Cumulative Frequency

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
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 lt w 10 3
10 lt w 20 5
20 lt w 30 5
30 lt w 40 9
40 lt w 50 11
50 lt w 60 15
60 lt w 70 14
70 lt w 80 8
80 lt w 90 6
90 lt 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
Cumulative Frequency
x
x
Weight (g) Frequency (f)
0 lt w 10 3
10 lt w 20 5
20 lt w 30 5
30 lt w 40 9
40 lt w 50 11
50 lt w 60 15
60 lt w 70 14
70 lt w 80 8
80 lt w 90 6
90 lt w 100 4
Cumulative Frequency
 
 
 
 
 
 
 
 
 
 
x
x
x
Cumulative frequency
x
x
x
x
x
Weight (g)
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 point are now joined with straight lines
The line always starts at the bottom of the
first class interval
This is because we dont know where in the class
interval 0 lt 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.
5
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
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
6
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
Upper quartile is 68g
Lower quartile is 38g
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
7
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 2   5  
11 - 15 4   11  
16 - 20 8   16  
21 - 25 16   10  
26 - 30 19   8  
31 - 35 10   8  
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
8
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 2   5  
11 - 15 4   11  
16 - 20 8   16  
21 - 25 16   10  
26 - 30 19   8  
31 - 35 10   8  
1
2
Median position is 30
3
7
Lower quartile position is 15
7
18
15
34
Upper quartile position is 45
31
44
50
52
60
60
France
Britain
LQ 20
LQ 13.5
Median 19
Median 25
UQ 26
UQ 29
IQR 12.5
IQR 9
The scores in Britain are higher with less
variation
9
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