Let x₁, x₂, x₃, x₄, x₅ be the observations with mean m and standard deviation s.

The standard deviation of the observations kx₁, kx₂, kx₃, kx₄, kx₅ is

Given observations are x₁, x₂, x₃, x₄, x₅

So the mean of the 5 observations is

⇒ ∑x_i= x₁+x₂+ x₃+x₄+x₅ =5m……..(i)

And the standard deviation of the 5 observations is

Substituting equation (i) in above equation we get

Now we will find the mean and standard deviation of the observations kx₁, kx₂, kx₃, kx₄, kx₅, we get

So the mean of these 5 observations is

Substituting value from equation (i), we get

m₁=mk…………(iii)

And the standard deviation of the 5 observations is

Substituting equation (iii) in above equation we get

Substituting the value from equation (ii), we get

σ=ks

Hence the standard deviation of new set of observations, i.e., kx₁, kx₂, kx₃, kx₄, kx₅ is ks.