Variance Calculator
Calculate population and sample variance, weighted variance for data with different importances, and pooled variance for combining multiple samples.
σ²Variance Calculator
Use when you have data for the entire population. Formula: σ² = Σ(x-μ)²/N
How to Use the Variance Calculator
Choose the type of variance calculation, enter your data values separated by commas or spaces, and click Calculate. Results include variance, standard deviation, and other related statistics.
Calculator Modes
Population Variance (σ²)
Use when your data represents the entire population you're studying. Divides by N (total count):
- Formula: σ² = Σ(x - μ)² / N
- Where μ is the population mean and N is the population size
Sample Variance (s²)
Use when your data is a sample from a larger population. Uses Bessel's correction (n-1) for an unbiased estimate:
- Formula: s² = Σ(x - x̄)² / (n-1)
- Where x̄ is the sample mean and n is the sample size
Weighted Variance
When data points have different importances (weights). Enter data values and corresponding weights in the same order.
- Formula: σ² = Σw(x - μw)² / Σw
- Where μw is the weighted mean and w are the weights
Pooled Variance
Combines variances from multiple independent samples assuming equal population variances. Useful in ANOVA and t-tests.
- Formula: s²p = Σ(nᵢ-1)sᵢ² / Σ(nᵢ-1)
- Enter each sample's variance and sample size
Understanding Variance
Variance measures how far data points are spread out from their mean. A low variance indicates data points are close together; a high variance indicates they're spread out over a larger range.
Relationship to Standard Deviation
Standard deviation (σ or s) is the square root of variance. While variance is in squared units, standard deviation is in the same units as the original data, making it easier to interpret.
Common Applications
- Quality control and process improvement
- Financial risk assessment and portfolio theory
- Scientific experiments and research
- Educational testing and assessment
- Manufacturing tolerance analysis
- Machine learning and data science
Key Properties of Variance
- Variance is always non-negative (≥ 0)
- Variance of a constant is zero
- Var(aX + b) = a²Var(X) for constants a and b
- For independent variables: Var(X + Y) = Var(X) + Var(Y)