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Statistics



                      Notes
                                                                           
                                    On  the basis  of  the  above, we  can consider   X   X  Y  Y    as  an  absolute measure  of
                                                                               i
                                                                                     i
                                    correlation. This measure, like other absolute measures of dispersion, skewness, etc., will depend
                                    upon (i) the number of observations and (ii) the units of measurements of the variables.
                                    In order to avoid its dependence  on the  number of observations, we  take its  average,  i.e.,
                                     1   X   X  Y   Y  . This term is called covariance in statistics and is denoted as Cov(X,Y).
                                     n    i     i

                                    To eliminate the effect of units of measurement of the variables, the covariance term is divided
                                    by the product of the standard deviation of X and the standard deviation of  Y. The resulting
                                    expression is known as the Karl Pearson's coefficient of linear correlation or the product moment
                                    correlation coefficient or simply the coefficient of correlation, between X and Y.
                                                    Cov X,Y 
                                                r XY                                        .... (1)
                                                        Y
                                                       X

                                                   1
                                                      X   X  Y   Y 
                                                              i
                                                        i
                                    or    r      n                                          .... (2)
                                           XY
                                                1         2  1       2
                                                   X   X     Y   Y 
                                                n    i      n    i
                                              1
                                    Cancelling    from the numerator and the denominator, we get
                                              n
                                                         X   X  Y   Y 
                                                r         i     i                           .... (3)
                                                 XY           2         2
                                                       X   X    Y   Y 
                                                                    i
                                                         i
                                             
                                    Consider   X  X  Y   Y      X   X  Y   Y   X   X 
                                                                      i
                                                                              i
                                                                 i
                                                       i
                                                 i
                                                             X Y  X  Y i  (second term is zero)
                                                                i
                                                                  i
                                                             X Y  nXY   Y   nY 
                                                                             i
                                                                i
                                                                  i
                                                                2     2    2
                                                        
                                    Similarly we can write    X   X     X   nX
                                                                      i
                                                            i
                                                                  2     2    2
                                                          
                                                and              Y   Y    Y  nY
                                                                        i
                                                              i
                                    Substituting these values in equation (3), we have
                                                            X Y   nXY
                                                                i
                                                              i
                                                r XY                                        .... (4)
                                                          2
                                                                          2
                                                              2
                                                                     2
                                                       X   nX    Y   nY  
                                                                
                                                          i
                                                                     i
                                                                    X i  Y
                                                           X Y          i
                                                                 n
                                                             i
                                                               i
                                                r XY                n    n
                                                              X   2        Y   2
                                                                        2
                                                         2
                                                      X      i    Y     i   
                                                           n
                                                                          n
                                                                  
                                                         i
                                                                        i
                                                              n             n  
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