Question

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

Item (x)	5	7	9	12	14	17	19	21
Frequency (f)	6	5	3	6	5	3	2	4

Answer 1

Item (xi)	Frequency (fi)	Cumulative Frequency	(xi)(fi)
5	6	6	30
7	5	11	35
9	3	14	27
12	6	20	72
14	5	25	70
17	3	28	51
19	2	30	38
21	4	34	84
	n = 34		$\sum _{}^{}{f}_i{x}_i=$407

$$\begin{gathered} \mathrm{The}\mathrm{number}\mathrm{of}\mathrm{observations}\mathrm{is}34,\mathrm{which}\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{number}. \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ \mathrm{Median}=\frac{1}{2}\left[{\left(\frac{\mathrm{n}}{2}\right)}^{\mathrm{th}}\mathrm{term}+{\left(\frac{\mathrm{n}}{2}+1\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[{\left(\frac{34}{2}\right)}^{\mathrm{th}}\mathrm{term}+{\left(\frac{34}{2}+1\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[{\left(17\right)}^{\mathrm{th}}\mathrm{term}+{\left(18\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[12+12\right] \\ =12 \end{gathered}$$

Also,
$$\begin{gathered} \mathrm{Mean}=\frac{{\displaystyle \sum _{}^{}}{f}_i{x}_i}{{\displaystyle \sum _{}^{}}{f}_i} \\ =\frac{407}{34}=11.97 \end{gathered}$$

By the empirical formula, we have:
Mode = 3 Median $-$ 2 Mean
Or,
Mode = 3(12) $-$ 2(11.97) = 12.06