Re: st: Why many things have Normal distribution

[Steve Samuels <sjsamuels@gmail.com>]

David,

Here is some empirical evidence: the book by Hampel et al.(1986, pp
22-23) cites several investigators, starting with Bessel in 1818, who
studied "very high quality" data sets. Most of the sets were
longer-tailed than the normal and were well-approximated by
t-distributions with 3-9 d.f. Slight skewness was also noted.

Steve


Reference:

Hampel, Frank, Elvezio Ronchetti, Peter Rousseeuw, and Werner Stahel.
1986. Robust Statistics: The Approach Based on Influence Functions
(Wiley Series in Probability and Mathematical Statistics). New York:
John Wiley and Sons.

Jeffereys, H. (1939,1961). Theory of Probability. Clarendon Press,
Oxford

 


On Aug 29, 2013, at 10:49 PM, David Hoaglin wrote:

Yuval,

The Central Limit Theorem (CLT) describes the behavior of the
distribution of the sample mean as the sample size becomes large.  In
order for the distribution of the sample mean to approach a normal
distribution, the underlying distribution of the data must satisfy
some conditions, but those conditions are not very stringent.  The CLT
provides no information on how the underlying distribution behaves.
One does, however, expect the behavior of samples to approach that of
the underlying distribution (whatever that happens to be).

I would have no special expectations of the distribution of heights in
a large army.  I would look at the actual distribution --- empirical
evidence, rather than a thought experiment.  Apart from any attempts
to avoid serving, one would expect recruiters to reject people who
were too short and people who were too tall.  Also the actual
distribution might be a mixture of components.  As I recall, in the
19th century Quetelet used a frequency distribution of the chest
circumference of Scottish soldiers to illustrate a method of fitting a
normal distribution.  In compiling the data he merged several
components and made a variety of mistakes.

The outcomes of tossing an actual "fair" die depend on how carefully
the die was manufactured.  Iversen et al. (1971) analyzed the results
of a large number of throws of various types of dice.

You didn't say how you would use a normal distribution to approximate
the outcomes of throwing a fair die.  The basic distribution is
discrete, with six equally likely outcomes.

David Hoaglin

Iversen GR, Longcor WH, Mosteller F, Gilbert JP, Youtz C (1971). Bias
and runs in dice throwing and recording: a few million throws.
Psychometrika 36:1-19.

On Thu, Aug 29, 2013 at 5:38 PM, Yuval Arbel <yuval.arbel@gmail.com> wrote:
> What about the central limit theorem? I was referring to physical
> human features - such as height - and the example of Napoleon's army
> candidates for draft. In an army of millions of soldiers - you would
> expect a normal distribution of heights. The problem is that those who
> tried to avoid drafting probably bribed somebody to write false
> heights, which is shorter than the minimal required height. In this
> case - you might get a skewed distribution of heights
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