The standard deviation of 1,2,3,....25 is?

1 Answer
May 24, 2018

The standard deviation is sigma = 7.2111. This is the population standard deviation, not the sample standard deviation. (The difference is explained here).

Explanation:

I did this by writing a formula into a spreadsheet to find the standard deviation of the integers from 1 to 25.

Alternatively, we could calculate it by hand - click here for details.

"Standard deviation " sigma = sqrt(frac{sum_(i=1)^(N)(x_i - mu)^2}{N})

We know the mean of the dataset is:
mu = frac{sum_(i=1)^25 (i)}{25} = frac{1 + 2 + 3 + ... +25}{25}

mu = frac{1+25}{2} = 13

Substitute values:
sigma = sqrt(frac{sum_(i=1)^(25)(x_i - 13)^2}{25})

sigma= sqrt(frac{(1-13)^2+(2-13)^2+(3-13)^2 + ... + (25-13)^2}{25})

sigma = sqrt(frac{12^2+11^2 + ... +2^2+1^2+0^2+1^2 + ... + 11^2+12^2}{25})

sigma = sqrt(1300/25) = sqrt52

sigma = 7.2111