Standard Deviation Calculator

Standard deviation measures how spread out numbers are around their mean. A small standard deviation means the values cluster tightly; a large one means they are scattered widely. Variance is simply the standard deviation squared.

Enter a comma-separated list. Both versions are shown: population standard deviation (your list is the complete set) and sample standard deviation (your list is a sample of a larger group, which divides by n − 1 instead of n and gives a slightly larger value).

The formula

σ = √( Σ(x − μ)² ÷ n ) for a population; s = √( Σ(x − x̄)² ÷ (n − 1) ) for a sample

Worked example

This classic textbook data set has a mean of 5, a population variance of exactly 4 and a population standard deviation of exactly 2. As a sample, the variance is about 4.57 and the standard deviation about 2.14.

The example is pre-loaded in the result panel — press Calculate with your own numbers to replace it.

Frequently asked questions

Should I use population or sample standard deviation?

Use the population version when your list is everything you care about (all 20 students in a class). Use the sample version when your list is a subset used to estimate a bigger group (20 students sampled from a whole school).

What does a standard deviation of zero mean?

Every value is identical, so there is no spread at all. A list like 10, 10, 10 has a standard deviation and variance of 0.

Why square the differences?

Squaring makes positive and negative deviations both count positively and gives extra weight to values far from the mean. Taking the square root at the end brings the result back into the original units.

Category: Math calculators

Mean	5.00
Population standard deviation (σ)	2.00
Sample standard deviation (s)	2.14
Population variance	4.00
Sample variance	4.57
Count	8