# Variance Calculator

Free variance calculator. Calculate population and sample variance, standard deviation, and standard error from summary statistics.

## What this calculates

Calculate both population and sample variance from summary statistics. Also computes standard deviation and standard error of the mean.

## Inputs

- **Sum of Squares (Σx²)** — The sum of each value squared.
- **Sum of Values (Σx)** — The sum of all values.
- **Number of Values (n)** — min 2 — The total number of values in the data set.

## Outputs

- **Mean (x̄)** — The arithmetic mean of the data.
- **Population Variance (σ²)** — Variance when data represents the entire population.
- **Sample Variance (s²)** — Variance when data is a sample (uses n-1).
- **Population Std Dev (σ)** — Square root of the population variance.
- **Sample Std Dev (s)** — Square root of the sample variance.
- **Standard Error (SE)** — Standard error of the mean (s / √n).

## Details

Variance measures how spread out values are from the mean.

Population Variance (σ²):
σ² = Σ(xᵢ - μ)² / N = (Σx² - (Σx)²/N) / N

Sample Variance (s²):
s² = Σ(xᵢ - x̄)² / (n-1) = (Σx² - (Σx)²/n) / (n-1)

Sample variance uses (n-1) instead of n (Bessel's correction) to provide an unbiased estimate of the population variance.

Standard Deviation is the square root of variance, in the same units as the original data.

Standard Error (SE = s/√n) estimates how much the sample mean varies from the true population mean.

## Frequently Asked Questions

**Q: When do I use population vs. sample variance?**

A: Use population variance (divide by N) when your data includes every member of the group you are studying. Use sample variance (divide by n-1) when your data is a subset of the population. In most practical situations, you have a sample, so use sample variance.

**Q: Why does sample variance use n-1?**

A: Dividing by (n-1) instead of n is called Bessel's correction. It corrects the bias that occurs because a sample tends to underestimate the true population variance. The sample mean is computed from the same data, which reduces one degree of freedom.

**Q: What is the relationship between variance and standard deviation?**

A: Standard deviation is the square root of variance. While variance is in squared units (e.g., cm²), standard deviation is in the original units (e.g., cm), making it more interpretable. Both measure the same thing: the spread of data around the mean.

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Source: https://vastcalc.com/calculators/statistics/variance
Category: Statistics
Last updated: 2026-04-21