Standard Deviation Calculator
Calculate sample or population standard deviation along with variance, mean, and the sum of squared deviations. The breakdown shows how the chosen denominator affects the result.
Standard Deviation Calculator tool
About the Standard Deviation Calculator
Use this tool for statistics practice, comparing repeated measurements, or summarizing variation around a mean. Choose population when the data are the full group of interest, and sample when they represent a larger group.
How Is Standard Deviation Calculated?
Standard deviation is the square root of variance. Both begin with squared differences between each observation and the arithmetic mean.
Population variance divides by n. Sample variance divides by n − 1, the usual correction when estimating population variance from a sample.
Mean = Σx ÷ n
SS = Σ(x − mean)²
Population variance = SS ÷ n
Sample variance = SS ÷ (n − 1)
Standard deviation = √variance
Example
For 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5 and SS is 32. Population variance is 4 and population standard deviation is 2.
How to Use the Standard Deviation Calculator
- Enter numeric observations separated by commas, spaces, or new lines.
- Select sample or population according to how the data were collected.
- Use at least two observations for sample calculations.
- Read the standard deviation, variance, mean, and squared-deviation total.
- Follow the steps from the mean through the square root of variance.
Standard Deviation Calculator Examples
Population spread
For 1, 2, 3, mean = 2 and SS = 2. Population variance = 2/3 and SD ≈ 0.816496581.
Sample spread
For 1, 2, 3, sample variance = 2/(3 − 1) = 1 and sample SD = 1.
Units
Standard deviation has the same units as the observations. Variance has squared units, so it should not be interpreted as an ordinary distance from the mean.
Interpretation
A larger standard deviation indicates more spread around the mean. It does not by itself establish a normal distribution or justify a particular probability statement.
Frequently Asked Questions
Should I choose sample or population?
Use population for the complete group being described. Use sample when estimating variability in a larger population from observed members.
Why does sample variance divide by n − 1?
Estimating the mean from the same observations uses one degree of freedom. The correction makes sample variance unbiased under standard sampling assumptions.
What does sum of squares mean here?
It means the sum of squared deviations from the mean, not the sum of the raw squared observations.
Can standard deviation be negative?
No. It is the nonnegative square root of a nonnegative variance.
What happens with one observation?
Population standard deviation is zero. Sample standard deviation is undefined because n − 1 is zero.