Cumulative Frequency DiagramsEdexcel IGCSE Maths: Revision notes

Section 1

What is a cumulative frequency table?

Cumulative frequency means the running total of frequencies up to the end of each class interval.

Given a grouped frequency table, you add each frequency to the sum of all previous frequencies.

Height, $h$ (cm)	Frequency	Cumulative Frequency
$140 < h \leq 150$	4	4
$150 < h \leq 160$	9	13
$160 < h \leq 170$	15	28
$170 < h \leq 180$	10	38
$180 < h \leq 190$	2	40

Always plot cumulative frequency against the upper bound of each class interval, never the midpoint.

Key termscumulative frequencyclass intervalupper bound

Common mistake

Do not plot cumulative frequency against the midpoint of the class — this is a common error carried over from histograms. Always use the upper class boundary.

Section 2

How do you draw the cumulative frequency graph?

Plot each point at (upper bound, cumulative frequency).
Also plot a point at the lower bound of the first class with cumulative frequency 0.
Join the points with a smooth curve (an S-shaped ogive) — do not join with straight lines unless the question specifies a polygon.
Label both axes clearly, with cumulative frequency on the $y$ -axis.

The curve should always be non-decreasing (never goes down), since cumulative totals can only increase or stay the same.

Key termsogivecumulative frequency curve

Exam tip

Sketch the curve lightly in pencil first — an ogive should look like a stretched 'S', never wiggle up and down.

Section 3

How do you estimate the median, quartiles and IQR from the graph?

For $n$ total items of data:

Statistic	Position on cumulative frequency axis
Lower quartile, $Q_1$	$\frac{n}{4}$
Median, $Q_2$	$\frac{n}{2}$
Upper quartile, $Q_3$	$\frac{3n}{4}$

Draw a horizontal line from the cumulative frequency value across to the curve, then a vertical line down to the $x$ -axis to read off the estimate.

The interquartile range (IQR) is calculated as: $IQR = Q_3 - Q_1$

The IQR measures the spread of the middle 50% of the data and is less affected by outliers than the range.

Key termsmedianlower quartileupper quartileinterquartile range

Example

If $n = 40$ , read off values at cumulative frequency 10 ( $Q_1$ ), 20 ( $Q_2$ , the median) and 30 ( $Q_3$ ). If these give 158 cm, 165 cm and 172 cm, then $IQR = 172 - 158 = 14$ cm.

Common mistake

Do not use $\frac{n+1}{2}$ (the discrete-data formula) here — for grouped continuous data estimated from a graph, always use $\frac{n}{2}$ , $\frac{n}{4}$ and $\frac{3n}{4}$ .

Section 4

How do you construct a box plot from cumulative frequency data?

A box plot (box-and-whisker diagram) displays five key values on a single scaled line:

Value	Source
Minimum	Smallest data value (or lower bound of first class)
$Q_1$	Lower quartile from the cumulative frequency graph
Median	$Q_2$ from the cumulative frequency graph
$Q_3$	Upper quartile from the cumulative frequency graph
Maximum	Largest data value (or upper bound of last class)

Draw a box from $Q_1$ to $Q_3$ with a line at the median, and whiskers extending out to the minimum and maximum values. The box plot must be drawn to scale against a labelled axis.

Key termsbox plotwhisker

Exam tip

Exam questions often give you the box plot values directly and ask you to draw it accurately with a ruler — marks are lost for a box or whisker drawn even slightly off-scale.

Section 5

How do you compare two distributions using box plots or cumulative frequency data?

When comparing two data sets, you must always make two comparisons: one about a measure of location (median) and one about a measure of spread (IQR or range).

Example sentence structure:

"The median for Group A (165 cm) is higher than the median for Group B (158 cm), so Group A students are typically taller."
"The IQR for Group A (14 cm) is smaller than the IQR for Group B (20 cm), so the heights in Group A are more consistent / less varied."

Always link your comparison back to the context of the question (e.g. heights, times, marks) — a comparison with no context loses marks.

Key termsmeasure of locationmeasure of spread

Common mistake

Never compare two distributions using only the median, or only the IQR — exam mark schemes require one comment on average AND one comment on spread, both written in context.

Think of it like this

Think of the median as 'where the typical value sits' and the IQR as 'how tightly packed the middle of the data is' — like comparing the average commute time between two cities versus how reliable (consistent) that commute time is.

Must Know

Plot cumulative frequency against the upper bound of each class interval, joined with a smooth S-shaped curve (ogive).
Estimate $Q_1$ , median and $Q_3$ at positions $\frac{n}{4}$ , $\frac{n}{2}$ and $\frac{3n}{4}$ on the cumulative frequency axis.
$IQR = Q_3 - Q_1$ ; the IQR measures the spread of the middle 50% of data.
A box plot shows minimum, $Q_1$ , median, $Q_3$ and maximum, drawn accurately to scale.
When comparing distributions, always give one comment on median (location) and one comment on IQR or range (spread), both in context.
All estimates from a graph are approximate — never state them as exact values.

That's the notes covered.

Carry on to the next subtopic.