Calculate population variance (σ²), sample variance (s²), standard deviation, sum of squares, Bessel correction, and CV. Includes deviation table and computation methods.
The population variance calculator computes both population variance (σ², dividing by n) and sample variance (s², dividing by n−1) from your data. Variance is the average squared deviation from the mean — the fundamental measure of spread that underlies most of classical statistics.
This tool shows both the definitional and computational formulas for the sum of squares, the Bessel correction factor, standard deviations (σ and s), coefficient of variation, and a complete deviation table with squared deviations. The visual comparison chart puts variance, SD, IQR, and range side by side.
Enter your data, choose population or sample context, and get a comprehensive variance analysis with step-by-step calculations. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results. Validate that outputs match your chosen standards. Run at least one manual sanity check before publishing.
This calculator provides both population and sample variance with full transparency — you see every squared deviation, both formula methods, and the Bessel correction factor. The deviation table shows exactly which values contribute most to the variance, helping identify outliers and understand the structure of your data's spread.
The visual comparison of variance, SD, IQR, and range gives immediate context for how different dispersion measures relate for your specific data.
Population variance σ² = Σ(xᵢ − μ)² / N. Sample variance s² = Σ(xᵢ − x̄)² / (n − 1). Sum of squares SS = Σ(xᵢ − x̄)² = ΣXᵢ² − (ΣXᵢ)²/n. Bessel correction: s² = σ² × n/(n−1).
Result: σ² = 72.9097, s² = 79.5379
Mean = 81.25. Sum of squares = 874.917. Population variance = 874.917/12 = 72.91. Sample variance = 874.917/11 = 79.54. The Bessel correction factor is 12/11 = 1.0909, making sample variance about 9% larger than population variance.
Variance is the second central moment of a distribution: E[(X − μ)²]. It appears everywhere in statistics: the central limit theorem involves variance, ANOVA partitions variance into components, regression minimizes variance of residuals, and confidence intervals are proportional to variance. Understanding variance is fundamental to understanding statistics.
In machine learning, variance has a specific meaning: how much a model's predictions change across different training sets. High variance means overfitting — the model is too sensitive to the specific data it was trained on. The bias-variance tradeoff is about finding models with low enough variance (stable) and low enough bias (accurate) for good generalization.
For non-normal data, variance can be misleading because it's heavily influenced by outliers (due to squaring deviations). Robust alternatives include the IQR, MAD (median absolute deviation), and Qn and Sn estimators. For heavy-tailed distributions (Cauchy, Pareto), variance may not even exist mathematically — the integral diverges.
Population variance (σ²) divides the sum of squared deviations by N (the total count) and is used when your data contains every member of the population. Sample variance (s²) divides by n−1 (Bessel's correction) and is used when your data is a sample from a larger population. The n−1 correction makes s² an unbiased estimator of the true population variance.
Dividing by n would systematically underestimate the true population variance because the sample mean is closer to the sample data than the true population mean. The n−1 correction (Bessel's correction) removes this bias. Intuitively, once you know the mean and n−1 deviations, the last deviation is determined — you have only n−1 independent pieces of information (degrees of freedom).
The sum of squares is Σ(xᵢ − x̄)² — the total squared deviation from the mean. It's the numerator of both variance formulas. SS can also be computed as ΣXᵢ² − (ΣXᵢ)²/n (the computational formula). SS appears throughout statistics in ANOVA, regression, and hypothesis testing.
Variance is mathematically more fundamental: it's additive for independent variables, it decomposes nicely in ANOVA, and it's the key parameter in many probability distributions. SD is more interpretable because it's in the original units. Use variance for mathematical operations and SD for reporting and interpretation.
A large variance means data values are widely spread from the mean. However, "large" is relative — a variance of 100 is large for heights in centimeters but small for distances in kilometers. Use the coefficient of variation (CV = SD/mean × 100%) to compare variability across datasets with different scales.
No. Variance is the average of squared deviations, and squares are always non-negative. A variance of zero means all values are identical. Any positive value indicates some spread. If you get a negative number, there's a computational error — possibly from catastrophic cancellation in the computational formula with very large numbers.