Question

The following table shows the weight (in kg.) of $50$ persons:

Weight (in kg)	$42$	$47$	$52$	$57$	$62$	$67$	$72$
Number of persons	$3$	$8$	$6$	$8$	$11$	$5$	$9$

Find the mean, median and mode.

Answer 1

Step 1: Find the mean

$x$	$f$	$fx$	Cumulative frequency
$42$	$3$	$126$	$3$
$47$	$8$	$376$	$11$
$52$	$6$	$312$	$17$
$57$	$8$	$456$	$25$
$62$	$11$	$682$	$36$
$67$	$5$	$335$	$39$
$72$	$9$	$648$	$48$
	$\sum _{}f=50$	$\sum _{}fx=2935$

For the grouped data, mean $=\frac{\sum _{}fx}{{\displaystyle \sum _{}}f}$

$$\begin{gathered} =\frac{2935}{50} \\ =58.7kg \end{gathered}$$

Step 2: Find the median

Here, the number of observations $n=50$. Therefore,

Median $=\frac{1}{2}\left({\left(\frac{n}{2}\right)}^{th}term+{\left(\frac{n}{2}+1\right)}^{th}term\right)$

$=\frac{1}{2}\left({25}^{th}term+{26}^{th}term\right)$

$$\begin{gathered} =\frac{1}{2}\left(57+62\right) \\ =59.5kg \end{gathered}$$

Step 3: Find the mode

Buy empirical formula, Mode $=3Median-2Mean$

$$\begin{gathered} =3\times 59.5-2\times 58.7 \\ =178.5-117.4 \\ =61.1kg \end{gathered}$$

Hence, the mean, median and mode is $58.7kg,59.5kgand61.1kg$ respectively.