Standard deviation measures how spread out a set of numbers is around its average. A small standard deviation means values cluster tightly around the mean; a large one means they're more spread out. This calculator finds the mean, variance, and standard deviation from any list of numbers.
The formulas
The "sample" version divides by (count − 1) instead of count — a correction called Bessel's correction, used when your data is a sample meant to estimate a larger population rather than the complete population itself.
Sample vs. population — which to choose
| Situation | Use |
|---|---|
| Test scores of a whole class (all students) | Population |
| Survey of 200 people meant to represent a country | Sample |
| Every transaction your business made last year | Population |
| A random subset of customers surveyed for feedback | Sample |
Worked example
For the data set 4, 8, 6, 5, 3, 7, 9: the mean is 42 ÷ 7 = 6. The squared differences from the mean are 4, 4, 0, 1, 9, 1, 9, summing to 28. Population variance = 28 ÷ 7 = 4, giving a standard deviation of 2. Sample variance = 28 ÷ 6 ≈ 4.67, giving a standard deviation of about 2.16.
Common mistakes
- Using the wrong divisor. Mixing up sample and population standard deviation is one of the most common statistics errors, and it matters more for smaller data sets.
- Confusing variance and standard deviation. Standard deviation is the square root of variance and is expressed in the same units as the original data, which is why it's usually more intuitive to interpret than variance.
Frequently asked questions
What's the difference between sample and population standard deviation?
Population standard deviation divides by the count of values; sample standard deviation divides by (count minus 1), a correction used when the data is a sample meant to estimate a larger population.
What does a high standard deviation mean?
It means the values in the data set are spread out more widely from the average, compared to a low standard deviation where values cluster closely around the mean.
Is standard deviation always positive?
Yes. Because it's derived from squared differences and then a square root, standard deviation is always zero or a positive number, never negative.