Variance Calculator
Free variance calculator. Compute sample variance, population variance, grouped data variance, weigh
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
Variance Calculator — Sample & Population with 5 Input Modes
Raw data, frequency table, grouped data, weighted data, or from summary. Step-by-step computation, deviation squares visualization, and distribution charts.
Real-World Scenarios — Click to Load
Data Input
Data Distribution with Variance Band (μ ± σ)
Deviation Squares Visualization
Each bar shows (xᵢ − μ)². Sum of all bars = numerator in variance formula.
Population vs Sample Variance Comparison
Step-by-Step Table: xᵢ, x̄, (xᵢ−x̄), (xᵢ−x̄)²
| i | xᵢ | x̄ | (xᵢ − x̄) | (xᵢ − x̄)² |
|---|---|---|---|---|
| 1 | 72.0000 | 84.0000 | -12.0000 | 144.0000 |
| 2 | 85.0000 | 84.0000 | 1.0000 | 1.0000 |
| 3 | 90.0000 | 84.0000 | 6.0000 | 36.0000 |
| 4 | 78.0000 | 84.0000 | -6.0000 | 36.0000 |
| 5 | 92.0000 | 84.0000 | 8.0000 | 64.0000 |
| 6 | 88.0000 | 84.0000 | 4.0000 | 16.0000 |
| 7 | 76.0000 | 84.0000 | -8.0000 | 64.0000 |
| 8 | 81.0000 | 84.0000 | -3.0000 | 9.0000 |
| 9 | 95.0000 | 84.0000 | 11.0000 | 121.0000 |
| 10 | 83.0000 | 84.0000 | -1.0000 | 1.0000 |
Calculation Breakdown
For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- • Population variance (σ²) divides by N — use when you have data for every member of the group.
- • Sample variance (s²) divides by n−1 (Bessel's correction) — use when your data is a subset. Produces an unbiased estimate.
- • Shortcut formula s² = [Σx² − (Σx)²/n] / (n−1) is computationally efficient for raw data.
- • Grouped data uses class midpoints and frequencies: s² = [Σfᵢmᵢ² − (Σfᵢmᵢ)²/Σfᵢ] / (Σfᵢ − 1).
- • Weighted variance applies when observations have different importance (weights).
- • Standard deviation (σ or s) = √variance. CV = SD/mean × 100% for relative comparison.
Why Use n−1 for Samples? (Bessel's Correction)
When you divide by n instead of n−1 for sample variance, the result systematically underestimates the true population variance. Dividing by n−1 corrects this bias.
Intuition: The sample mean x̄ is always closer to the sample data than the true population mean μ. So deviations from x̄ are smaller on average than deviations from μ, producing a biased (low) estimate. The n−1 denominator restores the degrees of freedom lost when estimating μ from the sample.
Variance Properties
Scaling: Var(aX + b) = a²Var(X) — variance is unaffected by adding a constant, but scaling by a multiplies variance by a².
Independence: Var(X + Y) = Var(X) + Var(Y) when X and Y are independent. For dependent variables, add the covariance term.
Non-negativity: Variance is always ≥ 0. Zero variance means all values are identical.
Step-by-Step Computation Table
For raw data, build a table: xᵢ | x̄ | (xᵢ − x̄) | (xᵢ − x̄)². Sum the squared deviations, then divide by n−1 (sample) or N (population).
Example: Data 2, 4, 6, 8, 10 → Mean = 6
Deviations: -4, -2, 0, 2, 4 → Squared: 16, 4, 0, 4, 16 → Sum = 40
Population σ² = 40/5 = 8. Sample s² = 40/4 = 10.
Grouped Data Variance
When data is in class intervals with frequencies, use midpoints mᵢ = (lower + upper)/2. Formula: s² = [Σfᵢmᵢ² − (Σfᵢmᵢ)²/Σfᵢ] / (Σfᵢ − 1). Enter as "lower-upper:frequency" per line.
Weighted Variance
When observations have different weights wᵢ, the weighted mean is x̄_w = Σwᵢxᵢ / Σwᵢ. Weighted sample variance: s²_w = Σwᵢ(xᵢ − x̄_w)² / (Σwᵢ − 1). Enter as value:weight per line.
From Summary Statistics
If you have Sum of Squares (SS) and n, variance = SS / (n−1) for sample or SS / N for population. If you have SD, variance = SD². Use the "From summary" mode.
Frequently Asked Questions
When should I use population vs sample variance?
Use population variance when your dataset includes every member of the group. Use sample variance (n−1) when your data is a subset — this is the default in most research and software.
Can variance be negative?
No. Variance is a sum of squared deviations, so it is always non-negative. Zero variance means all values are identical.
What is the difference between variance and standard deviation?
Variance is the average squared deviation; standard deviation is its square root. SD has the same units as your data, making it easier to interpret.
How do outliers affect variance?
Variance is highly sensitive to outliers because deviations are squared. A single extreme value can dramatically increase the variance.
What is the coefficient of variation?
CV = (SD / |mean|) × 100%. It allows comparison of variability across datasets with different units or scales.
Pooled Variance (Two or More Groups)
When combining variances from k groups: s²_p = Σ(nᵢ−1)sᵢ² / Σ(nᵢ−1). Used in two-sample t-tests and ANOVA. For a single dataset, use the modes above.
Official Data Sources
Disclaimer: This calculator provides accurate variance computations using standard formulas. For critical applications, verify results. Grouped data assumes midpoints represent class values.
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