For two data sets,each of size 5,the variance | vedclass

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(C) Given: $n_1 = 5, \sigma_1^2 = 4, \overline{x}_1 = 2$ and $n_2 = 5, \sigma_2^2 = 5, \overline{x}_2 = 4$. Combined mean $\overline{x}_c = \frac{n_1\

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$\frac{11}{2}$

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(C) Given: $n_1 = 5, \sigma_1^2 = 4, \overline{x}_1 = 2$ and $n_2 = 5, \sigma_2^2 = 5, \overline{x}_2 = 4$. Combined mean $\overline{x}_c = \frac{n_1\overline{x}_1 + n_2\overline{x}_2}{n_1 + n_2} = \frac{5(2) + 5(4)}{5 + 5} = \frac{30}{10} = 3$. Calculate deviations: $d_1 = \overline{x}_1 - \overline{x}_c = 2 - 3 = -1$ and $d_2 = \overline{x}_2 - \overline{x}_c = 4 - 3 = 1$. Combined variance $\sigma^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}$. Substituting the values: $\sigma^2 = \frac{5(4 + (-1)^2) + 5(5 + 1^2)}{5 + 5} = \frac{5(5) + 5(6)}{10} = \frac{25 + 30}{10} = \frac{55}{10} = \frac{11}{2}$.

Items of expenditure	Number of families
$Education$	$150$
$Food and clothing$	$400$
$House rent$	$40$
$Electricity$	$250$
$Miscellaneous$	$160$

Class	Frequency
$0 - 20$	$17$
$20 - 40$	$f_1$
$40 - 60$	$32$
$60 - 80$	$f_2$
$80 - 100$	$19$
Total	$120$

For two data sets,each of size $5$,the variances are given to be $4$ and $5$,and the corresponding means are given to be $2$ and $4$ respectively. The variance of the combined data set is

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