Population Variance and Standard Deviation Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The population standard deviation is calculated by summing the squared deviations from the mean, dividing by n (the total population size), then taking the square root. For the values 2, 4, 4, 4, 5, the mean is 3.8 and the population standard deviation is about 0.98.
Explanation
Population variance and standard deviation measure the dispersion of a dataset around its mean, exactly like their sample equivalents — but with one essential formula difference: the denominator here is n (the total count), not (n−1). This distinction is not a minor technical detail: it depends directly on the nature of your data. Use this population calculator only when your values represent the ENTIRE population studied, with no sampling uncertainty to correct — for example the grades of all students in a whole class, or the height of each of the five employees of a small company. In the far more common situation where your data represents only a subset drawn from a larger population, our sample variance and standard deviation calculator (with its Bessel's correction, denominator n−1) applies, because dividing by n in that case would systematically underestimate the true variance of the full population. On the same dataset, the population standard deviation is always slightly smaller than the sample standard deviation (since n is always greater than n−1) — a gap that becomes negligible as the number of values increases, but stays significant on the small samples typical of this calculator, limited to 5 values. The covariance, which measures the joint dispersion of two variables rather than one, has exactly the same distinction between a population version and a sample version.
Example: the values 2, 4, 4, 4, 5
Inputs
Values: 2, 4, 4, 4, 5 (treated here as the entire population).
Calculation
Mean = (2+4+4+4+5) ÷ 5 = 19 ÷ 5 = 3.8. Sum of squared deviations = (2−3.8)²+(4−3.8)²+(4−3.8)²+(4−3.8)²+(5−3.8)² = 3.24+0.04+0.04+0.04+1.44 = 4.8. Population variance = 4.8 ÷ 5 = 0.96. Population standard deviation = √0.96 ≈ 0.9798.
Result
This population has a mean of 3.8, a variance of about 0.96, and a standard deviation of about 0.98 — compared with the sample standard deviation of the same values, about 1.10, calculated with a smaller (n−1) denominator.
Frequently asked questions
How do you know whether your data forms a population or a sample?
Ask yourself: do these values cover absolutely every individual or element you care about, without exception, or only part of a larger set? If it is the whole (all employees of a 5-person team, for example), use the population variance. If it is only a representative subset of a larger group (a survey, a repeated experimental measurement), the sample variance applies, by far the most common situation in practice.
Why is the population standard deviation always smaller than the sample one?
Because the denominator n is always strictly greater than (n−1): dividing the same sum of squared deviations by a larger number mechanically gives a smaller result. Bessel's correction (n−1) precisely compensates for a slight systematic underestimation of a population's true variance when only a sample is available, a bias that need not be corrected if the data already covers the entire population.
Is the gap between the two formulas significant in practice?
It depends directly on the number of values: with a very large amount of data, the difference between dividing by n or by (n−1) becomes negligible. On a small number of values like those of this calculator (2 to 5 at most), however, the gap stays significant — dividing by 2 rather than by 1 (the most extreme case, with only two values) literally doubles the result, which makes choosing the right formula all the more important.