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Quick simple question as I must have missed the explanation. why $$\sum_{j=1}^n (x_j - \bar{x}) = \sum_{j=1}^n x_j - n\bar{x} = (n\bar{x}- n\bar{x})$$

I understand why $\bar{x}$ turns out to be $n\bar{x}$ but i do not get how $x_j$ turns out to be $n\bar{x}$? This questions comes from trying to study the distribution of the MLE of $\beta$ in a single variate linear regression.

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  • $\begingroup$ How would you write the mean as a sum? $\endgroup$
    – Michael M
    Commented Apr 8, 2020 at 19:33
  • $\begingroup$ I have removed the generalized-linear-models tag. $\endgroup$
    – Gordon Smyth
    Commented Apr 13, 2020 at 0:26

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By definition,

$$ \bar x = \frac{\sum_{j=1}^n x_j}{n} $$

So multiplying through by $n$ yields the result that $\sum_{j=1}^n x_j = n\bar x$.

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