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Representation of data

Cumulative frequency

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Explainer Video

Tutor: Bilal

Summary

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

$1.$	Add a cumulative frequency column to a grouped frequency table.
$2.$	Calculate cumulative frequency by adding frequencies up to and including current row.
$3.$	Draw axes where the $y$ axis is the cumulative frequency and $x$ axis is a given measurement.
$4.$	Plot points at the upper bound of each class at their respective cumulative frequency.
$5.$	Connect the points with a smooth curve or a straight line depending on what fits the graph best.

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.

Height (cm)	Frequency
$140< h \le 145$	$2$
$145< h \le 150$	$3$
$150< h \le155$	$13$
$155< h \le 160$	$21$
$160 < h \le 165$	$5$

Add a cumulative frequency column and calculate each value:

Height (cm)	Frequency	Cumulative frequency
$140< h \le 145$	$2$	$2$
$145< h \le 150$	$3$	$2+3=5$
$150< h \le155$	$13$	$5+13=18$
$155< h \le 160$	$21$	$18+21=39$
$160 < h \le 165$	$5$	$39+5=44$

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:

Maths; Representation of data; KS5 Year 12; Cumulative frequency

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.

Learn with Basics

Length:

Unit 1

Averages from grouped frequency tables

Unit 2