The Pearson product-moment correlation coefficient is a measure of a linear correlation between two variables X and Y, giving a value between +1 and -1 inclusive. It is widely used in the science as a measure of the strength of linear dependence between two variables. Pearson’s correlation coefficient between two variables is defined as the covariance of the two variables divided by the product of their standard deviation. The form involves a product moment that is the mean of the product of the mean-adjusted random variables; hence the modifier product moment in the name.

The absolute value of both the sample and population Pearson correlation coefficients are less than or equal to 1. Correlations equal to 1 or -1 correspond to data points lying exactly on a line, or to a bivariate distribution entirely supported on a line. A key mathematical property of the Pearson correlation coefficient is that it is invariant to separate changes in location and scale in the two variables.

The correlation coefficient ranges from -1 to 1. A value of 1 implies that a linear equation describes the relationship between X and Y perfectly with all data points lying on a line for which Y increases as X increases. A value of -1 implies that all data points lie on a line for which Y decreases as X increases. A value of 0 implies that there is o linear correlation between the variables.

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