In mathematics, and in particular statistics, the Pearson product-moment correlation coefficient (r) is a measure of how well a linear equation describes the relation between two variables X and Y measured on the same object or organism. It is defined as the sum of the products of the standard scores of the two measures divided by the degrees of freedom:
- Y = a score on a random variable Y
- Y' = corresponding predicted value of Y, given the correlation of X and Y and the value of X
- = mean of Y
The coefficient ranges from -1 to 1. A value of 1 shows that a linear equation describes the relationship perfectly and positively, with all data points lying on the same line and with Y increasing with X. A score of -1 shows that all data points lie on a single line but that Y increases as X decreases. A value of 0 shows that a linear model is inappropriate – that there is no linear relationship between the variables.
The linear equation that best describes the relationship between X and Y can be found by linear regression. If X and Y are both normally distributed, this can be used to "predict" the value of one measurement from knowledge of the other. That is, for each value of X the equation calculates a value which is the best estimate of the values of Y corresponding the specific value of X. We denote this predicted variable by Y.
Any value of Y can therefore be defined as the sum of Y and the difference between Y and Y:
r is a parametric statistic. It assumes that the variables being assessed are normally distributed. If this assumption is violated, a non-parametric alternative such as Spearman's ρ may be more successful in detecting a linear relationship.