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Identity in statistics that frequently pops up

  1. Aug 29, 2015 #1
    1. The problem statement, all variables and given/known data
    In my statistics notes/lectures my professor will oftentimes use an identity that looks like the following:
    x_i is a non random variable, the summand is from i=1 to n;
    This segment comes from notes on linear regression (y_0 = b_0 + b_1*x_i)

    I actually forgot to mention that x-bar is supposed to be squared on the LHS, sorry about that.
    ∑(x_i)^2 - n*x-bar^2 = Σ(x_i - x-bar)^2
    However I just do not see how this works out at all!
    When I work it It turns out that x-bar = 1 but this doesn't make sense to me at all in the context.

    Does anyone have some insight they could provide me?

    ********EDIT*****
    After working through it a few times I actually solved the identity myself, however I am still curious if there's perhaps a more elegant way to power through the proof compared to the way I personally did below. Sorry for the confusion.


    2. Relevant equations


    3. The attempt at a solution
    Σ(x_i - x-bar)^2
    = Σ(x_i^2 - 2*x-bar*x_i + x-bar^2)

    = Σ(x_i^2) - 2*x-barΣ(x_i) + n*x-bar^2 = ∑(x_i)^2 - n*x-bar^2 (by the above equation in part 1)

    ⇒-2*x-barΣ(x_i) + n*x-bar^2 = - n*x-bar^2

    If you divide both sides by x-bar^2;
    -2Σ(x_i)/(x-bar) + n = -n
    = -2*n*x-bar/(x-bar) + n = -n
    = -2n + n = -n
    or - n = - n
     
    Last edited by a moderator: Aug 31, 2015
  2. jcsd
  3. Aug 30, 2015 #2

    Ray Vickson

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    Homework Helper

    Your method is the standard one, and is about as elegant and simple as you can get. However, you carried on way past the point you needed to. You were already done at line (3) [the one with the statement "by the above equation in part 1"]. At that point you have obtained exactly what you started out wanting to prove, so what possible use would any of the remaining material (in lines 4--9) be to you?
     
    Last edited by a moderator: Aug 31, 2015
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