CHAPTER 39

Cumulative Frequency

Cumulative Frequency Cumulative Frequency TablesThe cumulative frequency is the running total of the frequency up to the end of each class interval.

Cumulative Frequency GraphTo 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 GraphsMedianWhen 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 RangeTo 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 GraphExampleThe following table shows the marks obtained in a GCSE mathematics exam:a) Draw a cumulative frequency curveb) Use your graph to

find an estimate for:(i) the median number of marks(ii) the interquartile range

of the marks(iii) the percentage of students

who scored less than 36 marksc) If the pass mark is set at 25 marks,

estimate how many students passd) If 65% of the students pass, estimate

what the pass mark is

Number of Marks

Frequency

1 - 10 15

11 – 20 58

21 - 30 92

31 – 40 30

41 – 50 5

Cumulative Frequency Graph

Answer

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

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

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 50Lower quartile = 17.5 MarksThe 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 MarksInterquartile 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.