Compute Standard Deviation: Sample and Population Formula Calculator
Sample (n − 1)
SD = √(Σ(x − mean)² ÷ (n − 1))
Values separated by commas or spaces
Locate each observation relative to the arithmetic mean.
SD = √(Σ(x − mean)² ÷ (n − 1))
The mean is the center of this scaled number line; squared deviations are computed arithmetically.
Worked example you can change
Standard deviation = √(sum of squared deviations ÷ (count − 1))
- Mean ≈ 5; squared-deviation sum ≈ 32; denominator = 7
- Sample variance ≈ 4.57142857142857; standard deviation ≈ 2.1380899352994
Calculator
Enter 1–1,000 decimal values separated by commas or ASCII whitespace, up to 32,768 characters total; each value has up to 15 significant digits and magnitude zero or 10⁻¹²–10¹².
- Count
- 8
- Mean
- 5
- Sample variance
- 4.57142857142857
- Sample standard deviation
- 2.1380899352994
Formula variants
| Shape or method | Formula | Meaning |
|---|---|---|
| Sample standard deviation | s = √[Σ(x − mean)² / (n − 1)] | Use for a sample estimating variation in a population; requires two values. |
| Population standard deviation | σ = √[Σ(x − mean)² / n] | Use when the observations comprise the whole population. |
| Variance | variance = standard deviation² | Variance has squared units; its square root has the original units. |
Questions
Should I choose sample or population?
Choose sample when estimating population spread from a sample; choose population when your list is the entire population being described.
Why does sample standard deviation use n − 1?
Estimating the mean from the same sample uses one degree of freedom, leaving n − 1 for the variance estimate.
Can standard deviation be zero?
Yes. If all observations are equal, every centered deviation is zero.
Is one observation allowed?
Population mode accepts one observation and returns zero spread. Sample mode requires at least two.
Sources and method
Calculate sample or population standard deviation and variance. Follow centered deviations, the correct denominator and a worked example you can edit.
We calculate with 40 significant digits and round only the displayed result. Diagrams are labelled explanations and are not measurement drawings.
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