While an undergradaute I heard was that N in the S formula slightly underestimates
sigma, so use N-1 when you are doing inferential statistics (inferring sigma from
S). But use the N version when you have a complete population and are not
inferring sigma.
I checked a few convenient sources:
(1) Lehman (Statistics in the Behavioral Sciences, Brooks Cole) claims it's a
complex issue but uses N-1 (Richard Lehman, I believe, is a fellow tipster).
(2) Moore (Basic Practice of Statistics, edition 2) uses N-1 and I cannot find a
discussion of the topic other than a paragraph that defends the N-1 in terms of
degrees of freedom (Moore is a _real_ statistician btw).
(3) Sprinthall (Basic Statistical Analysis, edition 5, Allyn & Bacon) uses N. I
vaguely remember Sprinthall once saying (or writing) that the jury was out on
whether the N-1 is a better estimator of sigma (I can't find the question
addressed in the fifth edition).
(4) Agresti and Finlay (Statistical Methods for the Social Sciences, edition3,
Prentice Hall) uses N-1 without a discussion.
(5) Rosenthal and Rosnow (Essentials of Behavioral Research, edition 2) says this
(and I quote, page 272).
(he gives the formula for S^2 which uses N-1. This version. "..... is the
unbiased estimator of the population value of sigma^2. Unbiased estimators of
population values such as sigma^2 are estimators that, in the long run, under
repeated sampling, are the most accurate estimates. Interestingly, it turns out
that S (note: the N-1 version) is not an unbiased estimator of the population
value of sigma, but that fact rarely works a hardship on us."
The "note" in ( ) above is mine, not Rosenthal's. A better definition of
"unbiased" is that is it equally likely to overestimate as underestimate. It
appears that N-1 is _not_ the unbiased estimator we thought it was, but Rosenthal
doesn't provide references. As N increases, the differences between the versions
dimishes.
On a purely practical note, when the N formula is used, there is a simple
computational formula for S^2: ((SIGMA X^2/N) - MEAN^2). This is handy when you
already have the Mean.
Btw, if people are looking for a terriffic stats book that takes a Data Analysis /
Exploratory Data Analysis (EDA) approach (i.e. look at data before you throw it
into an ANOVA) - try Moore's _Basic Practice of Statistics_. It's not
psychological, but it's very good.
--
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John W. Kulig [EMAIL PROTECTED]
Department of Psychology http://oz.plymouth.edu/~kulig
Plymouth State College tel: (603) 535-2468
Plymouth NH USA 03264 fax: (603) 535-2412
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"What a man often sees he does not wonder at, although he knows
not why it happens; if something occurs which he has not seen before,
he thinks it s marvel" - Cicero.
