Standard Deviation Calculator (Sample & Population)

This standard deviation calculator returns the sample and population standard deviation, both variances and the mean with every intermediate step shown. Standard deviation is the standard way to describe how much a set of numbers varies. Enter your data and this calculator returns the sample and population standard deviation, both variances and the mean, with the intermediate sums shown.

The formula

mean      x̄ = Σx ÷ n
variance  s² = Σ(x − x̄)² ÷ (n − 1)   [sample]
          σ² = Σ(x − x̄)² ÷ n         [population]
std dev   s  = √s² ,  σ = √σ²

The only difference between the two versions is the divisor: n − 1 for a sample, n for a whole population.

Reading the number

For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and 95% within two. That makes it a quick way to judge whether a value is typical or an outlier.

Worked example

For the default set 2, 4, 4, 4, 5, 5, 7, 9:

mean        x̄ = 40 ÷ 8 = 5
deviations    −3, −1, −1, −1, 0, 0, 2, 4
squared        9,  1,  1,  1, 0, 0, 4, 16
Σ(x − x̄)²   = 32

sample      s² = 32 ÷ 7 = 4.5714     s = √4.5714 = 2.1381
population  σ² = 32 ÷ 8 = 4          σ = √4 = 2

The population standard deviation comes out at exactly 2, which is why this particular set turns up in so many textbooks — it is one of the few data sets whose spread is a round number. Every figure above appears in the breakdown panel, so the result is checkable by hand rather than something you have to take on trust.

How far apart the two answers really are

They differ by a fixed factor that depends only on the count: s ÷ σ = √(n ÷ (n − 1)).

So Bessel's correction is a serious adjustment for a small sample and a rounding detail for a large one. If your two results are nearly identical your data set is big enough that the choice barely matters; if they are far apart you are working with very few points and should say which formula you used.

A single number cannot produce a sample standard deviation at all. There is no spread to measure against, and dividing by n − 1 would mean dividing by zero, so the calculator asks for at least two values before it will answer.

Using it to compare two sets

A standard deviation only means something next to the mean it belongs to. The sets 2, 4, 4, 4, 5, 5, 7, 9 and 1, 2, 5, 8, 9 have the same mean of 5, but their population standard deviations are 2 and 3.1623. The second is visibly more scattered even though it is centred in exactly the same place. That is the real use of the number: two classes, two machines or two suppliers with the same average and different consistency.

Because the standard deviation carries the same units as the data it is directly readable — 2 points on a test, 2 kg on a scale. To compare spread across different units, divide the standard deviation by the mean; that ratio is the coefficient of variation, and both figures are already on screen. For roughly bell-shaped data about 68% of values fall within one standard deviation of the mean and 95% within two, so a value more than two out is a legitimate candidate for an outlier rather than ordinary noise. Average Calculator supplies the mean, median and mode for the same list.

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