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.

Question

$\begin{array}{1 1}(A)\;0.99\\(B)\;5.59\\(C)\;8\\(D)\;1.25\end{array} $

Answer 1

Toolbox:

• Formula used to solve this problem are :
• Combined SD $=\sqrt {\large\frac{n_1\sigma_1 ^2 + n_2 \sigma_2 ^2}{n_1+n_2} + \frac{n_1n_2 (\bar{x_1} - \bar {x_2})^2}{(n_1+n_2)}}$
• Given $n_1=20 \qquad n_2=20$
• $\sigma_1=5 \qquad \sigma_2=20$
• $\bar {x_1}=17 \qquad \bar{x_2} =22$

Substituting the values in the formula

Combined SD $= \sqrt { \large\frac{20 \times 5^2+ 20 \times 5^2}{20+20}+\frac{20 \times 20 (17-22)^2}{(20+20)^2}}$

$\qquad= \sqrt {\large\frac{1000}{40}+\frac{400 \times 5^2}{40^2}}$

$\qquad= \sqrt {\large\frac{1000}{40} +\frac{400 \times 5^2}{40^2}}$

$\qquad= \sqrt {\large\frac{1000}{40} +\large\frac{10000}{40 \times 40}}$

$\qquad= \large\frac{1}{40} $$ \sqrt { 40000+10000}$

$\qquad= \large\frac{1}{40}$$ \sqrt {50000}$

$\qquad= \large\frac{223.606}{40}$

$\qquad= 5.59$

$\therefore $ The combined standard deviation of the set obtained are $5.59$

Hence B is the correct answer.