Variance Calculator

Sample & population variance · deviation chart · steps

Sample divides by n−1; population divides by n
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What Is Variance?

Variance measures how far data values are spread from their mean. It is the average of squared deviations — squaring ensures all deviations are positive and emphasizes larger deviations. A variance of zero means all values are identical.

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Squared Units

Variance is in squared units (e.g., cm²). Take the square root for standard deviation in original units.

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Additive Property

For independent variables, variances add: Var(X+Y) = Var(X) + Var(Y). This makes variance useful in probability theory.

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Outlier Sensitive

Squaring amplifies the effect of outliers. A single extreme value can dramatically increase variance.

The Variance Formula

Sample Variance:  s² = Σ(xi − x̄)² / (n − 1)
Population Variance:  σ² = Σ(xi − μ)² / n

Worked Example

Data: [4, 8, 6, 5, 3]
Step 1: Mean = (4+8+6+5+3)/5 = 26/5 = 5.2
Step 2: Deviations: −1.2, 2.8, 0.8, −0.2, −2.2
Step 3: Squared: 1.44, 7.84, 0.64, 0.04, 4.84
Step 4: Sum of squares = 14.8
Step 5 (sample): s² = 14.8 / 4 = 3.7
Step 5 (population): σ² = 14.8 / 5 = 2.96

Properties of Variance

PropertyFormulaMeaning
Non-negativeVar(X) ≥ 0Always zero or positive
Constant shiftVar(X + c) = Var(X)Adding a constant doesn’t change spread
ScalingVar(cX) = c² · Var(X)Multiplying by c scales variance by c²
IndependenceVar(X+Y) = Var(X) + Var(Y)Variances of independent variables add
Zero varianceVar(X) = 0 ⇔ X is constantAll values are identical

Where Variance Is Used

Finance & Risk

Portfolio variance measures investment risk. Lower variance = more stable returns.

ANOVA

Analysis of Variance partitions total variance into between-group and within-group components.

Quality Control

Manufacturing uses variance to monitor process consistency and detect deviations from standards.

Frequently Asked Questions

Variance measures how spread out data values are from their mean. It is the average of squared deviations. Sample variance divides by n−1 while population variance divides by n. Larger variance means more spread; zero variance means all values are identical.
Variance is algebraically convenient because variances of independent variables are additive. It is used in ANOVA, regression analysis, and hypothesis testing. Standard deviation is the square root of variance and is easier to interpret since it has the same units as the data.
No. Variance is always zero or positive because it is computed from squared deviations. A variance of zero means all data values are identical. If you get a negative variance, it indicates a calculation error.
The coefficient of variation (CV) equals the standard deviation divided by the mean, times 100%. It measures relative variability, allowing comparison across datasets with different scales. Use CV only when the mean is positive and meaningful.
Dividing by n−1 is Bessel’s correction. The sample mean constrains one degree of freedom, so dividing by n would underestimate the true population variance. Dividing by n−1 produces an unbiased estimate. For large samples the difference is negligible.
Variance is always non-negative. Adding a constant to all values does not change variance. Multiplying all values by c scales variance by c². For independent random variables, variances are additive. Variance is sensitive to outliers due to squaring.

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