Standard Deviation Calculator

Standard deviation measures how spread out a set of numbers is — the typical distance between each value and the mean. A small standard deviation means the data clusters tightly around the average; a large one means the values are scattered widely. This standard deviation calculator returns both the sample and the population standard deviation so you can pick the one that matches your data.

How to use this calculator

1. Enter your numbers in the values box, separated by commas, spaces, or new lines (for example, 2, 4, 4, 4, 5, 5, 7, 9).
2. Read the sample standard deviation if your data is a sample drawn from a larger group — this is the most common case and the default for most coursework and research.
3. Read the population standard deviation if your data covers the entire group you care about, with no one left out.

The only difference between the two is the divisor: the sample version divides by n − 1 (Bessel’s correction, which corrects the bias when estimating from a sample), while the population version divides by n.

Worked example

Take the data set 2, 4, 4, 4, 5, 5, 7, 9 (8 values).

mean (x̄) = (2+4+4+4+5+5+7+9) / 8 = 5

Subtract the mean from each value, square the result, and add them up:

Σ(xi − x̄)² = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32

Now divide and take the square root:

population σ = √(32 / 8) = √4   = 2.000
sample     s = √(32 / 7) = √4.571 ≈ 2.138

So a typical value sits about 2 units away from the mean of 5. For the precise formulas and conventions, see the NIST/SEMATECH e-Handbook on standard deviation and variance.

Frequently asked questions

Should I use sample or population standard deviation?

Use the sample standard deviation (divide by n − 1) when your numbers are a sample from a larger population — the usual situation in statistics homework, surveys, and experiments. Use the population standard deviation (divide by n) only when your data includes every member of the group.

Why does the sample version divide by n − 1?

Dividing by n − 1 instead of n corrects a downward bias: a sample tends to underestimate the true spread of the population it came from. The n − 1 adjustment, called Bessel’s correction, makes the estimate unbiased.

What is the difference between standard deviation and variance?

Variance is the average of the squared distances from the mean; standard deviation is the square root of the variance. Standard deviation is easier to interpret because it is expressed in the same units as your original data.