Sample vs Population Variance

The two formulas side by side

The top half of both formulas, the sum of squared deviations, is identical. Only the denominator changes. For the data set 2, 4, 4, 4, 5, 5, 7, 9 the sum of squares is 32, so the population variance is 32 / 8 = 4 and the sample variance is 32 / 7 ≈ 4.571.

A quick decision checklist

Ask yourself these questions in order:

1. Do I want to draw conclusions about anyone or anything not in my data?If yes, use sample variance.
2. Could I have collected more data from the same source? More survey respondents, more readings, more days. If yes, your data is a sample.
3. Is my data literally every member of the group, and I only care about that group? Only then use population variance.

If you are still unsure, choose sample. Using the population formula on a sample understates the spread, which makes results look more certain than they are. Using the sample formula on a full population overstates it only slightly.

Examples

Note the class example: the same 28 marks become a sample the moment you use them to talk about "students at this school" or "students who take this exam". The choice depends on the question you are asking, not on the numbers themselves.

Why n − 1? A proof you can check by hand

A sample's own mean is always the point that minimises the squared distances to that sample. The true population mean is somewhere else, so measuring distances from the sample mean makes the spread look a little smaller than it really is. Dividing by a smaller number,n − 1, pushes the estimate back up by exactly the right amount on average. This adjustment is called Bessel's correction.

Here is a small case where you can see it work. Take a population of just three values,1, 2, 3. Its mean is 2 and its true variance is ((1 − 2)² + 0² + (3 − 2)²) / 3 = 2/3 ≈ 0.667.

Now list every possible sample of two values drawn with replacement (there are nine) and compute the variance of each one both ways:

Dividing by n gives an average estimate of 1/3, half the true value of 2/3. Dividing by n − 1 gives an average of exactly 2/3. Any single sample can still be off, but across all possible samples the n − 1 version hits the true variance on the nose. That property is what statisticians mean by an unbiased estimator.

How big is the difference?

Sample variance always equals population variance multiplied by n / (n − 1). That factor shrinks quickly as the data grows:

With small data sets, typical of homework, lab work and pilot studies, the choice changes the answer a lot. With thousands of values it barely matters.

One value is a special case

With a single value, n − 1 is zero, so sample variance is undefined: one observation tells you nothing about how much a group varies. Population variance of a single value is 0. The variance calculator explains this instead of returning an error, and lets you switch modes with one click to compare both answers on your own data.