Question

Two sets each of 20 observations have the same standard deviation 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.

Answer 1

Given :
$n_1=20,{\sigma }_1=5,{\overline{x}}_1=17$ and
$n_2=20,{\sigma }_2=5,{\overline{x}}_2=22$

Using

$\sigma =\sqrt{\frac{n_1{\sigma }_1^2+n_2{\sigma }_2^2}{n_1+n_2}+\frac{n_1{n}_2\left(\right({\overline{x}}_1-{\overline{x}}_2){)}^2}{(n_1+n_2{)}^2}}$

$\sigma =\sqrt{\frac{\left(20\right)(5{)}^2+(20\left)\right(5{)}^2}{40}+\frac{\left(400\right)(17-22{)}^2}{(40{)}^2}}$

$\sigma =\sqrt{\frac{1000}{40}+\frac{400\times 25}{1600}}$

$\sigma =\sqrt{25+\frac{25}{4}}=\sqrt{\frac{125}{4}}=\sqrt{31.25}=5.59$

Hence, the standard deviation of the combined set is 5.59