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Alternative formula for variance problem

  1. Jan 31, 2013 #1
    1. The problem statement, all variables and given/known data
    Which of the following formulas represents the variance of the data set {x1,x2,x3,...,x10}?
    (μ denotes the mean of the data set)

    Here is a photo that I took of the problem for better understanding.

    http://i.imgur.com/PqEKajx.jpg?1?7734

    I understand why the answer I chose is wrong.
    What I don't understand is how e) is the answer. I did the calculations by hand with that formula and it is correct.

    Would someone please show me how that formula is derived from the variance formula we usually see:
    v2.GIF


    2. Relevant equations

    Formula for variance

    3. The attempt at a solution
     
  2. jcsd
  3. Jan 31, 2013 #2

    SammyS

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    attachment.php?attachmentid=55218&stc=1&d=1359614696.gif

    In your case this becomes
    [itex]\displaystyle \sigma^2=\frac{\sum_{i=1}^{10}(x_i-\mu)^2}{10}[/itex]​
    Expand the square.

    [itex]\displaystyle (x_i-\mu)^2=x_i^2-2x_i\mu+\mu^2[/itex]

    Now, take the sum of each term. Then recall how you calculate μ .
     

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  4. Jan 31, 2013 #3

    I get [itex]\frac{{x_1 + x_2 + x_3....+x_10} + { 2x_1μ + 2x_2μ...+2x_10μ} + 10μ^2}{10}[/itex]

    I can see where [itex]\displaystyle \sigma^2=\frac{\sum_{i=1}^{10}(x_i)^2}{10}[/itex] comes from and that 10μ^2 cancels out.

    What about the [itex]{2x_1\mu + 2x_2μ...+2x__10\mu}[/itex] ?


    p.s. Sorry, having a hard time with Latex. I hope you understand what I mean! :O
     
    Last edited: Jan 31, 2013
  5. Jan 31, 2013 #4

    SammyS

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    Divide that by N, i.e. 10 .

    That's [itex]\displaystyle \ \ 2\mu\frac{\sum_{i=1}^{10}x_i}{10}[/itex]
     
  6. Jan 31, 2013 #5

    Oh, stupid me. >.>
    Thanks for your help!

    I understand now:
    http://i.imgur.com/NpdcN86.jpg
     
  7. Jan 31, 2013 #6

    Ray Vickson

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    You should not get [itex]\frac{{x_1 + x_2 + x_3....+x_10} + { 2x_1μ + 2x_2μ...+2x_10μ} + 10μ^2}{10}[/itex], which is wrong. You should not get [itex]\displaystyle \sigma^2=\frac{\sum_{i=1}^{10}(x_i)^2}{10}[/itex] because that is also wrong unless μ = 0. Start over, and proceed carefully!
     
  8. Jan 31, 2013 #7
    Sorry, I was a bit lazy typing out my work. But in the post before this I solved it!
    http://i.imgur.com/NpdcN86.jpg
    Thanks anyways. :P
     
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