box plots

Proportions

Percentages

Interpret quarters, halves and cumulative proportions.

Revise percentages

Exact arithmetic

Fractions

Calculate quartile positions and proportional counts.

Revise fractions

Content standards

Curriculum and rights review

Curriculum references checked 4 September 2026. Constructing and interpreting cumulative-frequency graphs, estimating quartiles and comparing distributions are shared Higher-tier GCSE Mathematics skills. All questions, diagrams, data and wording are original Pass an Exam material.

Official specification references