Standard deviation calculator

What standard deviation tells you

Standard deviation measures how spread out a set of numbers is from its average. A small standard deviation means most values sit close to the mean; a large one means the values are scattered widely. Two datasets can have the exact same average and look completely different once you know how spread out each one is — which is exactly the gap standard deviation fills in.

Population vs. sample standard deviation

This distinction trips a lot of people up, but it comes down to one question: does your list of numbers represent an entire group you care about, or just a sample drawn from a bigger group? If you have every test score from every student in a class, that's population data. If you only have scores from 20 randomly selected students out of a much larger school, that's sample data.

• Population standard deviation divides by N (the total count).
• Sample standard deviation divides by N − 1 instead, which slightly increases the result to correct for the fact that a sample tends to underestimate the true spread of the full population.

The formulas

Both formulas start the same way: find the mean, then find how far each number is from that mean, then square each of those differences (so negative and positive differences don't cancel out).

• Population variance = Σ(x − mean)² ÷ N
• Sample variance = Σ(x − mean)² ÷ (N − 1)
• Standard deviation = √variance, for either case

A worked example

For the numbers 2, 4, 4, 4, 5, 5, 7, 9 — a classic textbook example — the mean is exactly 5. Treated as a population, the standard deviation comes out to exactly 2. Treated as a sample, it comes out slightly higher, at about 2.14, because of that N − 1 adjustment.