Variance Calculator
See how spread out your data really is.
Variance Calculator
Sample and population variance, standard deviation, mean, and range from a list of values.
Separate values with commas, spaces, or new lines.
Sample variance divides by n − 1 and is the usual choice for data drawn from a larger population. Population variance divides by n and describes the whole set.
Sample variance (s²)
48.3
- Sample standard deviation
- 6.9498
- Population variance
- 40.25
- Population standard deviation
- 6.3443
- Mean
- 20.5
- Count
- 6
- Range
- 19
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What This Calculator Does
The Variance Calculator measures how far a set of numbers spreads around its mean. Paste your data and it returns both sample and population variance, the matching standard deviations, the mean, count, and range.
How It Works
Variance is the average squared distance from the mean. For a sample, divide the sum of squared deviations by n − 1 (Bessel’s correction) so the estimate is unbiased for the wider population. For a complete population, divide by n. Standard deviation is the square root of variance, which returns the spread to the original units.
Example
For 12, 15, 18, 22, 25, and 31 the mean is 20.5. The squared deviations sum to 241.5, so the population variance is 241.5 / 6 = 40.25 and the sample variance is 241.5 / 5 = 48.3. The sample standard deviation is about 6.95.
Formula
s² = Σ(xᵢ − x̄)² / (n − 1)Use when the data are a sample drawn from a larger population.
σ² = Σ(xᵢ − μ)² / NUse only when the list contains the entire population.
s = √s²Reading the Results
Sample variance
The usual choice for experimental or survey data: it corrects for the fact that a sample mean sits closer to its own data than the true population mean does.
Population variance
The variance of the complete list you entered, without the n − 1 correction. It is smaller than the sample variance by the factor (n − 1)/n.
Sources and References
NIST/SEMATECH e-Handbook of Statistical Methods
s² = Σ(xᵢ − x̄)² / (n − 1)
Section 1.3.5.6 documents the sample variance and the reason for the n − 1 denominator.
References identify the method or data behind this calculator. Where a standard or tariff changes, the current edition applies, and the assumptions stated above take precedence over a remembered figure.
Frequently Asked Questions
Should I use sample or population variance?
Use sample variance whenever your numbers are a subset of a larger group you want to describe — which covers most experiments and surveys. Use population variance only when the list is the entire group, such as every score in one class.
Why does variance use squared differences?
Squaring removes the sign, so distances above and below the mean cannot cancel out. It also makes large deviations count disproportionately more, which is why variance is sensitive to outliers.
What is the difference between variance and standard deviation?
Standard deviation is the square root of variance. Variance is in squared units (for example, squared centimetres), while standard deviation is back in the original units and is easier to interpret.
Can variance be negative?
No. Squared deviations are never negative, so variance is zero or positive. A variance of zero means every value is identical.