Variance of a distribution is V. If each value of the variate be multiplied by a constant quantity k then new variance is

Question : What is new variance in the problem

Solution :

Let \(\mu\ \) be mean of the distribution then it’s variance V is defined as \[V=\frac{1}{n}\sum_{i=1}^{n}\left(x_i-\mu\right)^2\] If each value of the variate be multiplied by a constant quantity k then new variance is \[V_1=\frac{1}{n}\sum_{i=1}^{n}\left(kx_i-k\mu\right)^2=\frac{k^2}{n}\sum_{i=1}^{n}\left(x_i-\mu\right)^2=k^2V\] Thus new variance is k^2 times previous variance. Correct option is (3).