Standard Deviation Calculator (Population and Sample)
Enter your numbers and how many of them to use (at least 2). You get both the population and the sample standard deviation.
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Formula
For n values with mean m:
- Population variance:
σ² = Σ(x − m)² ÷ n - Sample variance:
s² = Σ(x − m)² ÷ (n − 1) - Standard deviation: the square root of the variance
Worked example
Values 2, 4, 4, 4, 5, 5, 7, 9: mean 5, squared distances 9, 1, 1, 1, 0, 0, 4, 16 add to 32.
- Population: 32 ÷ 8 = 4, so σ = 2.
- Sample: 32 ÷ 7 = 4.5714, so s = 2.1381.
Frequently asked questions
What is the difference between population and sample standard deviation?
Use the population version when your numbers are the whole group, and the sample version when they are a sample drawn from a larger group. The sample version divides by n − 1 instead of n, so it is slightly larger.
How do you calculate standard deviation by hand?
Find the mean, square each value's distance from it, add those squares, divide by n (population) or n − 1 (sample), then take the square root. For 2, 4, 4, 4, 5, 5, 7, 9 the squares add to 32, so the population variance is 32 ÷ 8 = 4 and the standard deviation is 2.
Why do I need at least two numbers?
The sample formula divides by n − 1, which is zero for a single value, and one value has no spread to measure. Two values such as 1 and 3 give a sample standard deviation of 1.4142.
What does a standard deviation of zero mean?
Every value is identical, so nothing deviates from the mean. For example 7, 7, 7 has a standard deviation of 0.