Question

Each observation of a raw data whose variance is $\sigma^2$, is multiplied by $n$. What is the variance of the new set?

(a) $\sigma^2$

(b) $n^2 \sigma^2$

(c) $n \sigma^2$

(d) $\frac{\sigma^2}{n}$

Answer 1

Suppose we have a raw data i.e., $x_1, x_2, \ldots, x_k$

Then,

$\sigma^2=\frac{1}{k} \sum_{i=1}^k\left(x_k-\bar{x}\right)^2$

If each value is multiplied by $n$, then the values are

$n x_1, n x_2, \ldots$

The AM of the new value is

$\frac{n x_1+n x_2+\ldots+n x_k}{k}=n \bar{x}$

Therefore, the variance of the new set of values is

$\frac{1}{k} \sum_{i=1}^k\left(n x_i-n \bar{x}\right)^2=n^2\left[\frac{1}{k} \sum_{i=1}^k\left(x_i-x\right)^2\right]=n^2 \sigma^2$

So, The correct option of this question will be (B).