Standard deviation calculator

How spread out your numbers are, for a sample or a whole population.

Formula Maths

Standard deviation

Rounded for display

Variance: ≈ 4.5714
Mean: 5
Count: 8
Sum of squared deviations: 32
Standard error of the mean: ≈ 0.755929

Step by step

x̄ = 40 ÷ 8 = 5
Σ(x − x̄)² = 32
s² = 32 ÷ 7 = 4.571428571
s = √4.571428571 = 2.138089935

How it works

First find the mean, then the deviation of each value from it. Square the deviations and add them up.

For a whole population divide that sum by n. For a sample divide by n − 1 (Bessel’s correction), which corrects for the sample mean being closer to the data than the true mean. The standard deviation is the square root of the variance.

The sums are calculated exactly before the final square root.

Formula

x̄ = Σx ÷ n
σ² = Σ(x − x̄)² ÷ n
s² = Σ(x − x̄)² ÷ (n − 1)
SE = s ÷ √n

Worked example

For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the squared deviations add up to 32. The population variance is 32 ÷ 8 = 4, so σ = 2. The sample variance is 32 ÷ 7 ≈ 4.571, so s ≈ 2.138.

Questions people ask

Should I use sample or population?

Use population when your list is every member of the group. Use sample when it is a subset used to estimate the whole group, which is the usual case.

What is the difference between variance and standard deviation?

Variance is the average squared deviation; the standard deviation is its square root, so it is in the same units as the data.

What does a standard deviation of zero mean?

All the values are identical: none of them differ from the mean.