Question

The standard deviation of the observations 6, 5, 9, 13, 12, 8, 10 is

(a) 6 (b) $\sqrt{6}$ (c) $\frac{52}{7}$ (d) $\sqrt{\frac{52}{7}}$

Answer 1

The given observations are 6, 5, 9, 13, 12, 8, 10.

Now,

$\sum _{}{x}_i=6+5+9+13+12+8+10=63$

$\sum _{}{x}_i^2=36+25+81+169+144+64+100=619$

∴ Standard deviation of the observations, $\sigma$

$$\begin{gathered} =\sqrt{\frac{1}{N}\sum _{}x{}_i^2-{\left({\displaystyle \frac{1}{N}}\sum _{}{x}_i\right)}^2} \\ =\sqrt{\frac{1}{7}\times 619-{\left(\frac{1}{7}\times 63\right)}^2} \\ =\sqrt{\frac{619}{7}-81} \\ =\sqrt{\frac{619-567}{7}} \\ =\sqrt{\frac{52}{7}} \end{gathered}$$

Hence, the correct answer is option (d).