Pearson Correlation Coefficient: How Did He Derive It?

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The Pearson Correlation Coefficient, developed by Karl Pearson, quantifies the linear relationship between two variables. The formula incorporates the discrete independent variable (x), the discrete dependent variable (y), and their respective standard deviations (Sx and Sy). This statistical measure is essential for understanding the strength and direction of the relationship between paired data sets. The discussion emphasizes the derivation of this coefficient as a formal definition rather than a mere conceptual tool.

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Now how exactly did Pearson derive his correlation coefficient?
 
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Is it simply a definition which reflects the linear relationship between two variables?
 
gimmytang said:
Is it simply a definition which reflects the linear relationship between two variables?

That's what it's supposed to do; I'm just wondering how did Pearson derived or came up with the formula for it, which I have in the attached GIF image file, (correlation.gif)

Where x=discrete independent variable, y=discrete dependent variable,
and Sx=standard deviation of x-set, and Sy=standard deviation of y-set
 

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