Question

The table given below shows the weights (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

Weight (xi)	Number of Persons (fi)	Cumulative Frequency	(xi)(fi)
42	3	3	126
47	8	11	376
52	6	17	312
57	8	25	456
62	11	36	682
67	5	41	335
72	9	50	648
	n = 50		$\sum _{}^{}{f}_i{x}_i=$2935

$$\begin{gathered} \mathrm{The}\mathrm{number}\mathrm{of}\mathrm{observations}\mathrm{is}50,\mathrm{which}\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{number}. \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ \mathrm{Median}=\frac{1}{2}\left[{\left(\frac{\mathrm{n}}{2}\right)}^{\mathrm{th}}\mathrm{term}+{\left(\frac{\mathrm{n}}{2}+1\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[{\left(\frac{50}{2}\right)}^{\mathrm{th}}\mathrm{term}+{\left(\frac{50}{2}+1\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[{\left(25\right)}^{\mathrm{th}}\mathrm{term}+{\left(26\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[57+62\right] \\ =59.5 \end{gathered}$$

Also,

$$\begin{gathered} \mathrm{Mean}=\frac{{\displaystyle \sum _{}^{}}{f}_i{x}_i}{{\displaystyle \sum _{}^{}}{f}_i} \\ =\frac{2935}{50}=58.7 \end{gathered}$$

By the empirical formula, we have:
Mode = 3 Median $-$ 2 Mean
Or,
Mode = 3(59.5) $-$ 2(58.7) = 61.1