Standard Deviation Calculator
Calculate the population or sample standard deviation of a list of numbers.
Standard deviation
- Count
- Mean
How we calculated this
Formula (population)
σ = √[ Σ(xᵢ − μ)² / N ]
In words: the square root of (the sum of squared differences from the mean, divided by the count).
Worked example
Inputs: 2, 4, 4, 4, 5, 5, 7, 9. Formula: σ = √(Σ(x − μ)² ÷ N). Substitution: the mean is 5 and the squared differences add up to 32, so σ = √(32 ÷ 8) = √4. Result: the population standard deviation is 2. Treated as a sample the same numbers give s = √(32 ÷ 7), which is about 2.13809.
How it works
Standard deviation measures how far the numbers typically sit from their mean. The two standard formulas differ only in the divisor: the population formula divides by the number of values, and the sample formula divides by one fewer (to correct for estimating a wider group from a sample). Which you need depends on your data, so it is a choice you make, and the result names it. The two are never mixed. A sample standard deviation needs at least 2 values, since dividing by one fewer than 1 is undefined; the population standard deviation of a single value is 0.
What this result means
Standard deviation measures how spread out your numbers are from the mean. A small value means the numbers cluster close to the mean, a large value means they are widely spread.
Common mistakes
- Using sample standard deviation when your numbers are actually the entire population. The two formulas can give noticeably different answers on small data sets.