Question

The variance of numbers $x_1,x_2,x_3,.....x_n$ is V. Consider the following statements :
1. If every $x_1$, is increased by 2, the variance of the new set of numbers is V.
2. If the numbers $x_1$ is squared, the variance of the new set is $V^2$.
Which of the following statements is/are correct ?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

The correct option is A 1 only

In case 1, a deterministic value of 2 units is being added to every $x_i$ and hence, the variance of the set remains unchanged as $V$. New variance would be $\frac{1}{n}\sum (x_i+2-(\sum \frac{x_i+2}{n}))=\frac{1}{n}\sum (x_i-\sum \frac{x_i}{n})=V$

In case 2, each $x_i$ is squared and hence, the mean becomes $\frac{1}{n}\sum x_i^2$. Hence, the new variance is $\frac{1}{n}\sum {(x_i^2-\frac{1}{n}\sum x_i^2)}^2$

and not $V^2=\frac{1}{n^2}\sum {(x_i-\frac{1}{n}\sum x_i)}^4$.