The result obtained is equivalent to dividing the covariance between the two variables by the product of their standard deviations. In general the quantity of a correlation coefficient[?] is the square root of the coefficient of determination[?] (r^{2}), which is the ratio of explained variation to total variation:
where:
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:
The variance of Y is equal to the sum of the variance of the two components of Y:
Since the coefficient of determination implies that s_{y.x}^{2} = s_{y}^{2}(1  r^{2}) we can derive the identity
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 .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 nonparametric alternative such as Spearman's ρ may be more successful in detecting a linear relationship.
Search Encyclopedia

Featured Article
