Question

The following table shows the weights of 12 persons.

Weight (in kg):	48	50	52	54	58
Number of persons:	4	3	2	2	1

Find the median and mean weights. Using empirical relation, calculate its mode.

Answer 1

Calculation of Mean

Weight (xi)	48	50	52	54	58	Total
Number of Persons (fi)	4	3	2	2	1	$\sum _{}{f}_i$=12
fixi	192	150	104	108	58	$\sum _{}{f}_i{x}_i$ = 612

Mean = $\frac{\sum _{}{f}_i{x}_i}{\sum _{}{f}_i}=\frac{612}{12}=51\mathrm{kg}$.

Here, n = 12
$$\begin{gathered} \mathrm{Median}=\frac{\mathrm{value}\mathrm{of}{\left({\displaystyle \frac{n}{2}}\right)}^{\mathrm{th}}\mathrm{observation}+\mathrm{value}\mathrm{of}{\left({\displaystyle \frac{n}{2}+1}\right)}^{\mathrm{th}}\mathrm{observation}}{2} \\ \Rightarrow \mathrm{Median}=\frac{\mathrm{value}\mathrm{of}{\left({\displaystyle 6}\right)}^{\mathrm{th}}\mathrm{observation}+\mathrm{value}\mathrm{of}{\left({\displaystyle 7}\right)}^{\mathrm{th}}\mathrm{observation}}{2} \\ \Rightarrow \mathrm{Median}=\frac{50+50}{2} \\ \Rightarrow \mathrm{Median}=50\mathrm{kg}. \end{gathered}$$

Now,
Mode = 3 Median - 2 Mean
⇒ Mode = 3 x 50 - 2 x 51
⇒Mode = 150 - 102
⇒ Mode = 48 kg.
Thus, Mean = 51 kg, Median = 50 kg and Mode = 48 kg.