How it works

The Variance Calculator helps you quickly find the population variance (σ²) and sample variance (s²) for a set of numbers. Variance is a key measure in statistics that tells you how spread out your data points are from the average (mean). A high variance means data points are very spread out, while a low variance indicates they are clustered closely around the mean.

The Formula

Population Variance (σ²): Σ(xi - μ)² / N
Sample Variance (s²): Σ(xi - x̄)² / (n - 1) Where:
xi = each data point
μ = population mean
x̄ = sample mean
N = total number of data points (for population)
n = number of data points (for sample)

Worked Example

1. Example: Test Scores

   Imagine a class of students scored these marks on a test: 85, 90, 78, 92, 88. To find the variance: 1. **Calculate the Mean:** (85 + 90 + 78 + 92 + 88) / 5 = 433 / 5 = 86.6 2. **Find Differences from Mean:** (85 - 86.6)² = (-1.6)² = 2.56 (90 - 86.6)² = (3.4)² = 11.56 (78 - 86.6)² = (-8.6)² = 73.96 (92 - 86.6)² = (5.4)² = 29.16 (88 - 86.6)² = (1.4)² = 1.96 3. **Sum of Squared Differences:** 2.56 + 11.56 + 73.96 + 29.16 + 1.96 = 119.2 4. **Population Variance (N=5):** 119.2 / 5 = 23.84 5. **Sample Variance (n-1=4):** 119.2 / 4 = 29.8

Tips, Assumptions & Limitations

• Enter your numbers separated by commas, spaces, or new lines.
• Variance is always a non-negative value. A variance of zero means all data points are identical.
• Use population variance when your data set includes every member of a defined group. Use sample variance when your data is just a subset of a larger population.
• Variance is measured in squared units of the original data, which can sometimes make it harder to interpret directly than standard deviation.