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Pearson product-moment correlation coefficient

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:

$r = \frac {\sum z_x z_y}{N - 1}$

The result obtained is equivalent to dividing the covariance between the two variables by the product of their standard deviations. In general the correlation coefficient is one of the two square roots (either positive or negative) of the coefficient of determination (r2), which is the ratio of explained variation to total variation:

$r^2 = {\sum (Y' - \overline Y)^2 \over \sum (Y - \overline Y)^2}$

where:

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
$\overline Y$ = sample mean of Y (i.e., the mean of a finite number of independent observed realizations of Y, not to be confused with the expected value of Y)

The correlation coefficient adds a sign to show the direction of the relationship. The formula for the Pearson coefficient conforms to this definition, and applies when the relationship is linear.

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 Pearson coefficient is a statistic which estimates the correlation of the two given random 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′:

$Y = Y^\prime + (Y - Y^\prime)$

The variance of Y is equal to the sum of the variance of the two components of Y:

$s_y^2 = S_{y^\prime}^2 + s^2_{y.x}$

Since the coefficient of determination implies that sy.x2 = sy2(1 − r2) we can derive the identity

$r^2 = {s_{y^\prime}^2 \over s_y^2}$

The square of r is conventionally used as a measure of the strength of the association between X and Y. For example, if the coefficient is 0.90, then 81% of the variance of Y is said to be explained by the changes in X and the linear relation between X 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.