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)
Worked Example
Take the data set 4, 8, 6, 5, 7. The mean is (4+8+6+5+7)/5 = 6. The squared deviations from the mean are 4, 4, 0, 1, and 1, which sum to 10.
- Population variance: σ² = 10 / 5 = 2
- Sample variance: s² = 10 / (5−1) = 2.5
- Sample standard deviation: s = √2.5 ≈ 1.58
Notice the sample variance is larger — dividing by n−1 (Bessel's correction) compensates for the fact that a sample mean underestimates the spread around the true population mean.
Frequently Asked Questions
What is the difference between population and sample variance?+
Population variance divides the sum of squared deviations by N and is correct when your data covers the entire group you care about. Sample variance divides by n−1 (Bessel's correction), which produces an unbiased estimate when your data is only a sample drawn from a larger population. If in doubt, sample variance is usually the right choice for real-world data.
How do I enter data into the variance calculator?+
Type or paste your values separated by commas or spaces, choose the calculation mode, and click Calculate. For weighted variance you also enter a weight for each value in the same order, and for pooled variance you enter each sample's variance and size instead of raw data.
What is variance used for in practice?+
Variance quantifies spread: in finance it measures portfolio risk, in manufacturing it tracks process consistency, and in research it underlies t-tests, ANOVA, and regression. Two data sets can share the same mean but very different variances, which is why reporting an average alone often hides the real story.
Why is standard deviation often preferred over variance?+
Variance is expressed in squared units (for example, dollars squared), which is hard to interpret directly. Standard deviation is simply the square root of variance, putting the measure of spread back in the same units as your data. This calculator reports both, plus the coefficient of variation for comparing spread across data sets with different scales.
Is this variance calculator free and private?+
Yes. It is free, requires no sign-up, and every calculation runs locally in your browser — your data is never uploaded to a server. It works on mobile and desktop, which makes it handy for statistics homework, lab work, and quick data checks.
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