Sample standard
deviation calculator

This sample standard deviation calculator measures how much a set of observations varies around its mean. A small value indicates that the observations are close together; a larger value indicates more spread. It does not, by itself, say whether the data is good, bad, or normally distributed.

The sample standard deviation formula

s = √[Σ(xᵢ − x̄)² ÷ (n − 1)]

Here, xᵢ is each observation, x̄ is the mean, and n is the number of observations. The NIST statistical handbook defines sample variance using n − 1; sample standard deviation is its square root.

1. Find the mean by adding all values and dividing by their count.
2. Subtract the mean from each value and square each difference.
3. Add those squared differences.
4. Divide by n − 1 to get sample variance.
5. Take the square root to get sample standard deviation.

A worked example

For 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4, and 16. Their sum is 32. With eight observations:

s² = 32 ÷ 7 ≈ 4.5714
s = √(32 ÷ 7) ≈ 2.1381

The variance has squared units. Taking its square root returns standard deviation to the original units.

Use the sample result when your observations are a sample used to estimate variation in a larger population. Use the population result when the observations are the entire population you want to describe.

Population: σ = √[Σ(xᵢ − μ)² ÷ n]

For the same example, population variance is 32 ÷ 8 = 4 and population standard deviation is 2. The difference is the denominator, not a different set of values. Dividing by n − 1 corrects the usual sample variance estimator under the relevant sampling assumptions; it does not make the square root an exactly unbiased estimator of population standard deviation.

Input and precision

Enter at least two observations. Negative values, zeros, decimal points, and scientific notation such as 1.2e3 work. Each value may have up to 15 meaningful numeric digits; nonzero magnitudes must lie between 1e−100 and 1e100. Avoid unit labels and thousands separators.

The calculation centers the data before summing squared differences and uses compensated sums. This follows NIST’s guidance on numerical stability rather than subtracting two large, nearly equal sums of squares. Display rounding is applied after calculation.