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Tricky Summation

  1. Aug 16, 2010 #1
    1. The problem statement, all variables and given/known data

    I have this nasty summation and I am close to finding a way to calculate it with my graphing calculator. I just need to iron out the details. If I can rewrite the summation on terms of [itex]\bar{x}[/itex], [itex]\bar{y}[/itex] and [itex]\sum x_iy_i[/itex] I will be all set. I will explain these terms in a moment. First, here is the sum:

    [tex]S_{xy} = \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})[/tex]

    Where [itex]x_i[/itex] and [itex]y_i[/itex] are the x and y (sample) averages.

    If I do out the multiplication I get


    [tex]S_{xy} =
    \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})[/tex]

    [tex] = \sum x_iy_i - x_i\bar{y} - y_i\bar{x} + \bar{x}\bar{y}[/tex]

    [tex] = \sum x_iy_i - \bar{y}\sum x_i - \bar{x}\sum y_i +\sum\bar{x}\bar{y}[/tex]

    Now I it is just the last term [itex]\sum\bar{x}\bar{y}[/itex] that is bothering me. I think the answer is obvious, but I would like confirmation: is the expression [itex]\sum\bar{x}\bar{y}[/itex] identical to [itex]n*(\bar{x}\bar{y})[/itex] ?


    It must be. I don't know why I am doubting this. :redface:
     
  2. jcsd
  3. Aug 16, 2010 #2

    rock.freak667

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    Yes

    [tex]\sum\bar{x}\bar{y} = n\bar{x}\bar{y}[/tex]

    Also remember that

    [tex]\sum x_i = n \bar{x}[/tex]

    [tex]\sum y_i = n \bar{y}[/tex]

    So what does

    [tex]- \bar{y}\sum x_i - \bar{x}\sum y_i +\sum\bar{x}\bar{y}[/tex]

    become?
     
  4. Aug 16, 2010 #3
    Oh neat. Looks like it reduces to [itex]-n\bar{x}\bar{y}[/itex]

    Yes?
     
  5. Aug 16, 2010 #4

    rock.freak667

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    Yes

    Which can also be written as (1/n)∑xi∑yi
     
  6. Aug 16, 2010 #5
    I like this. If I could write it all on terms of xbar, ybar, and n, that would be best. I think that the sum(xi*yi) term can be written in terms of these. I'll post back I'm a moment. Thanks again!!

    Hmmm... I am not seeing an easy way to write [itex]\sum x_iy_i[/itex] in terms of [itex]\bar{x}[/itex], [itex]\bar{x}[/itex], and n. Perhaps there isn't one?
     
    Last edited: Aug 16, 2010
  7. Aug 17, 2010 #6

    statdad

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    "I am not seeing an easy way to write in terms of [itex] \bar x[/itex], [itex] \bar y [/itex] , and n. Perhaps there isn't one?"

    No, there is not.
     
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