There seems to be some misunderstanding in the press about a fundamental
difference between a sample of a larger population and a complete
census.
J. A. Paulos in his NY Times article �We're Measuring Bacteria With a
Yardstick'
http://www.nytimes.com/2000/11/22/opinion/22PAUL.html
stated:
"Not to be too cryptic, let me simply state that the vote in Florida is
essentially a tie. The totals for Al Gore and George W. Bush, out of
nearly six million votes, are so close that the results are
statistically indistinguishable from what one would get by flipping a
coin six million times."
Paulos's solution:
"Flip a commemorative Gore-Bush coin in the Capitol Building in
Tallahassee. To avoid another tie, we'd better make sure it has rounded
edges."
Steven J. Gould has seconded this opinion in today's (11/30) Boston
Globe:
http://www.boston.com/dailyglobe2/335/oped/Heads_or_tails_+.shtml
"Unfortunately, in making a deadly serious, even prayerful, case for the
fairness of coin flipping in this circumstance, we must fight both the
greatest failure of education and the deepest foible of the human mind;
our propensity to misunderstand probability ... So flip a quarter ...
Let the fortunate man win, and the United States triumph."
Neither of these authors explicitly use the binomial distribution (but
Paulos certainly alludes to it), but in last Sunday's Boston Globe, two
letters to the editor made the argument that if the vote difference in a
state like Florida is within sqrt(n)/2 votes (about 1225 votes for
Florida), we should call the state race a tie and divide the electoral
votes equally. This embodies the false premise that an election is
somehow a random sample of a larger population. It isn't. An election
is a one-time only, finite, complete census. The votes cast by a
finite population of those that actually voted can, in theory at least,
be estimated without sampling error. Probability theory need not play
ANY role.
To repeat my main point. The votes now being shipped to Tallahasee
represent a one-time finite sample. Using the binomial distribution to
calculate the sampling error for this finite population is
inappropriate. We are NOT estimating what the relative proportions of
Bush vs. Gore voters might be in multiple Bernoulli trials of the likely
Florida voters. Those who didn't vote are irrelevant. We are simply
estimating the actual finite number of Bush vs. Gore votes.
E. C. Pielou faced this problem when dealing with calculating the
diversity of species in a sample. Many ecologists calculate diversity
using Shannon's H' statistic, but Pielou (1969, p. 231-233) points out
that for a fully censused population, Brillouin's H should be used and
"When H is used as the measure of the diversity of a completely censused
collection treated as a population, it is, of course, FREE OF SAMPLING
ERROR. <emphasis added>'
Undoubtedly, there are sources of error in the counting of ballots.
Last week, several studies described the measurement error from punch
card scanners. However, in a finite sample, even as large as 6 million
votes, I can identify no "intrinsic" or a priori source of error that
must be included in any propagation of error. What IS the error rate of
4 observers carefully inspecting each punched ballot? I can't find a
theorem related to that in any of my probability texts, and I warrant
nobody else can either. It seems to me that using probabilistic
arguments, especially erroneous arguments that elections are Bernoulli
trials, is not helping us resolve this electoral decision. When it
comes down to counting a finite and relatively small number of ballots,
with strict standards, there need be NO sampling error. Debate the
standards about dimpled chads and what counts as a vote, but don't throw
up your hands citing inappropriate probability theory and argue that
Supreme Court Justice Rhenquist should toss a quarter in the air to
decide who will be president.
Reference
Pielou, E. C. 1969. An introduction to mathematical ecology.
Wiley-Interscience, New York.
--
Dr. Eugene D. Gallagher
ECOS, UMASS/Boston
Sent via Deja.com http://www.deja.com/
Before you buy.
=================================================================
Instructions for joining and leaving this list and remarks about
the problem of INAPPROPRIATE MESSAGES are available at
http://jse.stat.ncsu.edu/
=================================================================
