Variance Calculator

About the Variance Calculator

The Variance Calculator computes both the population variance (σ²) and sample variance (s²) for any data set. Variance measures the average of the squared deviations from the mean.

Variance is a fundamental concept in statistics. It quantifies how far a set of numbers is spread out from their average value. A variance of zero means all values are identical; a large variance means they are widely scattered.

While variance is harder to interpret intuitively because it uses squared units, it is mathematically convenient and forms the basis for standard deviation, ANOVA, regression analysis, and many other statistical methods.

When This Page Helps

Variance is the foundation for many statistical tests. This calculator computes both population and sample variance and shows all intermediate steps.

How to Use the Inputs

1. Enter numbers separated by commas.
2. View both population and sample variance.
3. See the standard deviation (square root of variance).
4. Review the mean and sum of squared deviations.
5. Use the results in further analysis like ANOVA or regression.

Example Calculation

Tips & Best Practices

• Population variance divides by N; sample variance divides by n−1.
• Variance is always non-negative (zero means no variation).
• Variance has squared units, making it harder to interpret than std dev.
• Adding a constant to all values does not change the variance.
• Multiplying all values by k multiplies the variance by k².
• Variance is additive for independent random variables.

Variance in Portfolio Theory

Harry Markowitz's Modern Portfolio Theory uses variance as the measure of risk. The goal is to find portfolios that minimize variance for a given expected return.

ANOVA and Variance

Analysis of Variance (ANOVA) compares group means by partitioning total variance into within-group and between-group components, allowing statistical testing of mean differences.

Degrees of Freedom

Sample variance divides by n−1 (not n) because the sample mean constrains one degree of freedom. This Bessel's correction provides an unbiased estimate of population variance.

Variance also appears in quality control, forecasting, and experimental work where you need to compare how tightly different samples cluster around their mean.

Frequently Asked Questions

• What is variance?

  Variance is the average of squared deviations from the mean. It measures how spread out data values are around the center.

• Why are deviations squared?

  Squaring ensures all deviations are positive, so negatives do not cancel positives, and it gives extra weight to large deviations. That makes variance especially sensitive to outliers and to points that sit far from the mean.

• What is the difference between variance and standard deviation?

  Standard deviation is the square root of variance. It has the same units as the data, making it more interpretable. Variance is in squared units.

• When should I use population vs sample variance?

  Use population variance when you have data for every member of the group. Use sample variance when your data is a subset of a larger population.

• Can variance be negative?

  No, variance is always ≥ 0 because it is a sum of squared values. A variance of zero means all data values are identical.

• How is variance used in finance?

  In portfolio theory, variance measures investment risk. Lower variance means more predictable returns. Covariance and variance are used to optimize portfolio allocation.