Find standard deviation of two different sets of numbers when combined A set of 10 numbers has a mean of 10 and a standard deviation of 2.0 another set of 10 numbers have a mean of 4 and a standard deviation of 3.0 find the standard deviation of the 20 numbers
 A: $\newcommand{\+}{^{\dagger}}%
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$$
\angles{x} = {1 \over 20}\sum_{n = 1}^{20}x_{n}
= {1 \over 20}\pars{10\,{\sum_{n = 1}^{10}x_{n} \over 10}
+ 10\,{\sum_{n = 11}^{20}x_{n} \over 10}}
= \half\pars{\angles{x}_{1} + \angles{x}_{2}}
$$
Similarly,
$$
\angles{x^{2}} = \half\pars{\angles{x^{2}}_{1} + \angles{x^{2}}_{2}}
$$
Then,
\begin{align}
&\angles{x^{2}} - \angles{x}^{2}
=
\half\pars{\angles{x^{2}}_{1} + \angles{x^{2}}_{2}}
-
{1 \over 4}\pars{\angles{x}_{1}^{2} + 2 \angles{x}_{1}\angles{x}_{2} + \angles{x}_{2}^{2}}
\\[3mm]&=
\half\pars{\angles{x^{2}}_{1} - \angles{x}_{1}^{2}}
+
\half\pars{\angles{x^{2}}_{2} - \angles{x}_{2}^{2}}
+
{1 \over 4}\pars{\angles{x}_{1}^{2} - 2 \angles{x}_{1}\angles{x}_{2} + \angles{x}_{2}^{2}}
\end{align}
$$\color{#0000ff}{\large%
\sigma
={\root{2} \over 2}\root{%
\sigma_{1}^{2} + \sigma_{2}^{2}
+ \half\pars{\angles{x}_{1} - \angles{x}_{2}}^{2}}}
$$
$$
\sigma = {\root{2} \over 2}\root{2^{2} + 3^{2} + \half\pars{10 - 4}^{2}}
=
{\root{62} \over 2} \approx 3.9370
$$
