Standard Deviation Calculator

Paste or type your numbers, separated by commas, spaces or new lines. You'll get the standard deviation (both sample and population), the variance, the mean and the count.

Enter a list of numbers and press Calculate.

The standard deviation formula

Standard deviation measures how spread out a set of numbers is around their mean. You square each deviation from the mean, average those squares (the variance), then take the square root:

mean: μ = (Σ x) ÷ N population: σ = √( Σ(x − μ)² ÷ N ) sample: s = √( Σ(x − x̄)² ÷ (N − 1) )

The only difference between the two: the population formula divides by N, while the sample formula divides by N − 1 (Bessel's correction), which corrects the bias when you're estimating from a sample rather than measuring an entire population.

Worked example

For the set 10, 12, 23, 23, 16, 23, 21, 16 (N = 8):

Mean: 144 ÷ 8 = 18.
Sum of squared deviations: 178.
Population SD: √(178 ÷ 8) ≈ 4.717.
Sample SD: √(178 ÷ 7) ≈ 5.042.

Which one should I use?

Use the sample standard deviation when your numbers are a subset drawn from a larger group and you're estimating the spread of that group — the most common case in research and surveys. Use the population standard deviation only when your data covers every member of the group you care about.

Note: for large data sets the difference between dividing by N and N − 1 is tiny, but for small samples it matters, which is why statistics software defaults to the sample formula.

Frequently asked questions

What is the difference between sample and population standard deviation?

The population version divides the summed squared deviations by N; the sample version divides by N − 1 (Bessel's correction) to correct bias when estimating from a sample. Use the sample formula unless your data covers the entire population.

How is standard deviation calculated?

Find the mean, subtract it from each value and square the result, average those squares to get the variance, then take the square root.

What is variance?

Variance is the average of the squared deviations from the mean — the standard deviation squared. Standard deviation is in the original units, which is why it's usually reported instead.

How do I enter my data?

Type or paste your numbers separated by commas, spaces or new lines. The calculator ignores blank entries and reads everything else as a value.

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Mustafa Bilgic · Editor, Calcool
The variance and standard deviation formulas, and Bessel’s N − 1 correction for sample data, are standard descriptive statistics as defined by references such as the NIST/SEMATECH e-Handbook of Statistical Methods. All computation runs in your browser.

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