Standard Deviation Calculator
Calculate standard deviation, variance, mean, and sum of squares for a data set.
Separate values with commas, spaces, or new lines. Invalid values are ignored.
Standard Deviation
Variance
Mean (x̄)
Additional Statistics
Count (n)
Sum
Sum of Squared Deviations (SS)
Minimum
Maximum
Range
Step-by-Step Deviations
| xᵢ | xᵢ − x̄ | (xᵢ − x̄)² |
|---|
Population vs Sample Standard Deviation
Use the population standard deviation (σ) when your data includes every member of the group you care about. Use the sample standard deviation (s) when your data is only a subset drawn from a larger population. The sample formula divides by n − 1 instead of n (Bessel's correction) to give an unbiased estimate of the population variance.
What Standard Deviation Tells You
Standard deviation measures how spread out the values in a dataset are around the mean. A small standard deviation means the values cluster tightly around the mean, while a large standard deviation means they are more spread out. It is expressed in the same units as the original data, which makes it easier to interpret than the variance.
The Formula
- Mean: x̄ = (Σxᵢ) / n
- Sum of squared deviations: SS = Σ(xᵢ − x̄)²
- Population variance: σ² = SS / n
- Sample variance: s² = SS / (n − 1)
- Standard deviation: the square root of the variance
Example
- Dataset: 2, 4, 4, 4, 5, 5, 7, 9
- Mean: 5
- Population SD (σ): 2
- Sample SD (s): ≈ 2.14
Related Calculators
Note: This calculator automatically filters out non-numeric values. Ensure you select the correct data type (population or sample) for your analysis, as the two formulas produce different results for the same dataset.