Standard Deviation Calculator
Paste a list of numbers to measure how spread out they are, with both sample and population standard deviation and variance.
Result
Enter your values and select Calculate to see the result.
How to use this calculator
- Paste or type your numbers, separated by commas, spaces or new lines. Use a dot for decimals.
- The result shows the sample standard deviation first, with the population value underneath.
- Anything that is not a number is reported as an error rather than skipped.
Formula
mean (x̄) = Σx ÷ n
- Population variance:
σ² = Σ(x − x̄)² ÷ nandσ = √σ² - Sample variance:
s² = Σ(x − x̄)² ÷ (n − 1)ands = √s² - Standard error of the mean:
s ÷ √n
Dividing by n − 1 (Bessel’s correction) gives an unbiased estimate of the variance when the data is a sample.
Worked example
Eight values
Data: 2, 4, 4, 4, 5, 5, 7, 9. The mean is 40 ÷ 8 = 5, and the squared deviations add up to 32.
Population: σ² = 32 ÷ 8 = 4, so σ = 2.
Sample: s² = 32 ÷ 7 ≈ 4.571, so s ≈ 2.138.
Assumptions and limitations
- The sample standard deviation needs at least two values.
- A two-pass method (mean first) is used for numerical accuracy.
- Standard deviation is sensitive to outliers; check your data for typing errors.
Frequently asked questions
Should I use sample or population standard deviation?
Use the population value when your data includes every member of the group you care about. Use the sample value when your data is a sample used to estimate a larger group — this is the usual case in research and surveys.
What does standard deviation tell me?
It measures the typical distance of values from the mean. For roughly bell-shaped data, about 68% of values lie within one standard deviation of the mean and about 95% within two.