Covariance and Correlation Calculator

Covariance is a measure of how two random variables change together. For paired data (X, Y), if the values of Y tend to increase as the values of X increase, the covariance will be positive. If Y tends to decrease as X increases, covariance is negative. The correlation coefficient r (also called Pearson's correlation coefficient) indicates the strength of the relationship. When |r| is close to the 1, there is a strong linear relation between X and Y. See calculator and equations below.

Covariance Formula

Given a set of paired X and Y data {(X_i, Y_i)}_{i = 1...n}, the covariance of X and Y is

Covariance(X,Y) = (1/n)Σ (X_i - μ_X)(Y_i - μ_Y)

where μ_X and μ_Y are the population means of X and Y respectively. If μ_X and μ_X are replaced by the sample mean of X and Y, then the sample covariance is

Sample Covariance(X,Y) = (1/(n-1))Σ (X_i - x)(Y_i - y)

where x and y are the sample mean of X and Y respectively. These equations are analogous to the equations for variance and sample variance:

Var(X) = Cov(X,X) = (1/n)Σ (X_i - μ_X)²

Var(Y) = Cov(Y,Y) = (1/n)Σ (Y_i - μ_Y)²

Samp. Var(X) = Samp. Cov(X,X) = (1/(n-1))Σ (X_i - x)²

Samp. Var(Y) = Samp. Cov(Y,Y) = (1/(n-1))Σ (Y_i - y)²

Pearson's Correlation Coefficient

Pearson's correlation coefficient is a normalized measure of covariance that ranges from -1 to 1. Values at the extreme end indicate a strong linear relation between the two variables. The expression for the correlation coefficient is

r = Cov(X,Y)/√Var(X)Var(Y)

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