CHAPTER 39
Cumulative Frequency
Cumulative Frequency
Cumulative Frequency Tables
The cumulative frequency is the running total of the
frequency up to the end of each class interval.
Cumulative Frequency Graph
To draw a cumulative frequency graph:
1. Draw and label the variable on the horizontal axis and
the cumulative frequency on the vertical axis
2. Plot the cumulative frequency against the upper class
boundary of each class and join the points with a smooth
curve.
Using Cumulative Frequency Graphs
Median
When we draw a cumulative frequency graph we can use it to
calculate the MEDIAN of the frequency distribution. Work out what
half the total frequency is, go to this point on the cumulative
frequency axis (vertical axis). Read across to the curve and down to
the horizontal axis, this value is the median of the frequency
distribution.
The Interquartile Range
To find the LOWER QUARTILE, work out what one quarter of the
total frequency is, go to this point on the cumulative frequency axis.
Read across to the curve and down to the horizontal axis, this value
is the LOWER QUARTILE of the frequency distribution. To find the
UPPER QUARTILE, work out what three quarters of the total
frequency is, go to this point on the cumulative frequency axis. Read
across to the curve and down to the horizontal axis, this value is the
UPPER QUARTILE.
Interquartile Range = Upper Quartile – Lower Quartile
Cumulative Frequency Graph
Example
The following table shows the marks obtained in a GCSE
mathematics exam:
Number of
a) Draw a cumulative
frequency curve
Marks
b) Use your graph to
1 - 10
find an estimate for:
(i) the median number
11 – 20
of marks
21 - 30
(ii) the interquartile range
of the marks
31 – 40
(iii) the percentage of students
41 – 50
who scored less than 36 marks
c) If the pass mark is set at 25 marks,
estimate how many students pass
d) If 65% of the students pass, estimate
what the pass mark is
Frequency
15
58
92
30
5
Cumulative Frequency Graph
Answer
Number of Marks (less than)
Cumulative Frequency
10.5
15
20.5
15 + 58 = 73
30.5
73 + 92 = 165
40.5
165 + 30 = 195
50.5
195 + 5 = 200
b) (i) The median is the middle value. The total cumulative
frequency is 200. So the middle value will be at
approximately 100
Median = 22 marks
Cumulative Frequency Graph
(ii) The lower quartile is the one quarter value. The total
cumulative frequency is 200. So the one quarter value
will be at approximately 50
Lower quartile = 17.5 Marks
The upper quartile is the three quarters value. The total
cumulative frequency is 200. So the three quarters
value will be at approximately 150.
Upper quartile = 28 Marks
Interquartile range = 28 – 17.5
= 10.5 Marks
(iii) The number of students who score less than 36 Marks
= 185
% of students who score less than 36 marks = 185 x 100
200
= 92.5%
Cumulative Frequency Graph
c) The number of students who score less than 25 marks =
128
The number of students who pass = 200-128
= 72
d) 65% of 200 = 65 x 200
100
130 students score more than the pass mark
so 70 students score less than the pass mark
Pass mark = 20 Marks
Box Plots

BOX PLOTS (or BOX AND WHISKER DIAGRAMS) are
a convenient way of representing the MINIMUM VALUE,
LOWER QUARTILE, MEDIAN, UPPER QUARTILE and
MAXIMUM VALUE of a set of data.

It is easy to compare two (or more) distributions by
comparing their BOX PLOTS.

It is very easy to draw a BOX PLOT underneath the
CUMULATIVE FREQUENCY CURVE representing the
data.

A BOX is used to display the LOWER QUARTILE,
MEDIAN and UPPER QUARTILE. Lines join this box to
the Minimum Value and Maximum Value.

A BOX PLOT shows how the data is spread out and how
the middle 50% of the data is clustered.