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Math Help - simplifying the correlation coefficient

  1. #1
    Junior Member
    Joined
    Jan 2009
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    57

    simplifying the correlation coefficient

    Hi,

    I am trying to understand the following simplification.
    I cannot see how to get from the one step to the other.

    <br />
\sum (x_i - \bar{x})(y_i - \bar{y})<br />

    we get

    <br />
\sum (x_i y_i - y_i \bar{x} - x_i \bar{y} + \bar{x}\bar{y})<br />

    and then (how do they get to this step?)

    <br />
\sum (x y) -n\bar{x}\bar{y} -n\bar{x}\bar{y} +n\bar{x}\bar{y}<br />

    and then (very obviously)
    <br />
\sum (x y) -n\bar{x}\bar{y} <br />

    It can be seen on wolfram mathworld:
    Correlation Coefficient -- from Wolfram MathWorld

    I get the following.
    <br />
\sum (x_i y_i) + n\bar{x}\bar{y} -n(\bar{x}\sum y_i + \bar{y}\sum x_i)<br />

    How does the simplification from step 2 to step 3 happen.
    My simplification is not the same, and I don't see how to make it simpler.

    Thanks
    Regards
    Craig
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  2. #2
    Super Member
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    \displaystyle \overline{x}=\frac{\sum{x}}{n}, so \displaystyle n\overline{x}=\sum{x}.
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