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Cumulative frequency in reverse

On some occasions we may have to work with cumulative frequency ‘in reverse’. Sometimes we may start with a cumulative frequency curve and then have to work backwards to find the values that would go into a cumulative frequency table. The data in the table may be grouped, so we need to be careful when reading off values on the cumulative frequency curve.


Complete the following table for the grouped frequency of people’s times taken (in minutes) to read a small book using the cumulative frequency graph on the next page.

Time 0 – 60 60 – 90 90 – 120 120 – 180

Clearly the groups are differing sizes and we need to find the frequency of people that took the amounts of time that are stated in each.

Cumulative frequency

To fill in the table we must first use the graph above and read off the cumulative frequency at the points where the groups change. The table is shown below and it is clear that we need to find the cumulative frequency at the points 0, 60, 90, 120 and 180. This is done by finding these values on the horizontal axes and then drawing a line straight up to the curve and reading off the value on the vertical. Doing this we find the point on the curve as:

(0,0), (60,5), (90,16), (120,45), (180,57)

So these are the values of the cumulative frequency at each of the group boundaries. Putting the values for cumulative frequency into the table we get:

Time 0 – 60 60 – 90 90 – 120 120 – 180
Cumulative Freq 5 16 45 57

Then from this table we can find the frequencies of each category by finding the difference in the cumulative frequency as we move up. Cumulative frequencies are found by adding, so to work in reverse and find the frequencies we must subtract.

Five people had times that were up to and including 60 mins so this must be our first frequency and then others are found by taking the cumulative frequency and then subtracting the previous one. This gives the following:

Time 0 – 60 60 – 90 90 – 120 120 – 180
Frequency 5 11 29 12
Cumulative Freq 5 16 45 57
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