Question

Calculate the mode of each of the following using the empirical formula.

x	18	20	25	30	34	38	40
f	6	7	3	7	7	5	5

Answer 1

(xi)	(fi)	Cumulative Frequency	(xi)(fi)
18	6	6	108
20	7	13	140
25	3	16	75
30	7	23	210
34	7	30	238
38	5	35	190
40	5	40	200
	n = 40		$\sum _{}^{}{f}_i{x}_i=$1161

$$\begin{gathered} \mathrm{The}\mathrm{number}\mathrm{of}\mathrm{observations}\mathrm{is}40,\mathrm{which}\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{number}. \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ \mathrm{Median}=\frac{1}{2}\left[{\left(\frac{\mathrm{n}}{2}\right)}^{\mathrm{th}}\mathrm{term}+{\left(\frac{\mathrm{n}}{2}+1\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[{\left(\frac{40}{2}\right)}^{\mathrm{th}}\mathrm{term}+{\left(\frac{40}{2}+1\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[20\mathrm{th}\mathrm{term}+21\mathrm{st}\mathrm{term}\right] \\ =\frac{1}{2}\left[30+30\right] \\ =30 \end{gathered}$$

Also,
$$\begin{gathered} \mathrm{Mean}=\frac{{\displaystyle \sum _{}^{}}{f}_i{x}_i}{{\displaystyle \sum _{}^{}}{f}_i} \\ =\frac{1161}{40}=29.025\approx 29 \end{gathered}$$

By the empirical formula, we have:
Mode = 3 Median $-$ 2 Mean
Or,
Mode = 3(30) $-$ 2(29) = 32