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Date:         Tue, 29 Oct 2002 09:12:21 -0500
Reply-To:     Richard Ristow <wrristow@mindspring.com>
Sender:       "SPSSX(r) Discussion" <SPSSX-L@LISTSERV.UGA.EDU>
From:         Richard Ristow <wrristow@mindspring.com>
Subject:      Re: Some formulae needed
Comments: To: "Burleson,Joseph A." <burleson@up.uchc.edu>
In-Reply-To:  <7D25CCC35D3CD31192BF006008BFB1FE04F572C4@nsofs14.uchc.edu>
Content-Type: text/plain; charset="us-ascii"; format=flowed

At 03:27 PM 10/24/2002 -0400, Burleson,Joseph A. wrote:

>Now can anyone tell us why the harmonic [mean] is used instead of the >geometric? The harmonic clearly penalizes discrepancy more, e.g.: > >Overall N = 100, n1 = 30, n2 = 70 >Arithmetic n-bar = 50 >Geometric mean n = 45.8 >Harmonic n' = 42

That's why. Here's how it looks with a constant n1=10 as n2 grows larger:

N1: 10 N2 Total N Mean Geo. Harmonic ----------------------------------- 10 20 10 10.0 10.0 40 50 25 20.0 16.0 90 100 50 30.0 18.0 490 500 250 70.0 19.6 990 1000 500 99.5 19.8

Intuitively, you have two sources of random uncertainty: the mean of the larger group, and the mean of the smaller one. Those two uncertainties are equally important for the comparison. As the discrepancy in sample size gets larger, the uncertainty (the standard error of estimate) of the mean of the smaller group becomes much larger, and dominates the overall uncertainty.

Both the arithmetic mean (the "mean") and the geometric mean would have the effective total N grow indefinitely as N2 increases. The harmonic mean implies, correctly, that no matter how large N2 is, effective N is limited by N1, because the uncertainty in that group mean never diminishes.


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