Find the mean of the following distribution of marks using step deviation method.
| Marks | $30 - 40$ | $40 - 50$ | $50 - 60$ | $60 - 70$ | $70 - 80$ | $80 - 90$ | $90 - 100$ |
|---|---|---|---|---|---|---|---|
| Number of students | $8$ | $12$ | $24$ | $16$ | $9$ | $7$ | $4$ |
Class size $= h = 10$
Let assumed mean $= A = 65$
| Marks | Frequency $\left(f_i\right)$ | Mid-value $\left(x_i\right)$ | deviation $= x_i - A$ | $ t_i = \dfrac{x_i - A}{h} $ | $ f_i \times t_i $ |
|---|---|---|---|---|---|
| $30 - 40$ | $8$ | $35$ | $-30$ | $-3$ | $-24$ |
| $40 - 50$ | $12$ | $45$ | $-20$ | $-2$ | $-24$ |
| $50 - 60$ | $24$ | $55$ | $-10$ | $-1$ | $-24$ |
| $60 - 70$ | $16$ | $65$ | $0$ | $0$ | $0$ |
| $70 - 80$ | $9$ | $75$ | $10$ | $1$ | $9$ |
| $80 - 90$ | $7$ | $85$ | $20$ | $2$ | $14$ |
| $90 - 100$ | $4$ | $95$ | $30$ | $3$ | $12$ |
$\Sigma f_i = 80$, $\;$ $\Sigma f_i \times t_i = -37$
Mean $= A + \dfrac{\Sigma f_i \times t_i}{\Sigma f_i} \times h$
i.e. $\;$ Mean $= 65 + \dfrac{\left(-37\right)}{80} \times 10 = 65 - 4.625 = 60.375$