Standard Deviation Calculator

Find the standard deviation, variance and mean of a data set.

Sample standard deviation (s)

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Population SD (σ)

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Sample variance (s²)

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Mean

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Count

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Sample statistics need at least two values. Non-numeric entries are ignored.

Measuring spread

The mean tells you the centre of a data set; the standard deviation tells you how far the values typically stray from it. Two sets can share a mean yet look completely different — one tightly bunched, the other widely scattered.

σ = √( Σ(x − mean)² ÷ N )　·　s = √( Σ(x − mean)² ÷ (n − 1) )

The only difference between the two is the divisor. Population standard deviation divides by the full count; sample standard deviation divides by one less, to better estimate a wider population from a limited sample.

Worked example

For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5. The squared deviations sum to 32, so the population standard deviation is √(32 ÷ 8) = 2, while the sample standard deviation √(32 ÷ 7) is about 2.14.

Reading the number

Standard deviation shares the units of your data, which makes it intuitive: a test-score spread of 5 points means most results sit within a few points of the average. In roughly bell-shaped data, about two-thirds of values fall within one standard deviation of the mean.

Worth remembering

Pick the right one. Whole group → population; a sample of it → sample.
Outliers inflate it. A few extreme values can stretch the spread considerably.
Same units as the data. Unlike variance, which is in squared units.

Frequently asked questions

What is the difference between population and sample?

Use the population version when your data is the whole group. Use the sample version — which divides by n − 1 — when the data is a sample meant to represent a larger population.

What does the standard deviation tell me?

It measures how spread out the values are around the mean. A small value means the data clusters tightly; a large one means it is widely scattered.

How does it relate to variance?

Variance is the average of the squared deviations from the mean; the standard deviation is its square root. The square root brings the figure back into the original units.

Why n − 1 for a sample?

Dividing by n − 1 (Bessel’s correction) compensates for the tendency of a sample to underestimate the spread of the population it came from.