Covariance Calculator

Formulas

Sample covariance: cov(X, Y) = Σ((xᵢ - x̄)(yᵢ - ȳ)) / (n - 1)

Population covariance: cov(X, Y) = Σ((xᵢ - x̄)(yᵢ - ȳ)) / n

The n - 1 divisor (Bessel’s correction) makes the sample covariance an unbiased estimator of the population covariance. Statistical software defaults to the sample version.

The sign is what matters

The magnitude depends on the units: cov(height_cm, weight_kg) for adults is roughly +70, but change to height in metres and it drops to +0.7, same relationship, different number. This is why correlation is more commonly reported.

Covariance vs. correlation

correlation = covariance / (σ_X × σ_Y)

Correlation is the unit-free version, always in [-1, +1], comparable across datasets. Covariance is mostly useful inside formulas (portfolio variance, regression, PCA) rather than as a summary statistic.

Where covariance is used directly

• Portfolio theory: portfolio variance = Σ wᵢ wⱼ cov(rᵢ, rⱼ). You need covariance, not correlation, because the weights and variances matter.
• Linear regression: the slope estimate β = cov(X, Y) / var(X).
• Principal component analysis (PCA): the covariance matrix is what gets diagonalised.
• Signal processing: cross-covariance measures similarity between signals at different lags.

Common mistakes

• Mixing n and n-1: sample and population covariance can differ by a few percent on small datasets and cause reproducibility issues. Decide which you need and state it.
• Interpreting magnitude: a covariance of 50 is not “big” on its own; it depends entirely on the units and variance of X and Y.
• Assuming zero covariance = independence: zero covariance means no linear relationship. Two variables can have covariance 0 and still be perfectly predictable non-linearly (e.g., Y = X² on symmetric X).