How this instrument works
Population variance measures the same underlying idea as any variance: take every value's distance from the mean, square it so negative and positive deviations don't cancel out, and average the results. What sets it apart is the word 'population' — it applies specifically when your data set is not a slice of something bigger, but the complete group you actually care about. Every employee at a small company, every match a team played this season, every reading a single instrument took during one calibration run: if there is no larger group left unmeasured, you have a population.
Because you already hold every value that exists, there is no estimation happening and nothing to correct for — so population variance divides the sum of squared deviations by n, the plain count of values, instead of n − 1. That n − 1 adjustment (Bessel's correction) only matters when you are using a sample's mean to estimate an unknown population mean; here the mean, μ, is already the population mean exactly, calculated from the same complete data it describes.
Mixing the two up produces a real, if usually small, error: applying the sample formula to a true population needlessly inflates the result, and applying the population formula to a genuine sample understates how much the wider group actually varies. The choice depends entirely on what your data represents, not on how many values it contains — eight values can be a complete population or a tiny sample, and only you know which.
- Enter your numbers into Data set — separate them with commas, spaces or new lines in any order.
- Confirm first that your data is the entire group of interest, not a subset of a larger one — that's the condition this instrument assumes.
- Read Population variance (σ²) beneath the data field once you've entered at least two values.
- If your data is instead a sample representing a larger population you haven't fully measured, use this site's Variance (sample, s²) instrument, which divides by n − 1 rather than n.
- Want the result back in the original units? This site's Standard deviation instrument reports the square root of both the population and sample forms side by side.
Worked example — every score on one small quiz
Suppose 2, 4, 4, 4, 5, 5, 7, 9 are every score recorded for one specific eight-student seminar's quiz — a complete population, not a sample of a larger course. Enter those eight numbers into Data set. The population mean μ is 5 (sum 40 ÷ 8), and each value's squared deviation from 5 is 9, 1, 1, 1, 0, 0, 4, 16, which sums to 32.
Divide that sum of 32 by the full count n = 8, and Population variance (σ²) reads exactly 4.0. Compare that to what the same numbers would give under the sample formula — 32 ÷ 7 ≈ 4.571 — and you can see the population form returns a smaller figure, because it isn't correcting for any estimation the sample form assumes is happening.
Questions
How is population variance different from sample variance?
Population variance (σ²) divides the sum of squared deviations by n, the full count of values, and uses it when your data already covers everyone or everything in the group you're studying. Sample variance (s²) divides by n − 1 and is used when your data is a subset standing in for a larger, only partly measured population. The formulas differ only in that one divisor, but which one is correct depends on what your data represents.
How do I know if my data is a population or a sample?
Ask whether every member of the group you care about is included in your data, with nothing left out. If you measured all eight employees at a small office, that's a population. If you measured eight customers out of thousands who could have responded, that's a sample. The count of values tells you nothing by itself — only what the numbers represent decides which formula applies.
Why does the population formula divide by n instead of n − 1?
Because the mean used in the calculation, μ, is already the exact population mean — it isn't an estimate built from a subset that might understate the true spread. The n − 1 correction exists specifically to fix a bias that only appears when you estimate a population's mean from a sample of it, so it has nothing to correct for once you already have the whole population.
What happens if I use the wrong formula by mistake?
Using the sample formula (n − 1) on true population data slightly overstates the variance, since it applies a correction that wasn't needed. Using the population formula (n) on sample data slightly understates how much the wider population actually varies, because it skips the correction for estimating the mean from limited data. The gap shrinks as n grows large, but it is never exactly zero for a finite sample.
Is population variance always smaller than sample variance for the same data?
Yes, for the identical set of numbers run through both formulas, population variance is always smaller or equal, since dividing by n rather than n − 1 (a smaller number) shrinks the result. In the worked example above, the same eight scores give σ² = 4.0 under the population formula versus s² ≈ 4.571 under the sample formula.
Why is the result in squared units rather than my original units?
Because the formula squares every deviation from the mean before averaging, which keeps positive and negative deviations from cancelling to zero. To bring the figure back into your original units, take the square root — that gives the population standard deviation, σ, which this site's Standard deviation instrument computes directly.