Covariance Calculator

Covariance is a statistical measure that indicates the degree to which two variables change together. A positive covariance means the variables tend to increase or decrease together, while a negative covariance means one tends to increase as the other decreases. A covariance near zero suggests little to no linear relationship.

Population covariance is used when you have data for an entire population and divides the sum of products by n. Sample covariance is used when your data is a subset (sample) of a larger population and divides by n − 1 instead, which corrects for bias in estimating the true population covariance.

The sample covariance formula is: Cov(X,Y) = Σ(xᵢ − x̄)(yᵢ − ȳ) / (n − 1). The population covariance formula is: Cov(X,Y) = Σ(xᵢ − x̄)(yᵢ − ȳ) / n. Here x̄ and ȳ are the means of X and Y respectively, and n is the number of data points.

Select Sample or Population as the covariance type, then enter your comma-separated values in the Data Set X and Data Set Y fields. Both datasets must have the same number of values. The calculator will immediately display the covariance, means, and a step-by-step breakdown table.

Yes. A negative covariance means that when one variable is above its mean, the other tends to be below its mean — they move in opposite directions. For example, as temperature drops, heating costs tend to rise.

Unlike correlation, covariance has no fixed maximum or minimum — it is unbounded and depends on the scale of the variables. This makes it difficult to compare covariances across different datasets, which is one reason the correlation coefficient (which standardizes covariance) is often preferred.

Covariance measures the direction of the linear relationship between two variables but is affected by the scale of measurement. Correlation is a standardized version of covariance, scaled to always fall between −1 and +1, making it easier to interpret and compare across datasets.

A covariance of zero indicates no linear relationship between the two variables — knowing the value of one tells you nothing about the linear tendency of the other. However, a zero covariance does not rule out a non-linear relationship.