Standard Deviation Calculator
Calculate sample and population standard deviation from a list of numbers.
How to use this tool
- Enter numbers (comma-separated) in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your sample sd (s) and the full breakdown beneath it.
Enter a comma-separated list of numbers to calculate both the sample standard deviation (s) and population standard deviation (σ).
Formula
Sample SD: s = √[ Σ(xi − x̅)2 / (n − 1) ]
Population SD: σ = √[ Σ(xi − x̅)2 / n ]
How it works
The calculator parses a comma-separated list of numbers, computes the arithmetic mean, then sums the squared deviations from that mean. Sample SD divides by n − 1 (Bessel's correction) to give an unbiased estimate of the population standard deviation when only a sample is available; population SD divides by n and is appropriate when the data represents the entire population.
Worked example
Worked example
- Data: 2, 4, 4, 4, 5, 5, 7, 9. Count n = 8; mean = (2+4+4+4+5+5+7+9)/8 = 40/8 = 5.
- Squared deviations from mean: 9, 1, 1, 1, 0, 0, 4, 16. Sum of squared deviations = 32.
- Sample SD: s = √(32 / 7) = √4.5714 ≈ 2.1381.
- Population SD: σ = √(32 / 8) = √4 = 2.0.
Sample SD = 2.1381; population SD = 2.0; count = 8.
Key terms
- Standard deviation
- A measure of how spread out values are around the mean; larger values indicate greater variability.
- Bessel's correction
- Dividing by n−1 instead of n when estimating population variance from a sample, to remove the downward bias introduced by using the sample mean.
- Mean (x̅)
- The arithmetic average of all values, calculated as the sum of values divided by the count.
- Sum of squared deviations
- The sum of (xᵢ − x̅)² over all data points; the core quantity from which both variance and standard deviation are derived.
- Sample vs. population SD
- Use sample SD (divide by n−1) when your data is a subset of a larger population; use population SD (divide by n) when the data covers the entire population.
Frequently asked questions
- What is the difference between sample and population standard deviation?
- Sample SD (s) divides by n−1 (Bessel's correction) and is used when your data is a sample from a larger population. Population SD (σ) divides by n and applies when you have data for the entire population.