Let a, b, c, d, e be the observations with mean m and standard deviation s.

The standard deviation of the observations a + k, b + k, c + k, d + k, e + k is

Given observations are a, b, c, d, e

So the mean of the 5 observations is

⇒ ∑x_i=a+b+c+d+e=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 a + k, b + k, c + k, d + k, e + k, we get

So the mean of these 5 observations is

Substituting value from equation (i), we get

m₁=m+k…………(iii)

And the standard deviation of the 5 observations is

Substituting equation (iii) in above equation we get

Substituting the value from equation (i), we get

Cancelling like terms, we get

Comparing this with equation (ii), we get

σ=s

Hence the standard deviation of new set of observations, i.e., a + k, b + k, c + k, d + k, e + k is s, itself.