In Maths GCSE there are lots of ways which you are expected to handle data, one of these is understanding cumulative data.

Frequency polygons plot the frequency for each individual groups of data, whereas cumulative frequency graphs plot the total frequency from zero to the end of a particular interval.

Cumulative frequency means you add up the frequencies as you go down a cumulative frequency table, and for this reason, their shape always loosely resembles a stretched out S.

They can be used to find the median, lower quartile and upper quartile of a data set. They can also be used to find the inter-quartile range.

It is important to remember that you plot the cumulative frequency against the upper bound of a class interval. See here for more information on upper bounds

Constructing a cumulative frequency diagram:

From a table listing frequencies, we construct a cumulative frequency diagram by adding up the frequencies going along the list. 

The table below shows the heights of 50 people.

Height(cm) Frequency
140-144 1
145-149 2
150-154 5
155-159 6
160-164 8
165-169 13
170-174 8
175-179 5
180-184 2

Adding up the frequencies as we go down the table produces the table below

Height(cm) Frequency Cumulative frequency
140-144 1 1
145-149 2 3
150-154 5 8
155-159 6 14
160-164 8 22
165-169 13 35
170-174 8 43
175-179 5 48
180-184 2 50

We need to add a class with 0 frequency before the first class and then find the upper boundary for each class interval. An upper bound of the class interval 140-144 is 144.5, for example.

Height(cm) Frequency Upper bound  Cumulative frequency
140-144 1 144.5 1
145-149 2 149.5 3
150-154 5 154.5 8
155-159 6 159.5 14
160-164 8 164.5 22
165-169 13 169.5 35
170-174 8 174.5 43
175-179 5 179.5 48
180-184 2 184.5 50

Now all we have to do is plot the graph of cumulative frequency (always on the y-axis) against the upper bound. We need to draw a smoothed curve through all the points.

Interpreting a Cumulative frequency diagram:

From a cumulative frequency diagram you can find the median and quartiles.

  • To find the Lower Quartile, Go up the y-axis to 1/4 of the total frequency, then draw a line going horizontally till you hit the graph. Then draw another line going down to the x-axis from that point.
  • To find the Upper Quartile, Go up the y-axis to 3/4 of the total frequency, then draw a line going hortizontally till you hit the graph. Then draw another line going down to the x-axis from that point
  • To find the Median, Go up the y-axis to 1/2 of the total frequency, then follow the same pattern as before.
  • The Interquartile Range =  Upper Quartile - Lower Quartile

From the graph above we can see that the lower quartile's value is 158.2, the upper quartile's value is 170.8. The median is 165.8. So the Interquartile range => 170.8-158.2 = 12.6


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