Median from a Histogram (WJEC GCSE Maths & Numeracy (Double Award): Higher): Revision Note

The histogram shows the weight, in kg, of 60 new-born bottlenose dolphins.

Find an estimate for the median weight of the dolphins in the sample, giving your answer correct to two decimal places.

Answer:

$\frac{60}{2}=30$, so to estimate the median we need to find the weight of the 30th dolphin

To find the frequencies represented by the different bars, rearrange  $\mathrm{frequency}\mathrm{density}=\frac{\mathrm{frequency}}{\mathrm{class}\mathrm{width}}$  as  $\mathrm{frequency}=\mathrm{frequency}\mathrm{density}\times \mathrm{class}\mathrm{width}$

 
$\begin{array}{l}4-8\mathrm{kg}:\mathrm{frequency}=1\times 4=4\\ 8-10\mathrm{kg}:\mathrm{frequency}=8\times 2=16\\ 10-12\mathrm{kg}:\mathrm{frequency}=9.5\times 2=19\end{array}$
 

The first two classes have a cumulative frequency of $4+16=20$
So the median is going to be '10 dolphins into' the 10-12 kg class

The height (frequency density) of the 10-12 kg bar is 9.5
We need to find what width would give a frequency of 10
Use  $\mathrm{frequency}\mathrm{density}=\frac{\mathrm{frequency}}{\mathrm{class}\mathrm{width}}$   and solve for width
 

$\begin{array}{l}9.5=\frac{10}{\mathrm{width}}\\ \\ 9.5\times \mathrm{width}=10\\ \\ \mathrm{width}=\frac{10}{9.5}=\frac{20}{19}=1.0526...=1.05\left(2\mathrm{d}.\mathrm{p}.\right)\end{array}$

 
That means that the median lies 1.05 kg into the 10-12 kg class interval

$10+1.05=11.05$

 
Estimated median = 11.05 kg (2 d.p.)