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Q.

A group of 10 observations has mean 5 and standard deviation $2 \sqrt{6} .$ Another group of 20 observation has mean 5 and standard deviation $3 \sqrt{2},$ then the standard deviation of combined group of 30 observationsis

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a

$\sqrt{5}$

b

$3 \sqrt{5}$

c

20

d

$2 \sqrt{5}$

answer is A.

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Detailed Solution

$x_{1} = 5, σ_{1} = 2 \sqrt{6}, n_{1} = 10$
$x_{2} = 5 σ_{2} = 3 \sqrt{2}, n_{2} = 20$
combined variance $σ^{2} = \frac{n_{1} σ_{1}^{2} + n_{2} σ_{2}^{2}}{n_{1} + n_{2}} + \frac{n_{1} n_{2}}{{(n_{1} + n_{2})}^{2}} {({\bar{x}}_{1} - {\bar{x}}_{2})}^{2}$
$σ^{2} = \frac{(10) (24) + (20) (18)}{30} + \frac{200}{900} {(0)}^{2}$
$= \frac{240 + 360}{30} = \frac{600}{30} = 20$
$σ = \sqrt{20} = 2 \sqrt{5}$
standard deviation of combined group $= 2 \sqrt{5}$

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A group of 10 observations has mean 5 and standard deviation 26. Another group of 20 observation has mean 5 and standard deviation 32, then the standard deviation of combined group of 30 observationsis