Cumulative frequency diagrams

A cumulative frequency diagram creates a running total of the amounts within a table.

Example

The table below shows the lengths of 40 babies at birth.

To calculate the cumulative frequencies, add the frequencies together.

Length (cm)FrequencyCumulative frequency
30 \textless l \leq 3544
35 \textless l \leq 401014 ( 4 + 10 = 14)
40 \textless l \leq 451125 ( 14 + 11 = 25)
45 \textless l \leq 501237 ( 25 + 12 = 37)
50 \textless l \leq 55340 ( 37 + 3 = 40)

A cumulative frequency diagram is drawn by plotting the upper class boundary with the cumulative frequency. The upper class boundaries for this table are 35, 40, 45, 50 and 55.

Cumulative frequency is plotted on the vertical axis and length is plotted on the horizontal axis.

Length vs Cumulative frequency graph

Finding averages from a cumulative frequency

A cumulative frequency diagram is a good way to represent data to find the median, which is the middle value.

To find the median value, draw a line across from the middle value of the table. In the example above, there are 40 babies in the table. The middle of these 40 values is the 20th value, so go across from this value and find the median length.

Finding the interquartile range

A cumulative frequency diagram is also a good way to find the interquartile range, which is the difference between the upper quartile and lower quartile.

The interquartile range is a measure of how spread out the data is. It is more reliable than the range because it does not include extreme values. A high value for the interquartile range shows that the data is spread out. A low value for the interquartile range means the data is closer together or more consistent.

Example

There are 40 babies in the table, so to find the lower quartile, find \frac{1}{4} of 40, which is the 10th value. Reading from the graph, the lower quartile is 38.

To find the upper quartile, find \frac{3}{4} of 40, which is the 30th value. Reading from the graph, the upper quartile is 47.

The interquartile range is the upper quartile – the lower quartile, so for this data the interquartile range is 47 – 38 = 9.

Using a graph to identify lower, median and upper interquartile ranges