Standard Deviation Calculator

Find the mean, variance, and population and sample standard deviation of a set of numbers.

Separate values with commas, spaces, or newlines. Invalid entries are ignored.

Result—

Enter your values and calculate to see the result.

Formula used

Mean:                μ = (Σ xᵢ) / n
Population variance:  σ² = Σ (xᵢ − μ)² / n
Sample variance:      s² = Σ (xᵢ − μ)² / (n − 1)
Standard deviation:   σ = √σ²   (population),   s = √s²   (sample)
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Worked example

Data: 2, 4, 4, 4, 5, 5, 7, 9 (n = 8)

Mean: (2+4+4+4+5+5+7+9) ÷ 8 = 5

Population variance: Σ(xᵢ − 5)² ÷ 8 = 32 ÷ 8 = 4, so population SD = √4 = 2.

Sample variance: 32 ÷ 7 ≈ 4.571, so sample SD ≈ 2.138.

Frequently asked questions

What is the difference between population and sample standard deviation?

Population standard deviation divides by n and is used when your numbers are the entire group. Sample standard deviation divides by n − 1 (Bessel's correction) and is used when your numbers are a sample drawn from a larger population. The sample figure is slightly larger and is the more common choice for real-world data.

How should I separate the numbers I enter?

Any mix of commas, spaces, and newlines works. You can paste a column from a spreadsheet, a comma-separated list, or type values with spaces — the calculator parses them all and ignores anything that is not a valid number.

Why does the sample standard deviation show "n/a"?

Sample standard deviation needs at least two values because it divides by n − 1. With a single number there is no spread to measure, so the sample figure is not defined while the population figure is zero.

What is variance versus standard deviation?

Variance is the average of the squared differences from the mean. Standard deviation is the square root of the variance, which returns the measure to the same units as your original data and is usually easier to interpret.