Standard Deviation Calculator Sample and Population

About this tool

This free standard deviation calculator finds the sample and population standard deviation of a list of numbers, with the variance, mean, median, mode, range and standard error. It also shows every step of the working, so you can check homework or learn how the formula works.

Quick answer: Paste your numbers (separated by commas, spaces or new lines), choose Sample or Population, and the standard deviation appears at once with the step-by-step working below it.

Sample or population standard deviation?

• Population (σ): use it when your numbers are the whole group you care about, such as the scores of every student in one class. It divides by N, the number of values.
• Sample (s): use it when your numbers are a sample taken from a bigger group, such as 50 people surveyed from a city. It divides by n − 1, which gives a better estimate for the whole group.

If you are not sure, most statistics classes and spreadsheets use the sample standard deviation by default.

Standard deviation formula

Population: σ = √( Σ(x − μ)² ÷ N )
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )

1. Find the mean: add all the values and divide by how many there are.
2. Subtract the mean from each value and square the result.
3. Add up the squares.
4. Divide by N (population) or n − 1 (sample). This is the variance.
5. Take the square root. This is the standard deviation.

Worked example

For the numbers 2, 4, 4, 4, 5, 5, 7, 9 the mean is 40 ÷ 8 = 5. The squared differences are 9, 1, 1, 1, 0, 0, 4, 16, which add up to 32. The population variance is 32 ÷ 8 = 4, so the population standard deviation is 2. The sample variance is 32 ÷ 7 = 4.571, so the sample standard deviation is about 2.138.

Need just the mean, median and mode? Use the average calculator. For other maths, try the scientific calculator.

FAQ

What does standard deviation tell you?

It measures how spread out the numbers are around the mean. A small standard deviation means the values sit close to the average; a large one means they are spread widely.

What is the difference between variance and standard deviation?

Variance is the average of the squared differences from the mean. Standard deviation is its square root, so it is in the same units as your data, which makes it easier to read.

Can I paste numbers from a spreadsheet?

Yes. Copy a column or row from Excel or Google Sheets and paste it in. Tabs, spaces, commas and new lines all work, and anything that is not a number is ignored and counted.