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Proof that Sample Variance is Unbiased

# Proof that Sample Variance is Unbiased - Proof that Sample...

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Proof that Sample Variance is Unbiased Plus Lots of Other Cool Stuff Scott D. Anderson http://www.spelman.edu/~anderson/teaching/437/unbiased/unbiased.html Fall 1999 Expected Value of S 2 The following is a proof that the formula for the sample variance, S 2 , is unbiased. Recall that it seemed like we should divide by n , but instead we divide by n -1. Here's why. First, recall the formula for the sample variance: 1 ) ( ) var( 2 1 2 = = = n x x S x n i i Now, we want to compute the expected value of this: [] = = 1 ) ( 2 1 2 n x x E S E n i i [] = = 2 1 2 ) ( 1 1 n i i x x E n S E Now, let's multiply both sides of the equation by n -1, just so we don't have to keep carrying that around, and square out the right side, just like we did with that shortcut formula for SSX, above. [] + = = n i i i x x x x E S E n 1 2 2 2 2 ) 1 (

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Proof that Sample Variance is Unbiased - Proof that Sample...

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