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  • The mean and standard deviation of a set of n1 observations are ba and s1, respectively while the mean and standard deviation of another set of n2 observations are and s2, respectively. Show that the standard deviation of the combined set of (n1 + n2)

The mean and standard deviation of a set of n_1 observations are \bar{X_1} and s_1, respectively while the mean and standard deviation of another set of n_2 observations are \bar{X_2} and s_2, respectively. Show that the standard deviation of the combined set of (n_1 + n_2) observations is given by

\mathrm{SD}=\sqrt{\frac{\mathrm{n}_{1}\left(\mathrm{s}_{1}\right)^{2}+\mathrm{n}_{2}\left(\mathrm{s}_{2}\right)^{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}+\frac{\mathrm{n}_{1} \mathrm{n}_{2}\left(\overline{\mathrm{x}}_{1}-\overline{\mathrm{x}}_{2}\right)^{2}}{\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)^{2}}}

Answers (1)

\\\\ \text{It is given that mean and Standard deviation of a set of }n_{1}\text{ observations are }\overline{x_{1}}~and~ \sigma _{1}~respectively~ \\\\ \\ \text{while the mean and standard deviation of naother set of }n_{2}\text{observations are }\overline{x_{2}}~and~ \sigma _{2}respectively \] \\ \overline{x_{1}}=\frac{1}{n_{1}} \sum _{i=1}^{n_{1}}x_{i} \\\\ \\ \overline{x_{2}}=\frac{1}{n_{2}} \sum _{j=1}^{n_{2}}y_{j} \\\\ \\ \overline{x}=\frac{1}{ \left( n_{1}+n_{2} \right) } \left[ \sum _{i=1}^{n_{1}}x_{i}+ \sum _{j=1}^{n_{2}}y_{j} \right] =\frac{n_{1}\overline{x_{1}}+n_{2}\overline{x_{2}}}{n_{1}+n_{2}} \\\\ \\ \sigma _{1}^{2}=\frac{1}{n_{1}} \sum _{i=1}^{n_{1}} \left( x_{i}-\overline{x} \right) ^{2} \\\\

 


\\ \sigma _{2}^{2}=\frac{1}{n_{2}} \sum _{j=1}^{n_{2}} \left( y_{j}-\overline{x} \right) ^{2} \\\\ \\ \sigma ^{2}= \sigma _{1}^{2}+ \sigma _{2}^{2}=\frac{1}{n_{1}+n_{2}} \left[ \sum _{i=1}^{n_{1}} \left( x_{i}-\overline{x} \right) ^{2}+\frac{1}{n_{2}} \sum _{j=1}^{n_{2}} \left( y_{j}-\overline{x} \right) ^{2} \right] \\\\ \\ \sum _{i=1}^{n_{1}} \left( x_{i}-\overline{x} \right) ^{2}= \sum _{i=1}^{n_{1}} \left( x_{i}-\overline{x_{j}}+\overline{x_{j}}-\overline{x} \right) ^{2} \\\\ \\ = \sum _{i=1}^{n_{1}} \left( x_{i}-\overline{x_{j}} \right) ^{2}+n_{1} \left( \overline{x_{j}}-\overline{x} \right) ^{2}+2 \left( \overline{x_{j}}-\overline{x} \right) \sum _{i=1}^{n_{1}} \left( x_{i}-\overline{x_{j}} \right) \\\\ \\ \sum _{i=1}^{n_{1}} \left( x_{i}-\overline{x_{j}} \right) =0 \\\\


\\ \sum _{i=1}^{n_{1}} \left( x_{i}-\overline{x} \right) ^{2}=n_{1}s_{1}^{2}+n_{1} \left( \overline{x_{1}}-\overline{x} \right) ^{2} \\\\ \\ d_{1}=\overline{x_{1}}-\overline{x} \\\\ \\ =\overline{x_{1}}-\frac{n_{1}\overline{x_{1}}+n_{2}\overline{x_{2}}}{n_{1}+n_{2}} \\\\ \\ =\frac{n_{2} \left( \overline{x_{1}}-\overline{x_{2}} \right) }{n_{1}+n_{2}} \\\\ \\ \sum _{i=1}^{n_{1}} \left( x_{i}-\overline{x} \right) ^{2}=n_{1}s_{1}^{2}+\frac{n_{1}n_{2}^{2} \left( \overline{x_{1}}-\overline{x_{2}} \right) ^{2}}{ \left( n_{1}+n_{2} \right) ^{2}} \\\\ \\ \sum _{j=1}^{n_{2}} \left( y_{j}-\overline{x} \right) ^{2}= \sum _{j=1}^{n_{2}} \left( y_{j}-\overline{x_{i}}+\overline{x_{i}}-\overline{x} \right) ^{2} \\\\
\\ = \sum _{j=1}^{n_{2}} \left( y_{j}-\overline{x_{i}} \right) ^{2}+n_{2} \left( \overline{x_{i}}-\overline{x} \right) ^{2}+2 \left( \overline{x_{i}}-\overline{x} \right) \sum _{j=1}^{n_{2}} \left( y_{j}-\overline{x_{i}} \right) \\\\ \\ \sum _{j=1}^{n_{2}} \left( y_{j}-\overline{x_{i}} \right) =0 \\\\ \\ \sum _{j=1}^{n_{2}} \left( y_{j}-\overline{x} \right) ^{2}=n_{2}s_{2}^{2}+n_{2} \left( \overline{x_{2}}-\overline{x} \right) ^{2} \\\\ \\ d_{2}=\overline{x_{2}}-\overline{x} \\\\ \\ =\overline{x_{2}}-\frac{n_{1}\overline{x_{1}}+n_{2}\overline{x_{2}}}{n_{1}+n_{2}} \\\\ \\ =\frac{n_{1} \left( \overline{x_{2}}-\overline{x_{1}} \right) }{n_{1}+n_{2}} \\\\
\\ \sum _{j=1}^{n_{2}} \left( x_{i}-\overline{x} \right) ^{2}=n_{2}s_{2}^{2}+\frac{n_{1}^{2}n_{2} \left( \overline{x_{2}}-\overline{x_{1}} \right) ^{2}}{ \left( n_{1}+n_{2} \right) ^{2}} \\\\ \\ \sigma ^{2}= \sigma _{1}^{2}+ \sigma _{2}^{2}=\frac{1}{n_{1}+n_{2}} \left[ \sum _{i=1}^{n_{1}} \left( x_{i}-\overline{x} \right) ^{2}+ \sum _{j=1}^{n_{2}} \left( y_{j}-\overline{x} \right) ^{2} \right] \\\\ \\ =\frac{1}{n_{1}+n_{2}} \left[ n_{1}s_{1}^{2}+\frac{n_{1}n_{2}^{2} \left( \overline{x_{1}}-\overline{x_{2}} \right) ^{2}}{ \left( n_{1}+n_{2} \right) ^{2}}+n_{2}s_{2}^{2}+\frac{n_{1}^{2}n_{2} \left( \overline{x_{2}}-\overline{x_{1}} \right) ^{2}}{ \left( n_{1}+n_{2} \right) ^{2}} \right] \\\\ \\ =\frac{1}{n_{1}+n_{2}} \left[ n_{1}s_{1}^{2}+n_{2}s_{2}^{2}+\frac{n_{1}n_{2} \left( \overline{x_{1}}-\overline{x_{2}} \right) ^{2}}{ \left( n_{1}+n_{2} \right) ^{2}} \left( n_{1}+n_{2} \right) \right] \\\\

\\ =\frac{n_{1}s_{1}^{2}+n_{2}s_{2}^{2}}{n_{1}+n_{2}}+\frac{n_{1}n_{2} \left( \overline{x_{1}}-\overline{x_{2}} \right) ^{2}}{ \left( n_{1}+n_{2} \right) ^{2}} \\\\ \\ \sigma =\sqrt {\frac{n_{1}s_{1}^{2}+n_{2}s_{2}^{2}}{n_{1}+n_{2}}+\frac{n_{1}n_{2} \left( \overline{x_{1}}-\overline{x_{2}} \right) ^{2}}{ \left( n_{1}+n_{2} \right) ^{2}}} \\\\

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