Population Variance Calculator

What the population variance calculator does

Population variance measures how spread out a set of values is, when that set is the entire group you care about. This calculator takes your data and returns the number of observations, the mean, the population variance, and the standard deviation, so you can see both the spread and the mean and deviations behind it.

Variance is one of the most important measures in statistics, the foundation that standard deviation and much else is built on. Below is how it works and when to use the population form rather than the sample one.

How to use it

1. Enter your dataset in the box, with values separated by commas, spaces, or new lines.
2. Press Calculate for the count, mean, variance, and standard deviation, or Reset to clear it.

How population variance is worked out

Variance is the average of the squared distances between each value and the mean. The calculator finds the mean first, then for every value measures how far it sits from that mean, squares each distance, and averages them over all the values:

Population variance (σ²) = the sum of (each value minus the mean)² ÷ N

Squaring the distances does two useful things: it makes every term positive, so distances above and below the mean do not cancel out, and it gives extra weight to values that lie far away. Dividing by N, the total number of values, turns that sum into an average. The result is a single number for how tightly or loosely the data clusters around its mean.

Dividing by N versus n minus 1

This is the distinction that defines the tool, and it is the one thing to get right. Population variance divides by N, the full count of values. Sample variance instead divides by one less than the count, n minus 1. Which is correct depends entirely on what your data represents.

Use population variance, dividing by N, when your data is the whole population, every member of the group you are studying, with no one left out. Use sample variance, dividing by n minus 1, when your data is only a sample drawn to estimate a larger population. The smaller divisor for a sample nudges the estimate upward, correcting for the fact that a sample tends to look a little less spread out than the full population it came from. So the choice is not about the numbers themselves but about whether they are everything, or just a slice.

Why the units are squared

Because variance squares every distance, it comes out in squared units. If your data is measured in metres, the variance is in metres squared, which is hard to picture directly. This is the one awkward thing about variance, and it is exactly why standard deviation exists.

The standard deviation is simply the square root of the variance, which undoes the squaring and brings the spread back into the original units. That is why the calculator reports both: the variance as the core quantity, and the standard deviation as the more readable version in the same units as your data. For most everyday interpretation, the standard deviation is the friendlier number, while the variance is the one that behaves neatly in the mathematics underneath.

A worked example

Suppose your population is the five values 2, 4, 6, 8, and 10. The mean is 30 divided by 5, which is 6. The distances from the mean are minus 4, minus 2, 0, 2, and 4, and squaring them gives 16, 4, 0, 4, and 16, which sum to 40.

Dividing by N, that is 40 divided by 5, the population variance is 8, and the standard deviation is the square root of 8, about 2.83. It is worth seeing the contrast: if these same five values were a sample rather than the whole population, you would divide by 4 instead of 5, giving a sample variance of 10. Same data, different divisor, because the question being asked is different.

Entering your data

Enter at least two values, separated by commas, spaces, or new lines. The calculator returns the number of observations, the mean, the population variance, and the standard deviation. Since this is the population form, use it when your data covers the entire group; if it is a sample of something larger, the sample variance is the right measure instead.

Questions people ask

What is population variance?

The average of the squared distances between each value and the mean, taken over an entire population. It measures how spread out the values are around their mean.

How is it different from sample variance?

Population variance divides the sum of squared distances by N, the full count. Sample variance divides by n minus 1. Use the population form when your data is the whole group, the sample form when it is a sample of a larger one.

Why is the variance in squared units?

Because each distance from the mean is squared. Taking the square root gives the standard deviation, which returns the spread to the original units of the data.

How does it relate to standard deviation?

The standard deviation is the square root of the variance. The calculator shows both, with the standard deviation being the more readable measure in the same units as your data.

References

A quick note on where the methods here come from. Variance, both population and sample, is 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 variance and standard deviation.

1. NIST/SEMATECH e-Handbook of Statistical Methods (variance and standard deviation). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (variance and standard deviation). https://openstax.org/details/books/introductory-statistics-2e