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# sum2 - Summation and Correlation The correlation coecient r...

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Summation and Correlation The correlation coefficient r can be written either 1 n n summationdisplay i =1 parenleftBigg ( x i - ¯ x ) SD x × ( y i - ¯ y ) SD y parenrightBigg or parenleftBig 1 n n i =1 x i y i parenrightBig - ¯ x ¯ y SD x × SD y The proof is as follows: 1 n n summationdisplay i =1 parenleftBigg ( x i - ¯ x ) SD x × ( y i - ¯ y ) SD y parenrightBigg = 1 n × SD x × SD y parenleftBigg n summationdisplay i =1 x i y i - x i ¯ y - ¯ xy i + ¯ x ¯ y parenrightBigg = 1 n × SD x × SD y parenleftBigg n summationdisplay i =1 x i y i - ¯ y n summationdisplay i =1 x i - ¯ x n summationdisplay i =1 y i + n summationdisplay i =1 ¯ x ¯ y parenrightBigg = 1 SD x × SD y parenleftBigg n i =1 x i y i n - ¯ y n i =1 x i n + - ¯ x n i =1 y i n + 1 n n summationdisplay i =1 ¯ x ¯ y parenrightBigg = parenleftBig 1 n n i =1 x i y i parenrightBig - ¯ y ¯ x - ¯ x ¯ y + ¯ x ¯ y SD x × SD y = parenleftBig 1 n n i =1 x i y i parenrightBig - ¯ x ¯ y SD x × SD y The first formula is like that in the text and is probably easier to understand intuitively.
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