Two sets each of 20 observations, have the same standard derivation 5. The first set has a mean 17 and the second a mean 22. Determine the standard deviation of the set obtained by combining the given two sets.

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Given, $n_{1}=20, \sigma_{1}=5, \bar{x}_{1}=17$ and $n_{2}=20, \sigma_{2}=5, \bar{x}_{2}=22$

We know that, $\sigma=\sqrt{\frac{n_{1} s_{1}^{2}+n_{2} s_{2}^{2}}{n_{1}+n_{2}}+\frac{n_{1} n_{2}\left(\bar{x}_{1}-\bar{x}_{2}\right)^{2}}{\left(n_{1}+n_{2}\right)^{2}}}$

$\begin{array}{l}=\sqrt{\frac{20 \times(5)^{2}+20 \times(5)^{2}}{20+20}+\frac{20 \times 20(17-22)^{2}}{(20+20)^{2}}} \\=\sqrt{\frac{1000}{40}+\frac{400 \times 25}{1600}}=\sqrt{25+\frac{25}{4}}=\sqrt{\frac{125}{4}}=\sqrt{31.25}=5.59\end{array}$

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