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Bias of the variance

librosdeholanda

New Member
Subscriber
Is there anyone who could help me understand why the bias of the variance is sigma squared divided by n instead of the same but as negative (-sigma squared divided by n)?
I am talking about the following expression:

1613925254686.png
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @librosdeholanda You make a good point, strictly speaking the bias of the sample variance is negative:
(n-1)/n*σ^2 - σ^2
=(n-1)/n*σ^2 - σ^2*n/n
= σ^2*[(n-1)-n]/n
= σ^2*(-1)/n = -σ^2/n
... although we barely need the math: intuitively, the biased (MLE) variance divides by n, while the unbiased variance divides by (n-1) such that the biased variance necessarily smaller than the unbiased variance. I think that because it's bias, we're not overly concerned by the direction of the bias, looks to me like the explanation. Nevertheless, I have to agree with you that strictly it should be negative. Thanks,
 
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