= a score on a random variable*Y**Y*= corresponding predicted value of*Y'**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 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*:

*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.