Variance Calculator Formula
Understand the math behind the variance calculator. Each variable explained with a worked example.
Formulas Used
Variance Pop
variance_pop = ((a - mean_val)^2 + (b - mean_val)^2 + (c - mean_val)^2) / 3Variance Sample
variance_sample = ((a - mean_val)^2 + (b - mean_val)^2 + (c - mean_val)^2) / 2Std Dev
std_dev = sqrt(((a - mean_val)^2 + (b - mean_val)^2 + (c - mean_val)^2) / 3)Mean
mean = mean_valVariables
| Variable | Description | Default |
|---|---|---|
a | Value 1 | 4 |
b | Value 2 | 8 |
c | Value 3 | 6 |
mean_val | Derived value= (a + b + c) / 3 | calculated |
How It Works
How to Calculate Variance
Formula (Population Variance)
σ² = Σ(xᵢ - μ)² / N
Steps
1. Find the mean (μ) 2. Subtract the mean from each value 3. Square each difference 4. Find the average of the squared differences
Variance is the square of the standard deviation.
Worked Example
Find the variance of 4, 8, and 6.
- 01Mean = (4 + 8 + 6) / 3 = 6
- 02Squared differences: (4-6)² = 4, (8-6)² = 4, (6-6)² = 0
- 03Variance (population) = (4 + 4 + 0) / 3 = 2.6667
- 04Variance (sample) = (4 + 4 + 0) / 2 = 4
When to Use This Formula
- Quantifying the spread of a dataset in squared units, which is needed directly in many statistical formulas including ANOVA, regression, and chi-squared tests.
- Decomposing total variability into explained and unexplained components in analysis of variance (ANOVA) to test whether group means differ significantly.
- Calculating portfolio variance in finance by combining individual asset variances and their covariances, which requires variance (not standard deviation) as the input.
- Comparing the dispersion of two processes or populations when feeding results into further statistical computations that require squared units.
- Estimating the power of a statistical test, where the effect size relative to the population variance determines how many observations you need.
Common Mistakes to Avoid
- Using population variance (÷ N) when you should use sample variance (÷ N-1) — for samples, dividing by N systematically underestimates the true variance. The correction matters most for small samples.
- Interpreting variance in the original units — variance is in squared units (e.g., cm² if the data is in cm), which is not intuitive. Take the square root to get standard deviation if you need a measure in the original units.
- Forgetting that variance is heavily influenced by outliers — because each deviation is squared, a single extreme value contributes disproportionately. Consider whether outlier-resistant measures are more appropriate.
- Adding variances of dependent variables as if they were independent — the variance of a sum is Var(X) + Var(Y) + 2Cov(X,Y). Ignoring the covariance term gives wrong results when the variables are correlated.
Frequently Asked Questions
What is variance?
Variance measures how far a set of numbers is spread out from their average value. It is the average of the squared differences from the mean.
Why do we square the differences?
Squaring removes negative signs (so deviations above and below the mean don't cancel out) and gives more weight to larger deviations.
Learn More
Guide
Understanding Standard Deviation - Complete Guide
Learn what standard deviation is, how to calculate it step by step, and why it matters. Covers population vs. sample standard deviation with clear examples.
Ready to run the numbers?
Open Variance Calculator