Question

Let \(x_1,x_2,x_3,x_4,x_5\), be the observations with mean 𝑚 and standard deviations 𝑠. The standard deviations of the observations \(kx_1,kx_2,kx_3,kx_4,kx_5\) is

Answer 1

For obeservation \(x_1,x_2,x_3,x_4,x_5\)

Given mean: \(m=\dfrac{x_1,x_2,x_3,x_4,x_5}{5}\)

\(\Rightarrow~m=\dfrac{\sum x_i^2}{5}\)

Given standard deviation

\(s=\sqrt{\dfrac{\sum x_i^2}{5}-\left ( \dfrac{\sum x_i}{5} \right )^2}...(i)\)

For observations \(kx_1,kx_2,kx_3,kx_4,kx_5\)

Standard deviation

\(\sigma=\sqrt{\dfrac{\sum(kx_i)^2}{5}-\left (\dfrac{\sum kx_i}{5} \right )^2}\)

\(\sigma=\sqrt{\dfrac{k^2\sum x_i^2}{5}-\left (\dfrac{k\sum x_i}{5} \right )^2}\)

\(\sigma=k\sqrt{\dfrac{\sum x_i^2}{5}-\left ( \dfrac{\sum x_i}{5} \right )^2}\)

\(\sigma=ks\) [From equation (i)]

Hence, the correct option is (𝐶).