Question

The ages of the employees of a company are given below:

Age (in years)	19	21	23	25	27	29	31
Number of persons	13	15	16	18	16	15	13

Find the mean, median and mode for the above data.

Answer 1

Age (xi)	Number of Persons (fi)	Cumulative Frequency	(xi)(fi)
19	13	13	247
21	15	28	315
23	16	44	368
25	18	62	450
27	16	78	432
29	15	93	435
31	13	106	403
	n = 106		$\sum _{}^{}{f}_i{x}_i=$2650

$$\begin{gathered} \mathrm{The}\mathrm{number}\mathrm{of}\mathrm{observations}\mathrm{is}106,\mathrm{which}\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{number}. \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ \mathrm{Median}=\frac{1}{2}\left[{\left(\frac{\mathrm{n}}{2}\right)}^{\mathrm{th}}\mathrm{term}+{\left(\frac{\mathrm{n}}{2}+1\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[{\left(\frac{106}{2}\right)}^{\mathrm{th}}\mathrm{term}+{\left(\frac{106}{2}+1\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[{\left(53\right)}^{\mathrm{rd}}\mathrm{term}+{\left(54\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[25+25\right] \\ =25 \end{gathered}$$

Also,
$$\begin{gathered} \mathrm{Mean}=\frac{{\displaystyle \sum _{}^{}}{f}_i{x}_i}{{\displaystyle \sum _{}^{}}{f}_i} \\ =\frac{2650}{106}=25 \end{gathered}$$

By the empirical formula, we have:
Mode = 3 Median $-$ 2 Mean
Or,
Mode = 3(25) $-$ 2(25) = 25