Question
Download Solution PDFIf x = X- \(\bar{X}\) and y = Y - \(\bar{Y}\) and the number of pairs (X, Y) is n, then the Karl Pearson's coefficient of correlation is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCalculation
Here, correlation coefficient = r
⇒ r = Cov(x,y)/SxSy
Here, Cov(x,y) = ∑(X - Y)(Y - Y)/(n - 1)
⇒ Sx = √∑(X - X̅) 2/(n - 1)
⇒ Sy = √∑(Y - Y̅ )2/(n - 1)
⇒ Cov(x,y)/SxSy = ∑(X - X̅)(Y - Y̅)/(n - 1)/√(∑(X - X̅)2)√(∑(Y - Y)2)/(n - 1)
⇒ Cov(x,y)/SxSy = ∑(X - X̅ )(Y - Y̅ )/√(∑(X - X̅ )2)√(∑(Y - Y̅ )2)
⇒ Cov(x,y)/SxSy = ∑xy/√∑x2 × √∑y2 Here, x = (X - \(̅{X}\)) and y = (Y - \(̅{Y}\))
∴ The Karl Pearson's coefficient of correlation is ∑xy/√[∑x2 × ∑y2]
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