Find the mean and variance for the following frequency distribution.

Classes $0-10$ $10-20$ $20-30$ $30-40$ $40-50$
Frequencies $5$ $8$ $15$ $16$ $6$

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Class Frequency ${f_i}$ Mid-point ${x_i}$ ${y_i} = \frac{{{x_i} - 25}}{{10}}$ ${y_i}^2$ ${f_i}{y_i}$ ${f_i}{y_i}^2$
$0-10$ $5$ $5$ $-2$ $4$ $-10$ $20$
$10-20$ $8$ $15$ $-1$ $1$ $-8$ $8$
$20-30$ $15$ $25$ $0$ $0$ $0$ $0$
$30-40$ $16$ $35$ $1$ $1$ $16$ $16$
$40-50$ $6$ $45$ $2$ $4$ $12$ $24$
  $50$       $10$ $68$

Mean, $\bar x = A + \frac{{\sum\limits_{i = 1}^5 {{f_i}{y_i}} }}{N} \times h$

$ = 25 + \frac{{10}}{{50}} \times 10 = 25 + 2 = 27$

Variance, $\left( {{\sigma ^2}} \right) = \frac{{{h^2}}}{{{N^2}}}\left[ {N\sum\limits_{i = 1}^5 {{f_i}{y_i}^2 - {{\left( {\sum\limits_{i = 1}^5 {{f_i}{y_i}} } \right)}^2}} } \right]$

$=\frac{(10)^{2}}{(50)^{2}}\left[50 \times 68-(10)^{2}\right]$

$=\frac{1}{25}[3400-100]=\frac{3300}{25}$

$=132$

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