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.