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Hello List
Thanks to all who responded to my recent request
for comments regarding an unbiased estimator of variance given N sample
values with associated weights. And a special thanks to Colin
Daly who took the time to write out a very nice mathematical
demonstration of an unbiased estimator. I have seen a number
of different equations for calculating the variance given a set of
sample values and associated weights and have often wondered which is the
"best" one. By "best, I mean the estimator which minimizes bias, or better
yet, is unbiased.
To summarize, an unbiased estimate of the
variance calculated from a set of samples with associated weights can be
obtained through the following equation:
s2 = Sum(w_i) / (Sum(w_i) - Sum(w_i * w_i)) * Sum(
w_i * (x_i - xbar) * (x_i - xbar))
where xbar = Sum(w_i * x_i) / Sum(w_i), i =
1,N.
If you are using an equation which provides
estimates different from those provided by the equation above, then
your estimate of variance is biased. :-)
Note, that the weights need not sum to 1.0.
However, if your weights do sum to 1.0, then the equation above simplifies
to:
s2 = 1 / (1 - Sum(w_i * w_i)) * Sum( w_i * (x_i -
xbar) * (x_i - xbar))
where xbar = Sum(w_i * x_i), i
=1,N.
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