Question

The mean and the standard deviation (s.d.) of five observations are $9$ and $0$, respectively. If one of the observations is changed such that the mean of the new set of five observations becomes $10$, then their s.d. is?

• A. $0$
• B. $4$
• C. $2$
• D. $1$

Answer 1

The correct option is C $2$

$\overline{x}=\frac{\sum x_i}{n}=9$

$s.d=\sqrt{\sum \frac{(x_i-\overline{x}{)}^2}{n}}=0$

$\Rightarrow x_i=\overline{x}\forall i$

$\therefore$ each term in the original observation is 9

$\Rightarrow$ observation $1=\{9,9,9,9,9\}$

${\overline{x}}_{new}=\frac{\sum x_i}{n}=\frac{9+9+9+9+x_5}{5}$

$\Rightarrow 10=\frac{36+x_5}{5}$

$\Rightarrow x_5=14$

$\Rightarrow$ new set of observations$=\{9,9,9,9,14\}$

$s.d_{new}=\sqrt{\frac{\sum (x_i-{\overline{x}}_{new}{)}^2}{n}}$

$=\sqrt{\frac{(9-10{)}^2+(9-10{)}^2+(9-10{)}^2+(14-10{)}^2}{5}}$

$=\sqrt{\frac{4+4^2}{5}}$

$=\sqrt{\frac{20}{5}}$

$=2$