Statistics · Standard Deviation

Standard Deviation Calculator

Find the standard deviation and variance of a data set.

Standard Deviation Calculator

Enter your numbers separated by commas or spaces, choose population or sample, then select Calculate.

Numbers

Separate numbers with commas or spaces.

Type

How this is calculated

Standard deviation measures how spread out the numbers are from the mean. Population: √(Σ(x − mean)² ⁄ n) Sample: √(Σ(x − mean)² ⁄ (n − 1)) Use "population" when your numbers are the entire group you care about; use "sample" when they're a subset used to estimate a larger group. The margin of error at 95% confidence is 1.96 × standard deviation ÷ √n — a common estimate of how much a sample's mean might differ from the true population mean.

Examples

For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5, and the population standard deviation is exactly 2. For that same data set, the margin of error at 95% confidence is about 1.386 (population) or 1.482 (sample).

About this calculation

This uses the standard, universally taught population and sample standard deviation formulas from introductory statistics.

Related tools

More calculators are on the way — browse everything planned on the Home page.

Frequently asked questions

What's the difference between population and sample standard deviation?

Population standard deviation divides by the count of numbers (n); sample standard deviation divides by one less (n − 1), which corrects for the fact that a sample tends to underestimate the true spread of the larger group it came from.

Why do I need at least 2 numbers for "sample"?

The sample formula divides by (n − 1); with only one number that would mean dividing by zero, so at least two numbers are required.

What does "margin of error" mean here?

It's a common estimate, at 95% confidence, of how far a sample's average might be from the true average of the whole group — calculated as 1.96 × standard deviation ÷ √(count). A smaller margin means the data more tightly pins down the average; it does not mean the data itself has less spread.