Question

Consider the first $10$ positive integers, if we multiply each number by $-1$ and then add $1$ to each number, then standard deviation of the numbers, so obtained is approximately equal to

• A. $8.25$
• B. $6.5$
• C. $3.87$
• D. $2.87$

Answer 1

The correct option is D $2.87$

The new numbers are
$-1+1,-2+1,\cdots ,-10+1$
i.e $0,-1,-2,-3,\cdots ,-9$
${\sigma }^2=\frac{\sum {x_i}^2}{n}-{\overline{x}}^2$

$\therefore {\sigma }^2=\frac{0^2+(-1{)}^2+\cdots +(-9{)}^2}{10}-{\left(\frac{0-1-2\cdots -9}{10}\right)}^2$
$\Rightarrow {\sigma }^2=\frac{9\times 10\times 19}{6\times 10}-\frac{9^2\times {10}^2}{4\times 100}=\frac{33}{4}$
$\therefore \sigma =2.87$