Cumulative frequency

Example

A teacher arranged the marks gained by all year 10 pupils in a mathematics test in a table as shown below:

This table shows the number of pupils (called the frequency) who gained marks in the various mark bands, eg 31-40. For example, the number of pupils who scored between 21 and 30 marks was 11. No pupil scored fewer than 11 marks or more than 90 marks.

To create a cumulative total for the frequency of pupils in each group (called the cumulative frequency) a third column is created as shown below:

The cumulative frequency column makes it easy to see at a glance that 68 pupils scored 50 marks or fewer, and that 32 pupils scored 40 marks or fewer.

A cumulative frequency graph is a way of presenting information visually, which allows other information to be deduced.

For example, from the graph we can obtain the median (or middle) mark. The median is the mark which half of all pupils exceed and half do not reach. As there are 160 pupils, we need to find 80 pupils, at the half-way point on the vertical axis, and then draw a line across until it meets the graph. Drawing a vertical line down from this point and reading the number of marks at that point shows that the median is 53 marks.

It is also possible to find the upper and lower quartile marks from the graph.

In this example, the lower quartile is the mark which one quarter of all pupils' scores do not reach.

In this example, the upper quartile is the mark which three quarters of all pupils' scores do not reach.

To find the lower quartile, find the point one quarter of the way up the vertical axis, which is 40 on the cumulative frequency axis. Draw a line across from the 40 mark until it meets the graph. Draw a vertical line down from this point. The lower quartile is 43 marks. Therefore a quarter of the marks lie below 43 marks.

To find the upper quartile, we need to find the point that is three quarters of the way up the vertical axis which is 120 on the cumulative frequency axis. Draw a line across from the 120 mark until it meets the graph. Draw a vertical line down from this point. An estimate of the upper quartile is 63. Therefore three-quarters of the marks lie below 63 marks.

The inter-quartile range is often used to give an idea of how widely the items of data are spread out.

The inter-quartile range is found by calculating the difference in value between the upper quartile and the lower quartile, that is:

upper quartile value - lower quartile value.

In this example, the estimate of the inter-quartile range is 63 - 43 = 20.

The marks of the middle 50% of pupils lie roughly between the lower quartile mark, 43, and the upper quartile mark, 63.

If this information was shown using a box and whisker diagram, the box would be drawn with the left-hand edge at 43 and the right-hand edge at 63.

Worked examples

This is a cumulative frequency graph illustrating, in percentages, the amount of time secondary school pupils spend on homework.

Example one

What is the median time that is spent on homework?

The median value is the number of hours spent on homework by the middle pupil(s). The middle value will be at 50% as the graph shows values from 0-100%.

From this point read across to the graph and draw a vertical line down to the horizontal axis, giving an estimate of the median as 5.4 hours.

The median is 5.4 hours, or 5 hours 24 minutes.

Example two

What is the inter-quartile range?

It is necessary to first find the upper and lower quartiles.

To find the lower quartile, it is necessary to find the number of hours of homework done by 25% or fewer of the pupils.

Reading across from 25% and down from the graph gives an estimate of the lower quartile as just under four hours.

Following the same process, the upper quartile at 75% is about 7½ hours.

From this an estimate of the inter-quartile range is found as

upper quartile - lower quartile

= 7½ hours

So the inter-quartile range is roughly 3½ hours.

The number of hours spent on homework by the middle 50% of the pupils ranges from 4 hours to 7½ hours.

Example three

What percentage of pupils spend six hours or less on homework per week?

Find the point marked 'six hours' on the horizontal axis. Move from the axis vertically up to the curve. Move from the curve horizontally to the vertical axis. Read the value from the graph. In this case the value is 60.

Therefore about 60% of the pupils spend six hours or less on homework.

Example four

What percentage of pupils spend more than seven hours on homework?

Find the point marked 'seven hours' on the horizontal axis and read the corresponding value from the graph. This shows that 70% of pupils spend seven hours or less on homework.

The question asked was how many pupils spend more than seven hours per week on homework, so therefore 100% - 70% = 30% spend more than seven hours on homework.

70% of pupils spend seven hours or less on homework per week.

So 30% of pupils spend more than seven hours on homework per week.

Note: percentages are sometimes referred to as percentiles in this context. Percentiles are defined in the glossary.

Avoiding common errors

• Ensure the cumulative frequency graph is always plotted using the correct axes, ie cumulative frequency is always plotted on the vertical axis
• Decide whether a question is asking for ‘more than’ or ‘less than’ a particular value (see Example four)
• Decide whether to read from the vertical axis to the horizontal axis or vice versa

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