What is the coefficient of variation?

The coefficient of variation (CV) equals the standard deviation divided by the mean times 100 percent. It measures relative variability allowing comparison across datasets with different scales. Use CV only when the mean is positive and meaningful.

What are the properties of variance?

Variance is always non-negative. Adding a constant to all values does not change variance. Multiplying all values by a constant c scales variance by c squared. For independent random variables variances are additive. Variance is sensitive to outliers due to squaring.

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Variance Type

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(x_i − x̄)² / (n − 1)

Population Variance: σ² = Σ(x_i − 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

Property	Formula	Meaning
Non-negative	Var(X) ≥ 0	Always zero or positive
Constant shift	Var(X + c) = Var(X)	Adding a constant doesn’t change spread
Scaling	Var(cX) = c² · Var(X)	Multiplying by c scales variance by c²
Independence	Var(X+Y) = Var(X) + Var(Y)	Variances of independent variables add
Zero variance	Var(X) = 0 ⇔ X is constant	All values are identical

Property

Formula

Meaning

Non-negative

Var(X) ≥ 0

Always zero or positive

Constant shift

Var(X + c) = Var(X)

Adding a constant doesn’t change spread

Scaling

Var(cX) = c² · Var(X)

Multiplying by c scales variance by c²

Independence

Var(X+Y) = Var(X) + Var(Y)

Variances of independent variables add

Zero variance

Var(X) = 0 ⇔ X is constant

All 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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Variance Calculator

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Deviation from Mean

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What Is Variance?

Squared Units

Additive Property

Outlier Sensitive

The Variance Formula

Worked Example

Properties of Variance

Where Variance Is Used

Finance & Risk

ANOVA

Quality Control

Frequently Asked Questions

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Result

Enter data and click Calculate

Deviation from Mean

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Python Compiler

Related Tools

What Is Variance?

Squared Units

Additive Property

Outlier Sensitive

The Variance Formula

Worked Example

Properties of Variance

Where Variance Is Used

Finance & Risk

ANOVA

Quality Control

Frequently Asked Questions

Support This Free Tool