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    How does the 1st line (of the summation operator) become the 2nd one?
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    You are asking why n\left(\sum_i x_i^2 + \sum_i x_i^2\right) = n \sum_i 2x_i^2...

    ...because x_i^2+x_i^2 = 2x_i^2 for each i and multiplication distributes over addition. There isn't really any way to break it down other than what they wrote unless you want to start defining addition and multiplication of plain numbers.

    Quit using stupid terms like summation operator and just remember that you are dealing with a simple sum of terms.

    Write \sum_{i=1}^n x_i^2 = x_1^2 + x_2^2 + \cdots + x_n^2 if it helps.
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    (Original post by Mark85)
    Quit using stupid terms like summation operator
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    (Original post by Mark85)
    You are asking why n\left(\sum_i x_i^2 + \sum_i x_i^2\right) = n \sum_i 2x_i^2...

    ...because x_i^2+x_i^2 = 2x_i^2 for each i and multiplication distributes over addition. There isn't really any way to break it down other than what they wrote unless you want to start defining addition and multiplication of plain numbers.

    Quit using stupid terms like summation operator and just remember that you are dealing with a simple sum of terms.

    Write \sum_{i=1}^n x_i^2 = x_1^2 + x_2^2 + \cdots + x_n^2 if it helps.
    Posted the wrong question, sorry. I meant I don't get the last line-- how the final result is obtained from the penultimate.
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    (Original post by zomgleh)
    Posted the wrong question, sorry. I meant I don't get the last line-- how the final result is obtained from the penultimate.
    \frac{1}{n}\sum\limits_{i=1}^n x_i=\bar{x}
    Sub this in. It is similar to the proof of equivalence of 2 forms of the standard deviation formula.
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    Well, the mean is just

    \frac{1}{n}\sum_{i=1}^n x_i

    so

    \sum_i x^2_i - n\overline{x}^2 = \sum_i x^2_i - \left(\sum_i x_i\right)^2

    Can you see what to do from there? Have you even tried anything? The bit you quoted mentions a 'trick' - it isn't a trick in any sense, just a standard rearrangement but in any case, the reference to it suggests you will have seen it subsequently in the notes or books or whatever it is.

    If you prefer, write it out without the 'summation operator' as you grandiosely style it or even just write it out explicitly for n=3 or something so you see it.
 
 
 
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