Statistics
Variance Calculator
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How to use it
Using the variance calculator
- 01
Paste your values
Commas, spaces or newlines all separate cleanly.
- 02
Choose the divisor
Population (N) for complete data; sample (n−1) for estimates from a subset.
- 03
Read the companion rows
The SD row converts back to your original units for easier interpretation.
Good to know
Why square the deviations?
Raw deviations sum to zero by construction; positive and negative errors cancel. Squaring keeps everything positive, penalizing large misses disproportionately and yielding mathematically tractable formulas used across all of statistics.
Where variance shows up outside class
Portfolio volatility is (a scaled) variance of returns; manufacturing tolerances are monitored through process variance; A/B tests compare variances before trusting mean differences. Low variance often signals reliability more powerfully than a good average does.
- Zero variance = perfectly consistent data.
- Squared units: dollars → dollars², kg → kg².
- Take the square root (SD row) for intuition.
How it's calculated
The math behind this calculator
σ² = Σ(x − μ)² / N s² = Σ(x − x̄)² / (n − 1)Variance averages the squared distances of each value from the mean. It is the engine behind standard deviation (its square root), ANOVA tests, R² in regression and portfolio risk math in finance.
As with standard deviation, choose the population form (divide by N) for complete datasets and the sample form (divide by n − 1) when estimating from a subset.
Assumptions & limitations
- Values are separated by commas, spaces or newlines.
- Sample variance needs at least two numbers.
- Variance is in squared units; compare spreads via its square root when units matter.
Worked example
For 2, 4, 4, 4, 5, 5, 7, 9 the population variance is exactly 4 (squared units), giving a standard deviation of 2.
FAQ
Frequently asked questions
- Is variance ever negative?
- Never. Squared deviations cannot be negative, so variance ranges from zero upward.
- How is this different from standard deviation?
- SD is simply √variance. Variance is convenient algebraically; SD is convenient interpretively because units match the data.
- Why divide by n−1 for samples?
- Bessel’s correction fixes the bias from using the sample mean instead of the true population mean, making the estimate unbiased on average.
- Does variance depend on scale?
- Yes, quadratically: doubling every value multiplies variance by four. Normalize data before comparing spreads across different scales.
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