The mean of the following distribution is $24$. Find the missing frequency '$a$'.
| Class Interval | $0 - 10$ | $10 - 20$ | $20 - 30$ | $30 - 40$ | $40 - 50$ |
|---|---|---|---|---|---|
| Frequency | $10$ | $6$ | $a$ | $12$ | $5$ |
| Class Interval | Mid value $x_i$ | Frequency $f_i$ | $f_i x_i$ |
|---|---|---|---|
| $0 - 10$ | $5$ | $10$ | $50$ |
| $10 - 20$ | $15$ | $6$ | $90$ |
| $20 - 30$ | $25$ | $a$ | $25 a$ |
| $30 - 40$ | $35$ | $12$ | $420$ |
| $40 - 50$ | $45$ | $5$ | $225$ |
| $\Sigma f_i = 33 + a$ | $\Sigma f_i x_i =785 + 25 a$ |
Mean $= \dfrac{\Sigma f_i x_i}{\Sigma f_i} = \dfrac{785 + 25 a}{33 + a}$
Given: $\;$ Mean $= 24$
i.e. $\;$ $\dfrac{785 + 25 a}{33 + a} = 24$
i.e. $\;$ $785 + 25 a = 792 + 24 a$
$\implies$ $a = 7$