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Hello List,
On October 7, 2005, I pointed out that the correct
formula for a set of test results determined in samples of variable weights
was implemented in several Excel templates posted under Documents on
ai-geostats.org. However, on October 20, 2005, a flawed formula
was accepted as the one most likely formula to give the least
biased or best unbiased variance estimate.
Please peruse "Variances variable weights" (also posted under Documents) in
which the correct formula is once again applied. The basic difference between
these formulae is that the term: (1/sum(wi^2))-1, in which sum(wi)=1
in the correct formula, unlike the term: 1/sum(wi^2) in the accepted formula, is
the number of degrees of freedom for a set of "n" samples of variable
weights. Evidently, the number of degrees of freedom in the correct
formula is no longer a positive integer but a positive irrational. I
introduced this formula in "Sampling in Mineral Processing", a
paper that was published in 2002 and is posted on my website.
Look at the variance estimates in each of the three columns, and note
that the least biased variance estimate in the first column is lower than
the correct variance estimate in the second column, which, in turn, is
lower than the variance estimate in the third column. Change the variable
weights of 1, 2, 3, 4 and 5 to the same weight and note that all
columns give the same variance estimates and the same central values. If
one were to worry about the veracity of Excel's VAR-function for the
variance of a randomly distributed or randomized set of test data with
identical weights, then the heuristic proof with constant weights indicates
that this is one Excel stat function that does
meet rigorous QC requirements.
The variance of the central value of any set of measured values (its
arithmetic mean or some weighted average) is obtained by multiplying var(x),
the variance of the set, with the term: sum(wi^2). Therefore,
var(xbar)=var(x)*sum(wi^2) applies to area-, count-, density-, DISTANCE-,
length-, mass- and volume-weighted averages. This formula converges on the
Central Limit Theorem when all of the weighing factors converge on
1/n.
Here are a few true facts that derive from mathematical
statistics:
1. Each and every distance-weighted average has
its own variance,
2. Distance-weighted averages became honorific
kriged estimates,
3. Each and every kriged estimate has its own
variance,
4. Variances and covariances of SETS of kriged
estimates are invalid.
All I want to know is which of these four facts is false and
why.
Kind regards,
Jan W Merks |
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