Re: Calculating the Standard Deviation of a Weighted List

From: Ray Koopman (koopman_at_sfu.ca)
Date: 11/05/04 

Date: 5 Nov 2004 14:20:44 -0800

John wrote:
> When calculating the standard deviation of a weighted list, should
> one normalize the weights such that the sum of the weights = 1, or
> such that the sum of the weights = the number of data points proper?
>
> The latter seems to be the only way by which the weighted standard
> deviation, when the weight variable has the same value for every data
> point, equals the non-weighted standard deviation.
>
> Yet nearly everything I find on the internet seems to suggest the
> former.
>
> BTW, I'm not a statistician by any stretch so please forgive any
> ignorance.
>
> TIA,
> John

Normalizing the weights is not necessary.
The weighted mean is

m = Sum Wi*Xi / Sum Wi.

The weighted variance that corresponds to using N
in the denominator of the unweighted variance is

Sum Wi(Xi - m)^2 / Sum Wi.

The weighted variance that corresponds to using N-1
in the denominator of the unweighted variance is
Sum Wi(Xi - m)^2 / (Sum Wi - (Sum Wi^2 / Sum Wi)).

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