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Explanation of Logic Behind the Sum of Products

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Explanation on what is the logic behind using the sum of the products of deviation scores as the numerator of the formula for the correlation coefficient? Is this logic sound? Why or why not?

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A correlation is a number between -1 and +1 that measures the degree of association between two variables (call them X and Y). A positive value for the correlation implies a positive association (large values of X tend to be associated with large values of Y and small values of X tend to be associated with small values of Y). A negative value for the correlation implies a negative or inverse association (large values of X tend to be associated with small values of Y and vice versa).Pearson's correlation reflects the degree of linear relationship between two variables.
The formula for the Pearson correlation
Suppose we have two variables X and Y, with means XBAR and YBAR respectively and standard ...

Solution Summary

The solution provides an explanation of logic behind the sum of products. Psychological statistics is examined.

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I need as much information as possible. Examples would be helpful. Thank you.

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