Calculating the Pearson's Product-Moment Correlation Coefficient (r)

So if we multiply your Z scores together, sum all these pairs and we get a positive sum, we have a positive correlation.

If you have a negative relationship you will have the sum of positive Zx's times negative Zy's and vice versa. Add these up for a negative sum.

If you have no correlation, then you get equal numbers of positive Zx times negative Zy, positive Zx times positive Zy, negative Zx times negative Zy, and negative Zx times positive Zy. Add these up and you get zero.