Find the mean and variance for the following frequency distribution:
Classes $0-30$ $30-60$ $60-90$ $90-120$ $120-150$ $150-180$ $180-210$
$f_i$ $2$ $3$ $5$ $10$ $3$ $5$ $2$

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(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 $\sigma^2 = \frac{h^2}{N^2} [N \sum f_i y_i^2 - (\sum f_i y_i)^2] = \frac{30^2}{30^2} [30(76) - (2)^2] = 2280 - 4 = 2276$.

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