Covariance Calculator

Measure how two variables change together and their relationship strength.

Last updated: March 2026

Calculate Covariance

Data (x,y pairs per line)

Sample covariance (divide by n-1)

Covariance

1.500000

Pairs (n)

Correlation (r)

0.774597

Mean X

3.0000

Mean Y

4.0000

Var(X)

2.5000

Var(Y)

1.5000

What is Covariance?

Covariance measures how two variables change together. A positive covariance means when one variable increases, the other tends to increase as well. A negative covariance means when one increases, the other tends to decrease. Zero covariance means no linear relationship.

Unlike correlation, covariance is unbounded — it depends on the units and scale of your variables. This makes it harder to interpret directly. For example, covariance in dollars × pounds looks completely different from covariance in cents × ounces, even though the relationship is the same.

Correlation standardizes covariance to fix this problem. In fact, correlation = covariance / (SD of X × SD of Y). This is why correlation is often preferred for comparison, while covariance is useful in statistical models and portfolio analysis.

How to Calculate Covariance

The Formula

Sample: Cov(X,Y) = Σ[(xᵢ - x̄)(yᵢ - ȳ)] / (n - 1)

Population: Cov(X,Y) = Σ[(xᵢ - μₓ)(yᵢ - μᵧ)] / N

Use sample covariance (n-1) for sample data, population (n) for complete populations

Interpretation

Cov > 0

Positive relationship: variables tend to increase/decrease together

Cov < 0

Negative relationship: as one increases, the other tends to decrease

Cov ≈ 0

No linear relationship

Covariance vs Correlation

Covariance:

Unbounded, depends on scale/units, harder to interpret across different datasets

Correlation:

Bounded [-1, +1], standardized, comparable across datasets, easier to interpret

Example: Study Hours vs. Test Scores

Calculate covariance between study hours (X) and test scores (Y) for 5 students:

Study Hours: 1, 2, 3, 4, 5 → mean = 3

Test Scores: 60, 65, 75, 85, 95 → mean = 76

Step 1:

Deviations: (1-3)=-2, (2-3)=-1, (3-3)=0, (4-3)=1, (5-3)=2

Step 2:

For Y: (60-76)=-16, (65-76)=-11, (75-76)=-1, (85-76)=9, (95-76)=19

Step 3:

Products: (-2)(-16)=32, (-1)(-11)=11, (0)(-1)=0, (1)(9)=9, (2)(19)=38

Step 4:

Sum = 32+11+0+9+38 = 90

Result:

Cov(X,Y) = 90 / (5-1) = 22.5 — Strong positive covariance!

Frequently Asked Questions

Why is covariance hard to interpret?

Covariance depends on the units and scale. If X is in dollars and Y is in kilograms, doubling units doubles covariance. Correlation solves this by standardizing.

When should I use covariance instead of correlation?

Correlation is better for understanding relationship strength. Use covariance in statistical models, portfolio analysis, or when you need the raw joint variance.

Can covariance be infinite?

No, but it can be very large if variables have large variances. Correlation is always bounded [-1, +1], making it more interpretable.

What's the covariance of a variable with itself?

Cov(X, X) = Var(X). The covariance of a variable with itself is its variance.

Why divide by n-1 for sample covariance?

Dividing by n-1 provides an unbiased estimate of population covariance. Dividing by n underestimates population covariance.

Is covariance symmetric? (Cov(X,Y) = Cov(Y,X))

Yes, absolutely. The order doesn't matter. Cov(X,Y) = Cov(Y,X) always.