Cumulative frequency 

​​In a nutshell

The data in grouped frequency tables can be represented in cumulative frequency graphs. From these graphs, the median, quartiles and percentiles of a data set can be estimated.

Drawing a cumulative frequency graph

Cumulative frequency graphs can be drawn using the information from a grouped frequency table. Follow the steps in the table below:

​​procedure

Calculating percentiles

Percentiles can be found from a cumulative frequency graph, where the appropriate frequency is taken and the value is found on the $$x$$ axis using the curve. This frequency can be found using the equation below where $$k$$ is the percentile and $$n$$ is the total frequency. This is only valid for continuous data.

​$$\boxed{\dfrac{k}{100}(n)}$$​​

Note:  The lower quartile can be considered the $$25^{th}$$ percentile, the median $$50^{th}$$​ and upper quartile $$75^{th}$$​.

Example 1

The grouped frequency table below shows the height of Mike's classmates. Draw a cumulative frequency graph to represent this data. Then find the lower quartile, median and upper quartile. 

Add a cumulative frequency column and calculate each value:

Draw a cumulative frequency graph. Plot points at the upper bound of each class ($$145\ cm, 150\ cm, 155\ cm, 160\ cm$$ and $$165\ cm$$) and connect the points using a smooth line:

Find the $$y$$ value for the lower quartile, median and upper quartile:

$$\begin{aligned} \text{LQ} &= 0.25(44) &= 11 \\ \text{Median} &= 0.5(44) &=22 \\ \text{UQ} &= 0.75(44) &=33 \end{aligned}$$​​

Use the dashed lines to extract the respective $$x$$ values:

The lower quartile is $$\underline{152.5\ cm}$$, the median $$\underline{155.5\ cm}$$ and the upper quartile $$\underline{158\ cm}$$.

Note: The value given at any given percentile is only an estimate as this method assumes an even distribution within each class.