MATH · STANDARD DEVIATION
Standard Deviation Calculator
Calculate population and sample standard deviation and variance from a list of numbers. Uses the numerically stable two-pass algorithm.
About This Calculator
Standard deviation measures how spread out a dataset is around its mean. A small standard deviation means the values cluster tightly; a large one means they are widely dispersed. This calculator shows both the population standard deviation (σ) for a complete dataset and the sample standard deviation (s) when the data is a sample from a larger population.
How It Works
Enter your numbers separated by commas or spaces. The calculator uses the two-pass algorithm — computing the mean first, then summing the squared deviations — to avoid the numerical instability of the one-pass formula (Σx² − nμ²) that can silently lose precision.
The Formula
σ = √(Σ(xᵢ − μ)² / n) s = √(Σ(xᵢ − μ)² / (n − 1))
- μ
- arithmetic mean of the dataset
- n
- count of values
Frequently Asked Questions
- Should I use population or sample standard deviation?
- Use population (σ) when your dataset IS the entire population — for example, exam scores of all students in a class. Use sample (s) when your dataset is drawn from a larger population and you are estimating the population's spread — for example, quality-control measurements from a production batch.
- Why does sample standard deviation divide by n−1?
- Sample variance systematically underestimates population variance when computed with n. Dividing by n−1 (Bessel's correction) removes this bias, producing an unbiased estimator.
- What is variance?
- Variance is the square of the standard deviation. It is often easier to compute and manipulate mathematically, but its units are the square of the original units — which is why standard deviation (the square root of variance) is usually reported.