3.4.2.1 - Formulas for Computing Pearson's r

There are a number of different versions of the formula for computing Pearson's $$r$$. You should get the same correlation value regardless of which formula you use. Note that you will not have to compute Pearson's $$r$$ by hand in this course. These formulas are presented here to help you understand what the value means. You should always be using technology to compute this value.

First, we'll look at the conceptual formula which uses $$z$$ scores. To use this formula we would first compute the  $$z$$ score for every $$x$$ and $$y$$ value. We would multiply each case's $$z_x$$ by their $$z_y$$.  If their  $$x$$ and  $$y$$ values were both above the mean then this product would be positive. If their x and y values were both below the mean this product would be positive. If one value was above the mean and the other was below the mean this product would be negative. Think of how this relates to the correlation being positive or negative. The sum of all of these products is divided by $$n-1$$ to obtain the correlation.

Pearson's r: Conceptual Formula

$$r=\dfrac{\sum{z_x z_y}}{n-1}$$
where $$z_x=\dfrac{x - \overline{x}}{s_x}$$ and $$z_y=\dfrac{y - \overline{y}}{s_y}$$

When we replace $$z_x$$ and $$z_y$$ with the $$z$$ score formulas and move the $$n-1$$ to a separate fraction we get the formula in your textbook: $$r=\frac{1}{n-1}\Sigma{\left(\frac{x-\overline x}{s_x}\right) \left( \frac{y-\overline y}{s_y}\right)}$$