1. Cumulative Frequency
Objectives:
B Grade Construct and interpret a cumulative frequency
diagram
Use a cumulative frequency diagram to estimate
the median and interquartile range

2. Cumulative Frequency
A cumulative frequency diagram is a graph that can be used to find
estimates of the median and upper and lower quartiles of grouped
data.
The median is the middle value when the data has been placed in
order of size
The lower quartile is the ‘median’ of the bottom half of the data set
and represents the value ¼ of the way through the data.
The upper quartile is the ‘median’ of the top half of the data set
and represents the value ¾ of the way through the data.

3. A pet shop owner weighs his mice every week to check their health.
Cumulative Frequency
A pet shop owner weighs his mice every week to check their health.
The weights of the 80 mice are shown below:
weight (g)
Frequency (f)
0 < w ≤ 10 3
10 < w ≤ 20 5
20 < w ≤ 30
30 < w ≤ 40 9
40 < w ≤ 50 11
50 < w ≤ 60 15
60 < w ≤ 70 14
70 < w ≤ 80 8
80 < w ≤ 90 6
90 < w ≤100 4
Cumulative Frequency 3 8 13 22 33 48 62 70 76 80
Cumulative means adding up, so a cumulative frequency diagram
requires a running total of the frequency.

4. The point are now joined with straight lines
Cumulative Frequency x 80 70 60 50 40 30 20 10
Weight (g)
Frequency (f)
0 < w ≤ 10 3
10 < w ≤ 20 5
20 < w ≤ 30
30 < w ≤ 40 9
40 < w ≤ 50 11
50 < w ≤ 60 15
60 < w ≤ 70 14
70 < w ≤ 80 8
80 < w ≤ 90 6
90 < w ≤100 4
Cumulative Frequency x x 3 8 13 22 33 48 62 70 76 80 x x
Cumulative frequency x x x x x
Weight (g)
The point are now joined with straight lines
The cumulative frequency (c.f.) can now be plotted on a graph
taking care to plot the c.f. at the end of each class interval.
The line always starts at the
bottom of the first class interval
This is because we don’t know where in the class interval
0 < w ≤ 10, the values are, but we do know that by the end of
the class interval there are 3 pieces of data
The resulting graph should look like this and is sometimes called an
‘S’ curve.

6. From this graph we can now find estimates of the median, and
Cumulative Frequency
From this graph we can now find estimates of the median, and
upper and lower quartiles
Upper quartile
There are 80 pieces of data 80 70 60 50 40 30 20 10 x
Cumulative frequency
Weight (g)
The lower quartile is the 20th
piece of data ¼ of the total
pieces of data
The middle is the 40th
The upper quartile is the 60th
piece of data ¾ of the total
pieces of data
Median position
Read across, then
Down to find the
median weight
Lower quartile
Lower quartile is 38g
Median weight is 54g
Upper quartile is 68g

7. Cumulative Frequency
The upper and lower quartiles can now be used to find what is called
The interquartile range and is found by:
Upper quartile – Lower quartile
In this example:
Lower quartile is 38g
Upper quartile is 68g
The interquartile range (IQR) = 68 – 38 = 30g
Because this has been found by the top ¾ subtract the bottom ¼
½ of the data (50%) is contained within these values
So we can also say from this that half the mice weigh between
38g and 68g

8. In an international competition 60 children from Britain and France
Cumulative Frequency
In an international competition 60 children from Britain and France
Did the same Maths test. The results are in the table below:
Marks
Britain Frequency
Britain c.f.
France Frequency
France c.f
1 - 5 1 2
6 - 10 5 4 11 8 16 10 19
Using the same axes draw the cumulative frequency diagram for each
country.
Find the median mark and the upper and lower quartiles for both
countries and the interquartile range.
Make a short comment comparing the two countries

9. Both have 60 pieces of data
Cumulative Frequency
Both have 60 pieces of data
Marks
Britain Frequency
Britain c.f.
France Frequency
France c.f
1 - 5 1 2
6 - 10 5 4 11 8 16 10 19 1 2
Median position is 30 3 7 7 18
Lower quartile position is 15 15 34 31 44
Upper quartile position is 45 50 52 60 60
Britain 60 50 40 30 20 10
Cumulative frequency
Marks x x
France
Britain
France
LQ = 20
LQ = 13.5
Median = 25
Median = 19
UQ = 26
UQ = 29
IQR = 12.5
IQR = 9
The scores in Britain are higher with less variation

10. Cumulative Frequency
Summary
B Grade Construct and interpret a cumulative frequency
diagram
Use a cumulative frequency diagram to estimate
the median and interquartile range
Make a running total of the frequency
Put the end points not the class interval on the x axis
Plot the points at the end of the class interval
Join the points with straight lines – if it is not an ‘S’ curve
****Check your graph****
Find the median by drawing across from the middle of the
cumulative frequency axis
Find the LQ and UQ from ¼ and ¾ up the c.f. axis