Covariance Calculator
Calculate the covariance between two features or variables in a dataset.
Inputs
Sample Covariance
21.0000
Population Covariance
16.8000
Mean of X
6.0000
Mean of Y
10.6000
Step by step
Mean of X, Mean of Y
= x̄=6.0000, ȳ=10.6000
Σ(x−x̄)(y−ȳ)
= 84.0000
Sample covariance: ÷ (n−1)
84.0000 ÷ 4
= 21.0000
Population covariance: ÷ n
84.0000 ÷ 5
= 16.8000
How it works
Covariance measures how two variables change together: cov(X,Y) = Σ(x − mean_x)(y − mean_y) / (n−1) for the sample covariance. A positive covariance means the variables tend to increase together; a negative covariance means one tends to increase as the other decreases; a covariance near zero suggests no linear relationship. Unlike correlation, covariance's magnitude depends on the scale of the input variables, which is why it's often normalized into the unitless correlation coefficient for interpretability across different feature scales.
Formula
cov(X, Y) = sum((x_i - mean_x)(y_i - mean_y)) / (n - 1)
- x_i
- Values of variable X
- y_i
- Values of variable Y
- mean_x
- Mean of X
- mean_y
- Mean of Y
- n
- Number of paired observations
Frequently Asked Questions
Why is covariance hard to interpret directly?
Covariance's magnitude depends on the units and scale of both variables — a covariance of 1000 could indicate a strong or weak relationship depending on whether the variables are measured in single units or thousands, which is why correlation (a normalized version) is usually preferred for interpretation.
What does a covariance of exactly zero mean?
It means there is no linear relationship between the two variables on average — but note this doesn't rule out a strong nonlinear relationship, since covariance only captures linear co-movement.
How is covariance used in ML?
The covariance matrix (pairwise covariances across all features) underlies techniques like PCA (Principal Component Analysis), Gaussian distributions in probabilistic models, and portfolio-style risk analysis of feature interactions.
Why must both series be the same length?
Covariance is computed from paired observations — each x_i must correspond to the same observation as y_i — so mismatched lengths mean the pairing is undefined and the calculation cannot proceed.