Standard Deviation and Variance Calculator
Enter your data to get both the sample and the population standard deviation and variance, along with the values needed to check the working.
Result
- Sample variance (s²)
- 4.571429
- Population variance (σ²)
- 4
- Mean
- 5
- Sum of squared deviations
- 32
- Count (n)
- 8
- Standard error of the mean
- 0.755929
- Coefficient of variation (sample)
- 42.7618%
Formula
- Population variance σ² = Σ (x − mean)² ÷ n
- Sample variance s² = Σ (x − mean)² ÷ (n − 1)
- Standard deviation = √variance
- Standard error = s ÷ √n
How the Standard Deviation and Variance Calculator works
Standard deviation measures how far values typically lie from the mean. Find each value's distance from the mean, square it, add the squares, divide, and take the square root.
Use the population formula when your data are the entire group of interest. Use the sample formula, which divides by n − 1, when the data are a sample used to estimate a larger population. Dividing by n − 1 corrects the tendency of a sample to understate the spread.
Worked example
For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5. The squared deviations add to 32. Population variance is 32 ÷ 8 = 4, so σ = 2. Sample variance is 32 ÷ 7 = 4.5714, so s = 2.138.
Frequently asked questions
Should I use sample or population standard deviation?
Use sample standard deviation when your numbers are a subset of a larger group, which is the usual case. Use population standard deviation only when you have every member of the group.
What does a standard deviation tell me?
How spread out the data are. For roughly bell-shaped data, about 68 percent of values fall within one standard deviation of the mean and about 95 percent within two.
What is variance?
The average squared distance from the mean, and the square of the standard deviation. Its units are the square of the data's units.
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Calculations run in your browser and nothing you enter is sent to a server. Results are provided for general information. Check critical figures independently before relying on them.