Find the mean and variance for the following frequency distribution.

Classes $0-30$ $30-60$ $60-90$ $90-120$ $120-150$ $50-180$ $180-210$
$f_i$ $2$ $3$ $5$ $10$ $3$ $5$ $2$

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Class

Freq

 ${f_i}$

Mid-point

 ${x_i}$

${y_i} = \frac{{{x_i} - 105}}{{30}}$ ${y_i}^2$ ${f_i}{y_i}$ ${f_i}{y_i}^2$
$0-30$ $2$ $15$ $-3$ $9$ $-6$ $18$
$30-60$ $3$ $45$ $-2$ $4$ $-6$ $12$
$60-90$ $5$ $75$ $-1$ $1$ $-5$ $5$
$90-120$ $10$ $105$ $0$ $0$ $0$ $0$
$120-150$ $3$ $135$ $1$ $1$ $3$ $3$
$150-180$ $5$ $165$ $2$ $4$ $10$ $20$
$180-210$ $2$ $195$ $3$ $9$ $6$ $18$
  $30$       $2$ $76$

Mean, $ \bar x = A + \frac{{\sum\limits_{i = 1}^7 {{f_i}{y_i}} }}{N} \times h$

$ = 105 + \frac{2}{{30}} \times 30 = 105 + 2 = 107$

Variance,  $\left( {{\sigma ^2}} \right) = \frac{{{h^2}}}{{{N^2}}}\left[ {N\sum\limits_{i = 1}^7 {{f_i}{y_i}^2 - {{\left( {\sum\limits_{i = 1}^7 {{f_i}{y_i}} } \right)}^2}} } \right]$

$=\frac{(30)^{2}}{(30)^{2}}\left[30 \times 76-(2)^{2}\right]$

$=2280-4$

$=2276$

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