Consider the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. If 1 is added to each number, the variance of the numbers so obtained is

Now given numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

But these are first 10 natural numbers,

And we also know the standard deviation of the first n natural numbers is

Here n=10, substituting this in the above equation of standard deviation we get

Now when 1 is added to each numbers of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10; we get new series as 1+1, 2+1, 3+1, 4+1, 5+1, 6+1, 7+1, 8+1, 9+1, 10+1.

Now we know, if standard deviation of x series is s, then standard deviation of k+x series is s,

So the standard deviation of 1+1, 2+1, 3+1, 4+1, 5+1, 6+1, 7+1, 8+1, 9+1, 10+1 series is also same as the standard deviation of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 series,

Hence

Now for variance we will square on both sides, we get

σ²=8.25

Hence the variance of the numbers so obtained is 8.25