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Math Help - correlation and r square

  1. #1
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    correlation and r square

    Hi all

    I have two questions relating to correlation:

    Firstly where does the first 'n' in the numerator come from in the far right hand expression of the equation below



    Secondly, I am trying to work out the derivation of correlation^2 = r^2. I have started with squaring the above expression and trying to simplify, but I don't feel like I'm going about it the right way. Could someone give me a shove in the right direction please?
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  2. #2
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    Quote Originally Posted by factfinder View Post
    Hi all

    I have two questions relating to correlation:

    Firstly where does the first 'n' in the numerator come from in the far right hand expression of the equation below



    Secondly, I am trying to work out the derivation of correlation^2 = r^2. I have started with squaring the above expression and trying to simplify, but I don't feel like I'm going about it the right way. Could someone give me a shove in the right direction please?
    \sum_{i=1}^n \left(X_i - \overline{X}\right) \left(Y_i - \overline{Y}\right) = \sum_{i=1}^n \left( X_i Y_i - \overline{Y} X_i - \overline{X} Y_i + \overline{X} ~ \overline{Y}\right)


    = \sum_{i=1}^n X_i Y_i - \overline{Y} \sum_{i=1}^n X_i - \overline{X} \sum_{i=1}^n Y_i + \sum_{i=1}^n \overline{X} ~ \overline{Y}


    = \sum_{i=1}^n X_i Y_i - \overline{Y} (n \overline{X}) - \overline{X} (n \overline{Y}) + n \overline{X} ~ \overline{Y}


    = \sum_{i=1}^n X_i Y_i - n \overline{X} ~ \overline{Y}.
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  3. #3
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    Hi Mr Fantastic

    sorry I was looking for how to get from this:

    <br />
\frac{\sum x_i y_i - n \overline{x} ~ \overline{y}}{(n-1) s_x s_y}<br />

    to this:

    <br />
\frac{n \sum x_i y_i - \sum x_i \sum y_i}{\sqrt{n \sum x_i^2 - (\sum x_i)^2}\sqrt{n \sum y_i^2 - (\sum y_i)^2}}<br />

    (with the first equation I have the same thing in a textbook but completely omitting the (n-1) term in the denominator, so I'm not sure how that is so for starters...)
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  4. #4
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    Quote Originally Posted by factfinder View Post
    Hi Mr Fantastic

    sorry I was looking for how to get from this:

    <br />
\frac{\sum x_i y_i - n \overline{x} ~ \overline{y}}{(n-1) s_x s_y}<br />

    to this:

    <br />
\frac{n \sum x_i y_i - \sum x_i \sum y_i}{\sqrt{n \sum x_i^2 - (\sum x_i)^2}\sqrt{n \sum y_i^2 - (\sum y_i)^2}}<br />

    (with the first equation I have the same thing in a textbook but completely omitting the (n-1) term in the denominator, so I'm not sure how that is so for starters...)
    n \overline{x} ~ \overline{y} = n \, \frac{\sum x_i}{n} \, \frac{\sum y_i}{n} = \frac{\sum x_i ~  \sum y_i}{n}

    so multiply the numerator and denominator by n.
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  5. #5
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    Thanks Mr Fantastic.

    Am I right in saying that the (n-1) term in the original denominator is discarded then?
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  6. #6
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    Quote Originally Posted by factfinder View Post
    Thanks Mr Fantastic.

    Am I right in saying that the (n-1) term in the original denominator is discarded then?
    No. The formulae for s_x and s_y have been substituted and simplification done.
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