Show that the correlation coeffcient of x i − x ¯ and x j − x ¯ is − ( n − 1 ) − 1

Question

Show that the correlation coeffcient of       x          i        −      x    ¯   and       x          j        −      x    ¯   is   −  (  n  −  1      )          −      1

Salma Bradley · Accepted Answer

Step 1                                    x          1                −                                            x              1                        +                          x              2                        +                          x              3                        +            ⋯            +                          x              n                                n                                    =                                              (          1          −                      1            n                    )                          x          1                +                                            x              2                        +                          x              3                        +            ⋯            +                          x              n                                n                                    So                     var        ⁡                  (                      x            1                    −                                                    x                1                            +                              x                2                            +                              x                3                            +              ⋯              +                              x                n                                      n                    )                                    =                                                          (            1            −                          1              n                        )                    2                var        ⁡        (                  x          1                )        +                  1                      n            2                          var        ⁡        (                  x          2                )        +        ⋯        +                  1                      n            2                          var        ⁡        (                  x          n                )            and all of these variances are the same number.                          cov        ⁡                  (                      x            1                    −                                                    x                1                            +              ⋯              +                              x                n                                      n                    ,                                x            2                    −                                                    x                1                            +              ⋯              +                              x                n                                      n                    )                                    =                                                          cov            ⁡            (                          x              1                        ,                          x              2                        )                                0                          −        cov        ⁡                  (                      x            1                    ,                                                    x                1                            +              ⋯              +                              x                n                                      n                    )                                                                  −        cov        ⁡                  (                      x            2                    ,                                                    x                1                            +              ⋯              +                              x                n                                      n                    )                +        var        ⁡                  (                                                    x                1                            +              ⋯              +                              x                n                                      n                    )                    And you seem to know what to do from there.Another way is to recall that      (          x      1        −                            x          1                +        ⋯        +                  x          n                    n        )    +  ⋯  +      (          x      n        −                            x          1                +        ⋯        +                  x          n                    n        )    =  0and so the variance of the above must be 0. Therefore                    0        =                                                      ∑                      k                        =                        1                    n                var        ⁡                  (                      x            k                    −                                                    x                1                            +              ⋯              +                              x                n                                      n                    )                                                                  −        2                  ∑                      i            ,            j                        :                        i                        &amp;lt;                        j                          cov        ⁡                  (                      x            i                    −                                                    x                1                            +              ⋯              +                              x                n                                      n                    ,                                x            j                    −                                                    x                j                            −              ⋯              +                              x                n                                      n                    )                                    =                                    n        var        ⁡                  (                      x            k                    −                                                    x                1                            +              ⋯              +                              x                n                                      n                    )                                                                  −        2                                            (                                            n            2                                              )                                      cov        ⁡                  (                      x            1                    −                                                    x                1                            +              ⋯              +                              x                n                                      n                    ,                                x            2                    −                                                    x                j                            −              ⋯              +                              x                n                                      n                    )                    and then from the value of that variance deduce the value of the covariances.

Show that the correlation coeffcient of x <mrow class="MJX-TeXAtom-ORD"> i

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