Covariance Calculator

What Is Covariance?

Covariance measures how two variables move together. A positive covariance means they tend to increase together; a negative covariance means one tends to rise when the other falls. The magnitude depends on the units of both variables, so covariance is not standardized like correlation.

Sample and Population Formulas

Population covariance divides by $n$:

$$\mathrm{Cov_{pop}}(X,Y) = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{n}$$

Sample covariance (an unbiased estimator of population covariance) divides by $n - 1$:

$$\mathrm{Cov_{sample}}(X,Y) = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{n - 1}$$

Covariance vs Correlation

Correlation scales covariance by the product of the standard deviations, producing a unitless value between $-1$ and $1$. This calculator also reports Pearson $r$ when both standard deviations are positive, so you can compare association strength independently of units.

How to Use This Calculator

• Choose sample or population covariance.
• Paste paired values for Dataset X and Dataset Y (same length).
• Read covariance, means, the sum of deviation products, and Pearson correlation.

Example

For $X = \{2,4,6,8,10\}$ and $Y = \{3,5,7,11,13\}$, the means are $\bar{x}=6$ and $\bar{y}=7.8$. The sample covariance is positive, showing that larger $X$ values pair with larger $Y$ values.

Related Tools

Explore related statistics tools:

• Correlation Coefficient Calculator
• Statistics Calculator
• Linear Regression Calculator