Histograms and cumulative frequency

Histograms and cumulative frequency

Histograms are best used for large sets of data, especially when the data has been grouped into classes. They look a little similar to bar charts or frequency diagrams.

In histograms, the frequency of the data is shown by the area of the bars and not just the height.

Histograms are most commonly used for continuous data.

Histograms often have bars of varying width, i.e. unequal class intervals. This is not a problem as we are dealing with area, not just the height.

The vertical axis of a histogram is labelled frequency density and is calculated by the following formula:

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You may be given a question like this:

The ages of people sunbathing on a beach somewhere on a Greek island were recorded and organised into the frequency table below. Draw a histogram of this data.

Ages (x): Frequency (f):
0 ≤ x < 15 15
15 ≤ x < 25 28
25 ≤ x < 40 30
40 ≤ x < 60 42
60 ≤ x < 100 20

To draw the histogram we need to calculate the frequency densities for each class interval. By extending the table using 2 extra columns we can easily calculate the frequency densities required. The workings have been shown for the first 2.

Ages (x): Frequency (f): Class width: Frequency density:
0 ≤ x < 15 15 15 15/15 = 1
15 ≤ x < 25 28 10 28/10 = 2.8
25 ≤ x < 40 30 15 2
40 ≤ x < 60 42 20 2.1
60 ≤ x < 100 20 40 0.5

All we now need to do is draw this onto graph paper and we have our histogram.

The ages will be on the x-axis (from 0 to 100 on a continuous scale).

Frequency density will be on the y-axis (from 0 to 3).


Cumulative frequency is kind oflike a running total. We add each frequency to the ones before to get an 'at least' total.

These cumulative frequencies ('at least' totals) are plotted against theupper class boundaries to give us a cumulative frequency curve.


The distances a javelin is thrown during a sports day is recorded. The data is organised into the frequency table shown below:

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The cumulative frequency column is the column you will be expected to add for yourself.

To draw the cumulative frequency curve we simply plot the cumulative frequencies against the upper end of each class interval.

Remember to always place the cumulative frequencies on the y-axis.

Cumulative frequency

One of the best reasons for drawing a cumulative frequency curve is that it is simple to read off estimates for the median and quartiles even when faced with a large amount of data.

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