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Maths

Inter-quartile range, cumulative frequency, box and whisker plots - Higher

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# Cumulative frequency

The cumulative frequency is obtained by adding up the frequencies as you go along, to give a 'running total'.

## Drawing a cumulative frequency diagram

The table shows the lengths (in cm) of 32 cucumbers.

Before drawing the cumulative frequency diagram, we need to work out the cumulative frequencies. This is done by adding the frequencies in turn.

LengthFrequencyCumulative Frequency
21-2433
25-28710 (= 3 + 7)
29-321222 (= 3 + 7 + 12)
33-36628 (= 3 + 7 + 12 + 6)
37-40432 (= 3 + 7 + 12 + 6 + 4)

The points are plotted at the upper class boundary. In this example, the upper class boundaries are 24.5, 28.5, 32.5, 36.5 and 40.5. Cumulative frequency is plotted on the vertical axis.

There are no values below 20.5cm.

Cumulative frequency graphs are always plotted using the highest value in each group of data, (because the table gives you the total that are less than the upper boundary) and the cumulative frequency is always plotted up a graph, as frequency is plotted upwards.

Cumulative frequency diagrams usually have this characteristic S-shape, called an ogive.

## Finding the median and quartiles

When looking at a cumulative frequency curve, you will need to know how to find its median, lower and upper quartiles, and the interquartile range.

By drawing horizontal lines to represent 1/4 of the total frequency, 1/2 of the total frequency and 3/4 of the total frequency, we can read estimates of the lower quartile, median and upper quartile from the horizontal axis.

Quartiles are associated with quarters. The interquartile range is the difference between the lower and upper quartile.

From these values, we can also estimate the interquartile range: 33 - 28 = 5.

Remember to use the total frequency, not the maximum value, on the vertical axis. The values are always read from the horizontal axis.