What does sum of the product of the deviations mean?

Answer 1

Suppose you conduct an experiment that yields a collection of pairs, (x1, y1), (x2, y2), (x3, y3), ... (xn, yn). In other words, yi is the value of variable y when variable x assumes the value xi.

Let us call the average of the xi values x-bar and the average of the yi values y-bar. Then an x-deviation is xi - x-bar and a y-deviation is yi - y-bar. One product of a pair of these deviations is ( xi - x-bar )( yi - y-bar ).

If you now sum these deviations with the i going from 1 to n you will have the 'sum of the product of the deviations'.