Question

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

Marks	10	11	12	13	14	16	19	20
Number of students	3	5	4	5	2	3	2	1

Answer 1

Marks (xi)	No. of Students (fi)	Cumulative Frequency	(xi)(fi)
10	3	3	30
11	5	8	55
12	4	12	48
13	5	17	65
14	2	19	28
16	3	22	48
19	2	24	38
20	1	25	20
	n = 25		$\sum _{}^{}{f}_i{x}_i=$332

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

^{$$\begin{gathered} ={\left(13\right)}^{\mathrm{th}}\mathrm{term} \\ =13 \end{gathered}$$}


Also,
$$\begin{gathered} \mathrm{Mean}=\frac{{\displaystyle \sum _{}^{}}{f}_i{x}_i}{{\displaystyle \sum _{}^{}}{f}_i} \\ =\frac{332}{25}=13.28 \end{gathered}$$

By the empirical formula, we have:
Mode = 3 Median $-$ 2 Mean
Or,
Mode = 3(13) $-$ 2(13.28) = 12.44