Cumulative Frequency
Objectives:
B Grade
Construct and interpret a cumulative frequency
diagram
Use a cumulative frequency diagram to estimate
the median and interquartile range
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.
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)
0 < w ≤ 10
10 < w ≤ 20
20 < w ≤ 30
30 < w ≤ 40
40 < w ≤ 50
50 < w ≤ 60
60 < w ≤ 70
70 < w ≤ 80
80 < w ≤ 90
90 < w ≤100
Frequency Cumulative
(f)
Frequency
3
3
5
8
5
13
22
9
33
11
15
48
62
14
8
70
76
6
80
4
Cumulative means adding up, so a cumulative frequency diagram
requires a running total of the frequency.
Cumulative Frequency
0 < w ≤ 10
10 < w ≤ 20
20 < w ≤ 30
30 < w ≤ 40
40 < w ≤ 50
50 < w ≤ 60
60 < w ≤ 70
70 < w ≤ 80
80 < w ≤ 90
90 < w ≤100
Frequency Cumulative
(f)
Frequency
3
3
5
8
5
13
22
9
33
11
15
48
62
14
8
70
76
6
80
4
x
x
x
70
Cumulative frequency
Weight (g)
80
x
60
50
x
40
x
30
x
20
x
10
0
x
x
0
10
20
30
40
50
60
Weight (g)
70
80
90
The point
are now
joined with
lines
cumulative
frequency
(c.f.)straight
can now
be plotted on a graph
The line
always
at the
taking
care
to plotstarts
the c.f.
at the end of each class interval.
bottom
of the first
class
interval
This
is
because
we
don’t
know
in the
interval called an
The resulting graph should
lookwhere
like this
andclass
is sometimes
0‘S’
< curve.
w ≤ 10, the values are, but we do know that by the end of
the class interval there are 3 pieces of data
100
Cumulative Frequency
x
x
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
The lower
is the 20th
x
70
middlequartile
is the 40th
x
piece of data ¼ of the total
60
pieces of data
50
x
th
The upper quartile is the 60
Median position
40
piece of data ¾ of the total
x
Read
across,
then
30
pieces of data
Down
find the
x
20
Lowertoquartile
median weight
x
10
x
0
Lower quartile is 38g
Median weight is 54g
Upper quartile is 68g
x
0
10
20
30
40
50
60
Weight (g)
70
80
90
100
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
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
1-5
6 - 10
11 - 15
16 - 20
21 - 25
26 - 30
31 - 35
Britain
Frequency
1
2
4
8
16
19
10
Britain
c.f.
France
Frequency
2
5
11
16
10
8
8
France
c.f
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
Cumulative Frequency
Marks
1-5
6 - 10
11 - 15
16 - 20
21 - 25
26 - 30
31 - 35
Britain
Frequency
1
2
4
8
16
19
10
Britain
c.f.
1
3
7
15
31
50
60
Britain
France
Frequency
2
5
11
16
10
8
8
Both have 60 pieces of data
France
c.f
2
7
18
34
44
52
60
Median position is 30
Lower quartile position is 15
Upper quartile position is 45
x
60
Britain
LQ = 20
Median = 25
UQ = 29
IQR = 9
Cumulative frequency
France
xx
50
x
France
x
LQ = 13.5
Median = 19
UQ = 26
IQR = 12.5
40
x
30
20
x
10
0
0
xx
5
x
x
10
x
x
15
20
Marks
25
30
35
The scores in Britain are higher with less variation
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