Population Variance Calculator
Enter your complete dataset and get the population variance σ², population standard deviation σ, sample variance s², and key descriptive statistics in one step.
Sum of squared deviations from the mean, divided by N
- 1
Mean (μ = Σxᵢ ÷ N)
40 ÷ 8 = 5 - 2
Sum of squared deviations
Σ(xᵢ − μ)² = 32Square each value's deviation from the mean, then sum all squares. - 3
Population variance (σ² = Σ(xᵢ − μ)² ÷ N)
32 ÷ 8 = 4
How does this calculator work?
Population variance σ² = Σ(xᵢ − μ)² ÷ N (divide by the full count, not N − 1). Standard deviation σ = √σ². Use population variance when the list is the whole population; use sample variance s² (÷ N − 1) for a drawn sample. Paste any list of numbers and get both plus median, range and coefficient of variation.
Formula
How this is calculated
Variance measures how spread out the values in a dataset are around their mean. The population variance divides by N (the full count) because it describes the exact spread of the entire population — every data point is included. The mean μ = Σxᵢ / N is computed first, then each value's squared deviation (xᵢ − μ)² is summed and divided by N to get σ². Taking the square root gives the population standard deviation σ, which has the same units as the original data and is easier to interpret.
The sample variance s² = Σ(xᵢ − x̄)² / (N − 1) uses N − 1 (Bessel's correction) in the denominator. This makes s² an unbiased estimator of the population variance when you only have a sample, not the full population. Use σ² when your list is the entire population; use s² when it is a sample drawn from a larger population.
The calculator also reports the median, range, min, max and coefficient of variation (σ / |μ| × 100%), which expresses the spread relative to the mean for comparing datasets with different units or scales. The bell-curve diagram centred at the mean and shaded ±1σ illustrates the region that would contain ≈68% of values in a normally distributed population.
Frequently asked questions
Use population variance (÷ N) when your data is the entire group you are describing — for example, all exam scores in one class. Use sample variance (÷ N − 1) when your data is a random sample drawn to estimate a larger population — for example, a survey of 100 customers out of 10,000.
Variance is in squared units (e.g. metres²), which is harder to relate to the original measurements. Standard deviation — the square root of variance — is in the same units as the data, making it directly interpretable as the typical distance of a value from the mean.
The coefficient of variation (CV = σ / |μ| × 100%) expresses dispersion as a percentage of the mean, making it useful for comparing spread between datasets with different units or very different means. A CV above roughly 30% is often considered high variability.
Also known as
TG we-Calculate Editorial Team. (2026). Population Variance Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/population-variance-calculator
TG we-Calculate Editorial Team. "Population Variance Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/population-variance-calculator.
TG we-Calculate Editorial Team, "Population Variance Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/population-variance-calculator
@misc{wecalculate_population_variance_calculator, title = {Population Variance Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/population-variance-calculator}}, year = {2026}, note = {TG we-Calculate} }
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