Question

The marks obtained by 80 students in test are given below:

Marks	4	12	20	28	36	44
Number of students	8	10	16	24	15	7

Find the modal marks.

Answer 1

Marks (xi)	Number of Students (fi)	Cumulative Frequency	(xi)(fi)
4	8	8	32
12	10	18	120
20	16	34	320
28	24	58	672
36	15	73	540
44	7	80	308
	n = 80		$\sum _{}^{}{f}_i{x}_i=$1992

$$\begin{gathered} \mathrm{The}\mathrm{number}\mathrm{of}\mathrm{observations}\mathrm{is}80,\mathrm{which}\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{number}. \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ \mathrm{Median}=\frac{1}{2}\left[{\left(\frac{\mathrm{n}}{2}\right)}^{\mathrm{th}}\mathrm{term}+{\left(\frac{\mathrm{n}}{2}+1\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[{\left(\frac{80}{2}\right)}^{\mathrm{th}}\mathrm{term}+{\left(\frac{80}{2}+1\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[{\left(40\right)}^{\mathrm{th}}\mathrm{term}+{\left(41\right)}^{\mathrm{st}}\mathrm{term}\right] \\ =\frac{1}{2}\left[28+28\right] \\ =28 \end{gathered}$$

Also,
$$\begin{gathered} \mathrm{Mean}=\frac{{\displaystyle \sum _{}^{}}{f}_i{x}_i}{{\displaystyle \sum _{}^{}}{f}_i} \\ =\frac{1992}{80}=24.9 \end{gathered}$$

By the empirical formula, we have:
Mode = 3 Median $-$ 2 Mean
Or,
Mode = 3(28) $-$ 2(24.9) = 34.2