From: cmg@ferrari.cray.com (Charles Grassl)
Subject: Re: Geometric Mean or Median
Reply-To: cmg@magnet.cray.com
Organization: Cray Research, inc.
Distribution: comp.benchmarks
Date: 14 Aug 92 15:12:45 CDT

In article <1992Aug14.155857.6561@riacs.edu>, lamaster@pioneer.arc.nasa.gov (Hugh LaMaster) writes:
>In article <Aug14.142126.38458@yuma.ACNS.ColoState.EDU>, shafer@CS.ColoState.EDU (spencer shafer) writes:
>|> 
>|>                               A discussion of this, and an offered proof
>|> of the geometric mean as preferred method is in the March 1986 issue of 
>|> Communications of the ACM, "How Not to Lie With Statistics: The Correct
>|> Way to Summarize Benchmark Results," by Fleming and Wallace.
>
>Yes, and there was a rebuttal to this "proof" in CACM by, I believe,
>J.E. Smith, in October of 1988.  {If I have the reference correct,}
>it is proved that the harmonic mean is the correct measure of rates, 
>if you want to examine a fixed workload and characterize the performance
>on that workload.

The references are below:

[FL,WA]  Fleming, P.J., Wallace, J.J,  "How Not to Lie With Statistics:
         The Correct Way to Summarize Benchmark results",
         Communications of the ACM, P. 218-221, March, 1986, Volume 29,
         no. 3.

[SM]     Smith, J.E., "Characterizing Computer Performance With a Single
         Number", Communications of the ACM, P. 1202-1206, October,
         1988, Volume 31, no. 10.

[GU]     Gustafson, J. et. al., "SLALOM", Supercomputing Review, P.
         52-59, July, 1991.

In [FL,WA], Fleming and Wallace advocate the use of a geometric mean
for characterizing computer performance based on benchmarks.  In [SM],
Smith advocates the use of a harmonic mean, though he states that "the
most obvious single number performance measure is the total time".  The
total (elapsed) time is not only accurate, but has considerable
intuitive appeal.

Neither of the articles, [FL, WAL] or [SM], offer "proofs" in the
mathematical sense.  (If Smith's "proof" is correct, then is Fleming's
and Wallace's "proof" incorrect?)  Why do two articles advocate
different metrics?  The answer lies in the underlying assumptions in
each article.

Fleming and Wallace stress the the geometric mean only applies to
normalized performance results.  The assumption that individual results
are normalized leads to the use of the geometric mean.  Smith, in his
article, assumes that "work" is measured by floating point operations
and that these operations are all equivalent pieces of the workload.
This assumption leads to the use of a harmonic mean.

The article have two distint and different assumptions:
 1.  Results normalized to a specific machine [FL,WA]
 2.  Work is measured by floating point operations [SM]

Some benchmarks fit assumption (1) above.  Some benchmarks fit
assumption (2) above.  Some benchmarks do not fit either assumption.

Not all benchmark tests, especially those with a broad range of
performance characteristics, have realistic machines to normalize
against.  For example, a VAX 11/780, which is used for normalization of
the original SPEC benchmarks, is not appropriate for normalizing
performance of large floating point simulations.  We might ask, is the
VAX 11/780 reasonable for calibrating RISC workstations?

Not all computer "work" is measured by the number of floating point or
integer operations.  For example, the SLALOM benchmark [GU] does not
have an accurate operation count.  The authors of this benchmark do not
count the number of floating point operations performed, rather, speed
is measured by the number of "patches" covered in one minute of
computation.  Different algorithms have different numbers of
operations, but as long as the same number of patches are computed in
one minute, the speed is judged to be the same.

It is the situation, or the constraints, of a particular benchmark with
dictates the proper summarizing statistic.  The table below lists the
interpretation of various means.  (Note that the usual referred to
harmonic mean is often a -uniform- harmonic mean.  Smith, in his
article [SM], emphasizes the the use of weighted harmonic means.)

Geometric mean:            A measure of the distance in "performance
                           space" from the reference machine to the
                           tested machine.

(Uniform) harmonic mean:   The average performance if all benchmarks
                           were adjusted so that each performed the
                           same number of floating point operations.

(Uniform) arithmetic mean: The average performance if all benchmarks
                           were adjusted so that each ran for the
                           same amount of time.



Charles Grassl
Cray Research, Inc.
Eagan, Minnesota  USA
Newsgroups: comp.benchmarks