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The computational formula for Pearson's $$r$$ defines it as the mean cross-product of $$z$$ scores. To calculate it, each score for both variables is first transformed into a $$z$$ score. Then, each individual's two $$z$$ scores are multiplied to create a cross-product. The mean of all these cross-products represents Pearson's $$r$$, expressed by the formula: $$r = \frac {\sum(z_xz_y)}{N}$$.

Pearson's r Formula

To compute Pearson's $$r$$ manually, start by transforming the raw scores of two quantitative variables into $$z$$ scores. For example, consider an $$X$$ variable with a mean of $$4.00$$ and a standard deviation of $$1.90$$, and a $$Y$$ variable with a mean of $$40.00$$ and a standard deviation of $$11.78$$. If an individual's $$X$$ score is $$4$$ (a $$z$$ score of $$0.00$$) and their $$Y$$ score is $$30$$ (a $$z$$ score of $$-0.85$$), their cross-product is $$0.00$$. Another individual might have an $$X$$ score of $$7$$ ($$z = 1.58$$) and a $$Y$$ score of $$54$$ ($$z = 1.19$$), resulting in a cross-product of $$1.88$$. By taking the mean of these cross-products for the entire sample, we arrive at the value for Pearson's $$r$$. In this example, the mean of the cross-products is $$+0.53$$, indicating a positive relationship.

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