Sigma Notation

  • Thread starter bob1182006
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  • #1
492
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Homework Statement


Show:
[tex]\sum_{i=1}^n (x_i - \overline{x}) = 0[/tex]


Homework Equations


Sigma notation


The Attempt at a Solution



[tex]\sum_{i=1}^n x_i - \sum_{i=1}^n \overline{x} = \sum_{i=1}^n x_i - \frac{1}{n}\sum_{i=1}^n \sum_{i=1}^n x_i = 0[/tex]
[tex]\sum_{i=1}^n x_i = \frac{1}{n}\sum_{i=1}^n \sum_{i=1}^n x_i [/tex]

By Inspection I know i need to show that:
[tex]\sum_{i=1}^n \frac{1}{n}=1[/tex]

Since the LHS has no [itex]x_i[/itex] how can i show that the sum will result in n/n =1?

Is it just:
[tex]\sum_{i=1}^n 1 = n?[/tex]
 

Answers and Replies

  • #2
consider that [tex]\overline{x}[/tex] is idependent of [tex]i[/tex] therefore [tex]\sum_{i=1}^n\overline{x}[/tex] simply equals [tex]n \overline{x}[/tex]. Also IMPORTANT NOTE: In the second line of your attempted solution you use the fact that [tex]\sum_{i=1}^n (x_i - \overline{x}) = 0[/tex] how ever that's what you're trying to SHOW so you can't assume it's true. You have to work entirely on the LHS and get it equal to 0.
 
  • #3
492
1
Ah, I get it now thanks alot!

So on the first step it'd just be the n*xbar/n leaving only sum of x - sum of x which =0.

Thank You!
 
  • #4
Err no it'd be
[tex]\sum_{i=1}^n (x_i - \overline{x}) = 0[/tex]
[tex]\sum_{i=1}^n x_i - n\overline{x} = 0[/tex]
[tex]n \sum_{i=1}^n \frac{x_i}{n} - n\overline{x} = 0[/tex]

[tex]n\overline{x} - n\overline{x} = 0[/tex]

which gives you zero.... hope he comes back to read that.
 
  • #5
492
1
ah okay, I was expanding the xbar so there would be 2 equal sums being subrated which would just =0.

[tex]\sum_{i=1}^n x_i - n \frac{1}{n} \sum_{j=1}^n x_j = 0[/tex]

since n/n =1, and both sums start and end at the same place they are both equal and thus will give 0 as the answer.
 

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