Question

Find the mode of the following distribution of marks obtained by 80 students:

Marks obtained	0-10	10-20	20-30	30-40	40-50
Number of students	6	10	12	32	20

Answer 1

As the class 30 - 40 has maximum frequency, it is the modal class.
Therefore,
x_k = 30, h = 10, f_k = 32, f_k-1 = 12, f_k+1 = 20
$\mathrm{Mode},{\mathrm{M}}_0={\mathrm{x}}_{\mathrm{k}}+\left\{\mathrm{h}\times \frac{\left({\mathrm{f}}_{\mathrm{k}}-{\mathrm{f}}_{\mathrm{k}-1}\right)}{\left(2{\mathrm{f}}_{\mathrm{k}}-{\mathrm{f}}_{\mathrm{k}-1}-{\mathrm{f}}_{\mathrm{k}+1}\right)}\right\}$
$=30+\left\{10\times \frac{\left(32-12\right)}{\left(2\times 32-12-20\right)}\right\}$
$$\begin{gathered} =30+\left\{10\times \frac{20}{\left(64-32\right)}\right\} \\ =30+\left\{10\times \frac{20}{32}\right\} \\ =\left(30+\frac{25}{4}\right) \\ =\left(\frac{145}{4}\right) \\ =36.25 \end{gathered}$$