Population Standard Deviation Calculator
Calculate population standard deviation from raw data. Paste values to get mean, variance, and sigma for the full population, not a sample.
Enter the Details
Calculate the population standard deviation of a set of numbers.
Enter numbers separated by comma , space or line break
Precision:
Result will appear here...
What the population standard deviation calculator does
The population standard deviation measures how spread out a set of numbers is, when those numbers are the entire group you care about. This calculator works it out from your data, along with the count of values used.
It is the version written with the Greek letter sigma, and it divides by the full count, N. That one detail is what separates it from the sample version, and the section below explains when it is the right choice.
How to use it
- Enter your numbers in the box, separated by commas, spaces, or line breaks.
- Choose a precision if you want more or fewer digits in the answer.
- Press Calculate for the population standard deviation, or Reset to clear it.
How the population standard deviation is worked out
It comes in four steps. Find the mean of your numbers. For each number, measure how far it is from the mean and square that distance. Average those squared distances, dividing by the full count. Then take the square root to return to your original units.
Population standard deviation = √( sum of squared distances from the mean ÷ N )
Dividing by N, the total number of values, is the mark of the population version. It works because when your data is the whole group, the mean you calculate is the true mean, so there is no bias to correct for.
When your data is a population
Use this calculator when your numbers are the complete set, with nothing left outside them. The test scores of every student in a class. Every sale you made last month. Every part that came off a line. In each case there is no larger group you are trying to reach, so the spread you measure is the real spread, not an estimate.
If instead your numbers are a sample, a smaller group meant to represent a larger population, you want the sample standard deviation, which divides by one less than the count to correct for the gap between your sample and the whole. Our sample standard deviation calculator handles that case. When in doubt about which you have, the sample version is the safer default, since most real data is a sample of something.
A worked example: eight numbers
Take the eight numbers 2, 4, 4, 4, 5, 5, 7, 9, treating them as a whole population. Their mean is 40 ÷ 8 = 5, and the squared distances from 5 add up to 32.
Dividing by the full count of 8 gives a variance of 32 ÷ 8 = 4, and the square root of that is a population standard deviation of 2. So on average these numbers sit two units away from their mean.
The precision option, and entering data
You can separate your numbers with commas, spaces, or line breaks, in any mix, and the order does not matter. The precision menu lets you set how many digits the answer shows, from a handful up to a very long decimal, which is handy if you need to feed the result into another calculation without rounding it early.
Questions people ask
What is the population standard deviation?
A measure of how spread out a full set of data is, found by taking the square root of the average squared distance from the mean, dividing by the total count N.
How is it different from the sample standard deviation?
The population version divides by N and is used when your data is the whole group. The sample version divides by N minus 1 and is used when your data is a sample estimating a larger group.
When should I use the population version?
When your numbers cover every member of the group you care about, so nothing sits outside your data. In that case the mean is exact and no correction is needed.
What does sigma mean here?
Sigma, the Greek letter, is the standard symbol for the population standard deviation. The sample standard deviation is written with the letter s instead.
References
A quick note on where the methods here come from. The definition of the population standard deviation, and how it differs from the sample version, are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering the same ground.
- NIST/SEMATECH e-Handbook of Statistical Methods (measures of spread). https://www.itl.nist.gov/div898/handbook/
- OpenStax, Introductory Statistics (measures of the spread of the data). https://openstax.org/details/books/introductory-statistics-2e
Ankit Khatiwada is a researcher and graduate student in Computer Science at Saarland University, with strengths in statistics, data analysis, data engineering, and full stack development. His work sits at the intersection of quantitative reasoning and applied technology, making him a strong fit for tools that depend on clear numerical logic. At Eon Tools, he reviews number and statistical tools.