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The Pearson product-moment correlation coefficient, also known as r, R, or Pearson's r, is a measure of the strength and direction of the linear relationship between two variables that is defined as the covariance of the variables divided by the product of their standard deviations. [4]
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
The most familiar measure of dependence between two quantities is the Pearson product-moment correlation coefficient (PPMCC), or "Pearson's correlation coefficient", commonly called simply "the correlation coefficient". It is obtained by taking the ratio of the covariance of the two variables in question of our numerical dataset, normalized to ...
Third, a zero Pearson product-moment correlation coefficient does not necessarily mean independence, because only the two first moments are considered. For example, Y = X 2 {\displaystyle Y=X^{2}} ( y ≠ 0) will lead to Pearson correlation coefficient of zero, which is arguably misleading. [ 2 ]
For this reason, covariance is standardized by dividing by the product of the standard deviations of the two variables to produce the Pearson product–moment correlation coefficient (also referred to as the Pearson correlation coefficient or correlation coefficient), which is usually denoted by the letter “r.” [3]
The point-biserial correlation is mathematically equivalent to the Pearson (product moment) correlation coefficient; that is, if we have one continuously measured variable X and a dichotomous variable Y, r XY = r pb. This can be shown by assigning two distinct numerical values to the dichotomous variable.
Pearson developed the Pearson product-moment correlation coefficient, defined as a product-moment, [45] the method of moments for the fitting of distributions to samples and the Pearson distribution, among many other things. [46]
The most common of these is the Pearson product-moment correlation coefficient, which is a similar correlation method to Spearman's rank, that measures the “linear” relationships between the raw numbers rather than between their ranks.