Find the mean and variance for the following frequency distribution:

Classes	$0-30$	$30-60$	$60-90$	$90-120$	$120-150$	$150-180$	$180-210$
$f_i$	$2$	$3$	$5$	$10$	$3$	$5$	$2$

(A) To find the mean and variance,we use the step-deviation method.

Class	$f_i$	$x_i$	$y_i = \frac{x_i - 105}{30}$	$f_i y_i$	$f_i y_i^2$
$0-30$	$2$	$15$	$-3$	$-6$	$18$
$30-60$	$3$	$45$	$-2$	$-6$	$12$
$60-90$	$5$	$75$	$-1$	$-5$	$5$
$90-120$	$10$	$105$	$0$	$0$	$0$
$120-150$	$3$	$135$	$1$	$3$	$3$
$150-180$	$5$	$165$	$2$	$10$	$20$
$180-210$	$2$	$195$	$3$	$6$	$18$
Total	$N=30$	-	-	$\sum f_i y_i = 2$	$\sum f_i y_i^2 = 76$

Mean $\bar{x} = A + \frac{\sum f_i y_i}{N} \times h = 105 + \frac{2}{30} \times 30 = 107$.
Variance $\sigma^2 = \frac{h^2}{N^2} [N \sum f_i y_i^2 - (\sum f_i y_i)^2] = \frac{30^2}{30^2} [30(76) - (2)^2] = 2280 - 4 = 2276$.