Standard Deviation Calculator
Paste or type a list of numbers to instantly get the mean, variance, and standard deviation — calculated both as a population and as a sample, since the two use slightly different formulas.
How the Standard Deviation formula works
Standard deviation measures how spread out a data set is from its mean:
Population: σ = √( Σ(x − mean)² / N ) Sample: s = √( Σ(x − mean)² / (N − 1) )
The sample version divides by N − 1 instead of N (Bessel's correction), which corrects for the tendency of a sample to underestimate the true population variance.
Step-by-step calculation
- Calculate the mean (average) of all values.
- Subtract the mean from each value and square the result.
- Sum all of those squared differences.
- Divide by N (population) or N − 1 (sample) to get the variance.
- Take the square root of the variance to get the standard deviation.
Worked example
Data set: 2, 4, 4, 4, 5, 5, 7, 9. Mean = 5. Squared differences sum to 32. Population variance = 32/8 = 4, so population standard deviation = √4 = 2. Sample variance = 32/7 ≈ 4.57, so sample standard deviation ≈ 2.14.
Frequently asked questions
Should I use population or sample standard deviation?
Use the population formula when your data set includes every member of the group you care about. Use the sample formula when your data is a subset drawn from a larger population you're trying to estimate.
What does a low versus high standard deviation mean?
A low standard deviation means values cluster tightly around the mean; a high standard deviation means values are spread widely. It's a measure of consistency, not of the average value itself.
Can standard deviation be negative?
No — because it comes from a square root of squared values, standard deviation is always zero or positive. A value of zero means every data point is identical.