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Math Help - variance and standard deviation

  1. #1
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    variance and standard deviation

    Why do we average by divivding by n-1 rather than n in calculating the variance? the formula I am using is s^2=[ sum of (x- X)^2]/(n-1) where X= mean. Thank you very much for your help!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by 0123 View Post
    Why do we average by divivding by n-1 rather than n in calculating the variance? the formula I am using is s^2=[ sum of (x- X)^2]/(n-1) where X= mean. Thank you very much for your help!
    Because you are using s^2 to estimate the population variance, and dividing
    by n results in a biased estimate while dividing by n-1 gives an unbiased
    estimator.

    RonL
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  3. #3
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    could you be more clear? because I do not understand why it is unbiased in the case on n-1. Thanks again.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by 0123 View Post
    could you be more clear? because I do not understand why it is unbiased in the case on n-1. Thanks again.
    Take my word for it, the proof is longer than I am prepared to reproduce here.

    I can illustrate it by example:

    Code:
    >x=normal(100,5);
    >m=sum(x)/5;         ..100 means of samples of size 5
    >v=sum(x^2)/5-m^2;   ..100 (1/n) variances of samples of size 5 
    >vv=sum(s')/100      .. average of 100 (1/n) variance of sample of size 5
             0.80403236 
    >vv*5/4              .. corrected to 1/(n-1) variances
             1.00504045 
    >
    Here we see the average variance of samples of size 5 from a population of
    variance 1 is ~0.8, while the corrected average variance is ~1, as required.

    RonL
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