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Math Help - Prove the sample variance formula.

  1. #1
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    Prove the sample variance formula.

    Basically I have to prove that the sample variance of a sample is given by:

    s^2 =  {\sum_{j=1}^nx_j^2-{1\over n}}(\sum_{j=1}^nx_j)^2\over n-1}

    For my purposes it's sufficient to show that {\sum_{j=1}^n(x_j-x_j^2) = {\sum_{j=1}^nx_j^2-{1\over n}}(\sum_{j=1}^nx_j)^2

    Help is appreciated. Thanks in advance.

    If I made any mistakes inputting the sums please let me know.
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  2. #2
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    Quote Originally Posted by Iceflash234 View Post
    Basically I have to prove that the sample variance of a sample is given by:

    s^2 =  {\sum_{j=1}^nx_j^2-{1\over n}}(\sum_{j=1}^nx_j)^2\over n-1}

    For my purposes it's sufficient to show that {\sum_{j=1}^n(x_j-x_j^2) = {\sum_{j=1}^nx_j^2-{1\over n}}(\sum_{j=1}^nx_j)^2

    Help is appreciated. Thanks in advance.

    If I made any mistakes inputting the sums please let me know.
    This is a standard proof found in most textbooks on mathematical statistics. Have you consulted any?
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    This is a standard proof found in most textbooks on mathematical statistics. Have you consulted any?
    I checked all the ones I have access to. I even tried googling it.
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  4. #4
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    Quote Originally Posted by Iceflash234 View Post
    Basically I have to prove that the sample variance of a sample is given by:

    s^2 =  {\sum_{j=1}^nx_j^2-{1\over n}}(\sum_{j=1}^nx_j)^2\over n-1}

    For my purposes it's sufficient to show that {\sum_{j=1}^n(x_j-x_j^2) = {\sum_{j=1}^nx_j^2-{1\over n}}(\sum_{j=1}^nx_j)^2

    Help is appreciated. Thanks in advance.

    If I made any mistakes inputting the sums please let me know.
    Your starting point should be the definition of sample variance: Sample Variance -- from Wolfram MathWorld

    What progress have you made from this?
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