Question

The outcome of each of $30$ items was observed; $10$ items gave an outcome $\frac{1}{2}-d$ each, $10$ items gave outcome $\frac{1}{2}$ each and the remaining $10$ items gave outcome $\frac{1}{2}+d$ each. If the variance of this outcome data is $\frac{4}{3},$ then $|d|$ equals

• A. $\sqrt{2}$
• B. $2$
• C. $\frac{\sqrt{5}}{2}$
• D. $\frac{2}{3}$

Answer 1

The correct option is A $\sqrt{2}$

Outcome of $10$ items $=\frac{1}{2}-d$
Outcome of other $10$ items $=\frac{1}{2}$
Outcome of rest of the $10$ items $=\frac{1}{2}+d$
We know that the variance of the data is independent of addition or subtraction of a constant in each outcome.
Let the outcomes be $-d,0,d$ which occur $10$ times each.
$\therefore$ We shift the given outcomes by $\frac{1}{2}$
So, we get
${\sigma }^2=\frac{10(-d{)}^2+10\times 0^2+10d^2}{30}-0^2=\frac{4}{3}\Rightarrow \frac{2}{3}{d}^2=\frac{4}{3}\therefore |d|=\sqrt{2}$