Standard Deviation Calculator
Sample and population standard deviation, with working
Sample Standard Deviation
Working
- Mean = 144 ÷ 8 = 18
- Squared differences: (10 − 18)² + (12 − 18)² + (23 − 18)² + (23 − 18)² + (16 − 18)² + (23 − 18)² + (21 − 18)² + (16 − 18)²
- Σ(x − mean)² = 192
- Variance = 192 ÷ (n − 1 = 7) = 27.428571
- SD = √27.428571 = 5.2372294
- 1Paste your numbersCommas, spaces or new lines
- 2Sample or population?Most data is a sample
- 3See the spreadSD, variance and the working
Standard deviation formulas
Standard deviation measures how spread out numbers are around their mean. The two versions differ only in what you divide by:
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )
Population: σ = √( Σ(x − μ)² ÷ N )
Variance = SD²
Sample or population?
Use population when your data covers every member of the group you care about, such as the marks of every student in one class. Use sample when it's a subset used to estimate a bigger group, such as 50 households surveyed from a city. Dividing by n − 1 (Bessel's correction) stops a sample from underestimating the spread.
Reading the result
For roughly bell-shaped data, about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three.
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Frequently asked questions
What does a high standard deviation mean?
The values are spread far from the mean. A low standard deviation means they cluster tightly around it.
Why divide by n − 1 for a sample?
A sample's values sit closer to their own mean than to the true population mean, so dividing by n would underestimate the spread. Dividing by n − 1 corrects that bias.
What is the difference between variance and standard deviation?
Variance is the average squared distance from the mean; standard deviation is its square root, which puts it back in the same units as the data.
Last reviewed: October 2026