Variance Calculator

Calculate population variance σ² and sample variance s² from the accepted finite values, using n and n − 1 denominators.

The Variance Calculator reports population variance σ² and sample variance s² from the accepted numeric values. Each value is the average squared distance from the appropriate mean, with a denominator that depends on whether the list represents a complete population or a sample. The page reports both so you can select the one that matches the data’s role.

Population variance divides the sum of squared deviations by n. Sample variance divides by n − 1. The sample denominator is used for the usual sample-based variance estimate; it does not mean that every observed dataset should automatically be described with s². State the selected interpretation with the result.

The input accepts finite numeric tokens separated by commas or whitespace. Empty separators are not values, and malformed or non-finite tokens are shown as ignored. Variance is measured in squared input units: if the observations are in metres, variance is in square metres. A standard deviation returns to the original unit, but it is not an output on this page.

Enter at least two finite values measured on a consistent scale.
Check the ignored-token message and the input list before interpreting either variance.
Select population σ² when the list is the complete group of interest; select sample s² when it is an observed sample.
Keep the squared-unit interpretation when reporting the result; do not describe variance as if it were a distance in the original unit.
If the measurements are converted to a new unit, remember that variance changes with the square of the conversion factor.

Population and sample variance formulas

For a population with mean μ, σ² = Σ(xᵢ − μ)² / n. Each accepted observation is centred at μ, the deviations are squared, and their sum is divided by the population count.

For a sample with mean x̄, s² = Σ(xᵢ − x̄)² / (n − 1). The n − 1 divisor is used for the sample variance calculation. Taking the square root of either variance would produce its corresponding standard deviation, but this page displays variances only.

The calculation uses unweighted observations. It does not accept a precomputed mean, sample weights, or frequency counts, and it does not adjust for a study design. If values have unequal importance or arise from a complex sampling scheme, these two formulas may not describe the required estimator.

Reading population and sample variance

The denominator and squared unit both matter

The two results use the same accepted values but divide by different quantities. Population variance describes the entered complete group; sample variance uses n − 1. Keep the σ² or s² label with the number because the label records which calculation was made.

For the default values 4, 8, 15, 16, 23, 42, the mean is 18 and the squared deviations sum to 910. Population variance is 910 / 6 = 151.6667, while sample variance is 910 / 5 = 182. Their units are the square of the original measurement unit.

Variance is sensitive to large deviations because each difference from the mean is squared. A single distant value can have a noticeable effect. The result alone does not identify which observation contributed most, test for outliers, or say whether the spread is acceptable for a particular purpose.

A useful consistency check is how the result behaves under a change of scale. Adding the same constant to every observation shifts the mean but leaves every deviation from that mean unchanged, so variance stays the same. Multiplying each observation by a conversion factor c multiplies every squared deviation, and therefore variance, by c². This is why changing metres to centimetres changes the numerical variance by the square of the unit conversion, not by the conversion factor itself.

For a sample and population computed from the same list, the squared-deviation total is shared; the divisor creates the difference. With six accepted values, n is 6 and n − 1 is 5. Keeping the divisor visible makes the two values auditable and helps prevent a correct calculation from being reported under the wrong interpretation.

Statistics tips and best practices

Write σ² or s² and state the denominator convention when sharing a variance.
Keep the squared unit in tables and reports; variance is not in the same unit as the raw values.
Use the sample formula only when the observations are being treated as a sample, not just because the list is short.
Inspect data quality and extreme observations before interpreting a large variance.
When rescaling measurements by a factor c, variance rescales by c²; convert units deliberately.

Variance worked example

For the default list, the mean is 18 and Σ(x − 18)² = 910.
Population variance is 910 / 6 = 151.6667; sample variance is 910 / (6 − 1) = 182.
The variance values use squared units. They are not the standard deviations 12.3153 and 13.4907.

Common mistakes to avoid

Using the population and sample labels interchangeably.
Forgetting that the sample denominator is n − 1, not n.
Reporting variance in the original unit rather than its square.
Interpreting variance as the typical absolute distance from the mean; the calculation squares deviations.
Assuming the larger of two variances is wrong without checking the denominator and population definition.

The calculator provides population and sample variance summaries for the accepted values only. It does not choose the correct statistical model, estimate uncertainty, or make an inference about unobserved cases.

For research or operational reporting, document the input definition, units, and denominator. Confirm that an unweighted variance is suitable for the data-collection design.