Why is cumulative frequency always plotted against the upper class boundary?

Each running total counts all observations up to the end of that class. The total must therefore be positioned at the class's upper boundary, not at its midpoint.

Should a cumulative frequency graph be a curve or straight lines?

Follow the instruction in the question. GCSE questions commonly use a smooth increasing curve, although some tasks may ask you to join points or use a given graph. Accurate plotted points matter more than making the curve look decorative.

Can a cumulative frequency graph go down?

No. Cumulative frequency is a running total, so it cannot decrease. A downward section indicates that a frequency was added incorrectly or a point was plotted inaccurately.

Why are quartiles from cumulative frequency graphs only estimates?

The original values are grouped into intervals, so their exact positions within each interval are unknown. Reading from a drawn curve introduces further measurement uncertainty.

What does a steep cumulative frequency graph mean?

A steep section means that many observations lie within a relatively narrow interval. A flatter section means that fewer observations lie in that part of the distribution.

How can I convert cumulative frequencies back into frequencies?

The first frequency equals the first cumulative frequency. After that, subtract consecutive cumulative totals. If the totals are $5$, $14$ and $31$, the frequencies are $5$, $14-5=9$ and $31-14=17$.

Cumulative Frequency GCSE Maths Revision Guide

Cumulative frequency records a running total of frequencies. In cumulative frequency GCSE Maths questions, you may need to complete a table, draw or interpret a graph, estimate the median and quartiles, calculate the interquartile range, or compare two distributions.

This Higher tier guide explains each skill step by step, with cumulative frequency worked examples and exam-style questions. You can also use the GCSE Maths topics by exam board and tier guide to check how the topic fits into your course.

What is cumulative frequency?

The frequency tells you how many values lie in one class interval. The cumulative frequency tells you how many values lie up to and including the end of that interval.

For example, suppose grouped times have frequencies of $4$ , $7$ and $13$ in the first three intervals. The cumulative frequencies are:

4,\quad 4+7=11,\quad 4+7+13=24.

The final cumulative frequency must equal the total number of observations. This is an important check in an examination.

A cumulative frequency graph is useful because it shows how the data build up across the whole distribution. It can be used to estimate the median, quartiles, percentiles and the number of observations above or below a particular value.

How to complete a cumulative frequency table

Add each new frequency to the running total. Do not treat each row as a separate calculation from zero.

Worked example 1: Finding cumulative frequencies

The table shows the times, $t$ minutes, taken by $40$ students to complete a task.

Time, $t$ minutes	Frequency	Cumulative frequency
$0<t≤100<t\leq10$	$4$	$4$
$10<t≤2010<t\leq20$	$7$	$11$
$20<t≤3020<t\leq30$	$13$	$24$
$30<t≤4030<t\leq40$	$10$	$34$
$40<t≤5040<t\leq50$	$6$	$40$

The running totals are:

4,\quad 4+7=11,\quad 11+13=24,\quad 24+10=34,\quad 34+6=40.

The final value is $40$ , which agrees with the total number of students.

To recover an ordinary frequency from cumulative frequencies, subtract consecutive totals. For example, the frequency in $20<t≤3020<t\leq30$ is:

24 - 11 = 13.

Drawing a cumulative frequency graph

Plot the upper class boundary against the cumulative frequency. For the worked example, the plotted points are:

(10,4),\ (20,11),\ (30,24),\ (40,34),\ (50,40).

Because there are no times at or below the lower boundary of $0$ , you can also begin with $(0, 0)$ . This gives the graph a meaningful starting point.

Use the following method:

Put the measured variable, such as time or height, on the horizontal axis.
Put cumulative frequency on the vertical axis.
Choose scales that use most of the grid and label both axes.
Plot each cumulative frequency at the upper boundary of its interval.
Join the points with a smooth increasing curve, or follow the precise instruction given in the question.

A cumulative frequency graph must never decrease. The running total can stay level if an interval has frequency zero, but it cannot go down.

The MathsGenie cumulative frequency revision lesson includes further teaching resources, while the revision guide provides focused topic notes and questions.

Estimating the median, quartiles and interquartile range

For $N$ observations, use these cumulative frequency positions:

Measure	Position on the cumulative frequency axis
Lower quartile, $Q_1$	$N4\frac{N}{4}$
Median	$N2\frac{N}{2}$
Upper quartile, $Q_3$	$3N4\frac{3N}{4}$

On a graph, mark the required position on the cumulative frequency axis. Draw horizontally to the curve, then vertically down to the horizontal axis. The value read from the horizontal axis is the estimate.

The interquartile range, or IQR, measures the spread of the middle half of the data:

\text{IQR}=Q_3-Q_1.

A smaller IQR means the middle $50%50\%$ of the values are more closely grouped, so the distribution is more consistent.

Worked example 2: Estimating quartiles

For the $40$ students in the earlier table, the required cumulative frequencies are:

Q_1:\frac{40}{4}=10,

\text{median}:\frac{40}{2}=20,

Q_3:\frac{3\times40}{4}=30.

Reading these positions from a cumulative frequency graph would give approximate values. If straight line segments are used between the plotted points, the estimates are approximately:

Q_1\approx18.6,

\text{median}\approx26.9,

Q_3\approx36.0.

Therefore:

\text{IQR}\approx36.0-18.6=17.4.

These are estimates because the original values within each grouped interval are unknown. A slightly different sensible reading from a hand-drawn curve may still be accepted within the tolerance stated by the mark scheme.

Reading other information from the graph

If the graph shows a total of $N$ observations and the cumulative frequency at $x$ is $C$ , then:

the estimated number at or below $x$ is $C$ ;
the estimated number above $x$ is $N - C$ ;
the estimated number between $a$ and $b$ is $C (b) - C (a)$ .

For example, suppose a graph represents $120$ runners. If the cumulative frequency is $44$ at $30$ minutes and $101$ at $50$ minutes, the estimated number taking between $30$ and $50$ minutes is:

101 - 44 = 57.

The estimated number taking more than $50$ minutes is:

120 - 101 = 19.

Subtraction is required because cumulative frequencies include all preceding values.

Comparing two cumulative frequency distributions

Comparisons normally need one statement about the average and one about the spread.

Use the median to compare typical values. Use the interquartile range to compare consistency or variability.

For example, suppose distribution A has median $28$ and IQR $12$ , while distribution B has median $31$ and IQR $7$ . Distribution B has the higher typical value because its median is higher. It is also more consistent because its IQR is smaller.

Avoid saying one set is simply “better”. State what the statistics show in the context of the question. The related MathsGenie box plots lesson is useful because box plots display the same median and quartile information directly.

Cumulative frequency exam question

The table shows the distances, $d$ kilometres, travelled by $80$ cyclists.

Distance, $d$ kilometres	Frequency
$0<d≤50<d\leq5$	$6$
$5<d≤105<d\leq10$	$14$
$10<d≤1510<d\leq15$	$28$
$15<d≤2015<d\leq20$	$20$
$20<d≤2520<d\leq25$	$12$

Complete the cumulative frequency values.
State the points that should be plotted.
Estimate the median, lower quartile and upper quartile using straight line segments between the points.
Calculate the interquartile range.

Answer

The cumulative frequencies are:

6,\quad20,\quad48,\quad68,\quad80.

The points are:

(0,0),\ (5,6),\ (10,20),\ (15,48),\ (20,68),\ (25,80).

For $80$ cyclists, the quartile positions are:

Q_1=20,\quad \text{median}=40,\quad Q_3=60.

The lower quartile occurs exactly at a cumulative frequency of $20$ , so:

Q_1=10.

The median lies between cumulative frequencies $20$ and $48$ :

\text{median}\approx10+\frac{40-20}{48-20}\times5\approx13.6.

The upper quartile lies between cumulative frequencies $48$ and $68$ :

Q_3\approx15+\frac{60-48}{68-48}\times5=18.

Therefore:

\text{IQR}=18-10=8.

Try 4 more cumulative frequency practice questions. Show all construction lines and calculations so that method marks are available even if a graph reading is slightly inaccurate.

Common mistakes in cumulative frequency questions

Plotting class midpoints: Cumulative frequency points are plotted at upper class boundaries, not midpoints.
Using ordinary frequencies as coordinates: Add the frequencies first, then plot the running totals.
Starting at the wrong point: When appropriate, begin at the lower boundary with cumulative frequency $0$ .
Finding quartiles on the horizontal axis first: Calculate $N4\frac{N}{4}$ , $N2\frac{N}{2}$ and $3N4\frac{3N}{4}$ on the cumulative frequency axis.
Adding when finding an interval frequency: Subtract cumulative frequencies at the two boundaries.
Reversing the IQR subtraction: Use $Q_3-Q_1$ , not $Q_1-Q_3$ .
Giving exact-looking answers: Values read from grouped data or a curve are estimates.
Comparing only medians: A complete comparison often needs both median and IQR.

How to revise cumulative frequency effectively

First practise completing cumulative frequency tables without drawing graphs. Then focus on plotting upper class boundaries accurately and reading quartiles with clear construction lines.

Next, practise questions that ask for counts above, below or between two values. Finish with questions comparing distributions or linking cumulative frequency graphs to box plots.

The cumulative frequency exam booklet provides concentrated practice. Check your work with the worked solutions and mark guidance, then apply the skill under timed conditions in past papers.

Conclusion

Cumulative frequency is a running total. To answer Higher tier questions successfully, complete the totals carefully, plot them at upper class boundaries, and use $N4\frac{N}{4}$ , $N2\frac{N}{2}$ and $3N4\frac{3N}{4}$ to estimate the quartiles and median.

Remember that graph readings from grouped data are estimates. Use the median to compare typical values and the IQR to compare spread. MathsGenie’s revision lesson, practice questions, exam booklet and solutions provide a structured way to develop these skills before attempting full past papers.

Sources and referenced URLs

What is cumulative frequency?

The frequency tells you how many values lie in one class interval. The cumulative frequency tells you how many values lie up to and including the end of that interval.

For example, suppose grouped times have frequencies of $4$ , $7$ and $13$ in the first three intervals. The cumulative frequencies are:

4,\quad 4+7=11,\quad 4+7+13=24.

The final cumulative frequency must equal the total number of observations. This is an important check in an examination.

How to complete a cumulative frequency table

Add each new frequency to the running total. Do not treat each row as a separate calculation from zero.

Worked example 1: Finding cumulative frequencies

The table shows the times, $t$ minutes, taken by $40$ students to complete a task.

Time, $t$ minutes	Frequency	Cumulative frequency
$0<t≤100<t\leq10$	$4$	$4$
$10<t≤2010<t\leq20$	$7$	$11$
$20<t≤3020<t\leq30$	$13$	$24$
$30<t≤4030<t\leq40$	$10$	$34$
$40<t≤5040<t\leq50$	$6$	$40$

The running totals are:

4,\quad 4+7=11,\quad 11+13=24,\quad 24+10=34,\quad 34+6=40.

The final value is $40$ , which agrees with the total number of students.

To recover an ordinary frequency from cumulative frequencies, subtract consecutive totals. For example, the frequency in $20<t≤3020<t\leq30$ is:

24 - 11 = 13.

Drawing a cumulative frequency graph

Plot the upper class boundary against the cumulative frequency. For the worked example, the plotted points are:

(10,4),\ (20,11),\ (30,24),\ (40,34),\ (50,40).

Because there are no times at or below the lower boundary of $0$ , you can also begin with $(0, 0)$ . This gives the graph a meaningful starting point.

Use the following method:

Put the measured variable, such as time or height, on the horizontal axis.
Put cumulative frequency on the vertical axis.
Choose scales that use most of the grid and label both axes.
Plot each cumulative frequency at the upper boundary of its interval.
Join the points with a smooth increasing curve, or follow the precise instruction given in the question.

A cumulative frequency graph must never decrease. The running total can stay level if an interval has frequency zero, but it cannot go down.

The MathsGenie cumulative frequency revision lesson includes further teaching resources, while the revision guide provides focused topic notes and questions.

Estimating the median, quartiles and interquartile range

For $N$ observations, use these cumulative frequency positions:

Measure	Position on the cumulative frequency axis
Lower quartile, $Q_1$	$N4\frac{N}{4}$
Median	$N2\frac{N}{2}$
Upper quartile, $Q_3$	$3N4\frac{3N}{4}$

The interquartile range, or IQR, measures the spread of the middle half of the data:

\text{IQR}=Q_3-Q_1.

A smaller IQR means the middle $50%50\%$ of the values are more closely grouped, so the distribution is more consistent.

Worked example 2: Estimating quartiles

For the $40$ students in the earlier table, the required cumulative frequencies are:

Q_1:\frac{40}{4}=10,

\text{median}:\frac{40}{2}=20,

Q_3:\frac{3\times40}{4}=30.

Reading these positions from a cumulative frequency graph would give approximate values. If straight line segments are used between the plotted points, the estimates are approximately:

Q_1\approx18.6,

\text{median}\approx26.9,

Q_3\approx36.0.

Therefore:

\text{IQR}\approx36.0-18.6=17.4.

Reading other information from the graph

If the graph shows a total of $N$ observations and the cumulative frequency at $x$ is $C$ , then:

the estimated number at or below $x$ is $C$ ;
the estimated number above $x$ is $N - C$ ;
the estimated number between $a$ and $b$ is $C (b) - C (a)$ .

For example, suppose a graph represents $120$ runners. If the cumulative frequency is $44$ at $30$ minutes and $101$ at $50$ minutes, the estimated number taking between $30$ and $50$ minutes is:

101 - 44 = 57.

The estimated number taking more than $50$ minutes is:

120 - 101 = 19.

Subtraction is required because cumulative frequencies include all preceding values.

Comparing two cumulative frequency distributions

Comparisons normally need one statement about the average and one about the spread.

Use the median to compare typical values. Use the interquartile range to compare consistency or variability.

Cumulative frequency exam question

The table shows the distances, $d$ kilometres, travelled by $80$ cyclists.

Distance, $d$ kilometres	Frequency
$0<d≤50<d\leq5$	$6$
$5<d≤105<d\leq10$	$14$
$10<d≤1510<d\leq15$	$28$
$15<d≤2015<d\leq20$	$20$
$20<d≤2520<d\leq25$	$12$

Complete the cumulative frequency values.
State the points that should be plotted.
Estimate the median, lower quartile and upper quartile using straight line segments between the points.
Calculate the interquartile range.

Answer

The cumulative frequencies are:

6,\quad20,\quad48,\quad68,\quad80.

The points are:

(0,0),\ (5,6),\ (10,20),\ (15,48),\ (20,68),\ (25,80).

For $80$ cyclists, the quartile positions are:

Q_1=20,\quad \text{median}=40,\quad Q_3=60.

The lower quartile occurs exactly at a cumulative frequency of $20$ , so:

Q_1=10.

The median lies between cumulative frequencies $20$ and $48$ :

\text{median}\approx10+\frac{40-20}{48-20}\times5\approx13.6.

The upper quartile lies between cumulative frequencies $48$ and $68$ :

Q_3\approx15+\frac{60-48}{68-48}\times5=18.

Therefore:

\text{IQR}=18-10=8.

Try 4 more cumulative frequency practice questions. Show all construction lines and calculations so that method marks are available even if a graph reading is slightly inaccurate.

Common mistakes in cumulative frequency questions

Plotting class midpoints: Cumulative frequency points are plotted at upper class boundaries, not midpoints.
Using ordinary frequencies as coordinates: Add the frequencies first, then plot the running totals.
Starting at the wrong point: When appropriate, begin at the lower boundary with cumulative frequency $0$ .
Finding quartiles on the horizontal axis first: Calculate $N4\frac{N}{4}$ , $N2\frac{N}{2}$ and $3N4\frac{3N}{4}$ on the cumulative frequency axis.
Adding when finding an interval frequency: Subtract cumulative frequencies at the two boundaries.
Reversing the IQR subtraction: Use $Q_3-Q_1$ , not $Q_1-Q_3$ .
Giving exact-looking answers: Values read from grouped data or a curve are estimates.
Comparing only medians: A complete comparison often needs both median and IQR.

How to revise cumulative frequency effectively

First practise completing cumulative frequency tables without drawing graphs. Then focus on plotting upper class boundaries accurately and reading quartiles with clear construction lines.

Next, practise questions that ask for counts above, below or between two values. Finish with questions comparing distributions or linking cumulative frequency graphs to box plots.

The cumulative frequency exam booklet provides concentrated practice. Check your work with the worked solutions and mark guidance, then apply the skill under timed conditions in past papers.

What is cumulative frequency?

How to complete a cumulative frequency table

Worked example 1: Finding cumulative frequencies

Drawing a cumulative frequency graph

Estimating the median, quartiles and interquartile range

Worked example 2: Estimating quartiles

Reading other information from the graph

Comparing two cumulative frequency distributions

Cumulative frequency exam question

Answer

Common mistakes in cumulative frequency questions

How to revise cumulative frequency effectively

Conclusion

Sources and referenced URLs

Frequently Asked Questions

Why is cumulative frequency always plotted against the upper class boundary?

Why is cumulative frequency always plotted against the upper class boundary?

Should a cumulative frequency graph be a curve or straight lines?

Should a cumulative frequency graph be a curve or straight lines?

Can a cumulative frequency graph go down?

Can a cumulative frequency graph go down?

Why are quartiles from cumulative frequency graphs only estimates?

Why are quartiles from cumulative frequency graphs only estimates?

What does a steep cumulative frequency graph mean?

What does a steep cumulative frequency graph mean?

How can I convert cumulative frequencies back into frequencies?

How can I convert cumulative frequencies back into frequencies?

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Ready to breeze through your GCSEs?IGCSEs?A-Levels?AS Levels?GCSEs?GCSEs?

What is cumulative frequency?

How to complete a cumulative frequency table

Worked example 1: Finding cumulative frequencies

Drawing a cumulative frequency graph

Estimating the median, quartiles and interquartile range

Worked example 2: Estimating quartiles

Reading other information from the graph

Comparing two cumulative frequency distributions

Cumulative frequency exam question

Answer

Common mistakes in cumulative frequency questions

How to revise cumulative frequency effectively

Conclusion

Sources and referenced URLs

Frequently Asked Questions

Why is cumulative frequency always plotted against the upper class boundary?

Why is cumulative frequency always plotted against the upper class boundary?

Should a cumulative frequency graph be a curve or straight lines?

Should a cumulative frequency graph be a curve or straight lines?

Can a cumulative frequency graph go down?

Can a cumulative frequency graph go down?

Why are quartiles from cumulative frequency graphs only estimates?

Why are quartiles from cumulative frequency graphs only estimates?

What does a steep cumulative frequency graph mean?

What does a steep cumulative frequency graph mean?

How can I convert cumulative frequencies back into frequencies?

How can I convert cumulative frequencies back into frequencies?

More from the blog

Year 10 Maths: How Parents Can Help Now

Active Recall for Maths: How to Do It

Spaced Practice for GCSE Maths: A Practical Guide

How Long Should You Revise Maths Each Day?

Ready to breeze through your GCSEs?

Ready to breeze through your GCSEs?