Re: st: Why many things have Normal distribution

[Yuval Arbel <yuval.arbel@gmail.com>]

Steve and David,

If I come to think about it - and as David previously mentioned -
income, for example, is not normally distributed, no matter how much
we increase the sample:

If we take the big corporations, for example - we find that most of
the workers earn a minimal wage, where senior managers earn at least
ten times more. I would therefore anticipate that the income variable
distribution will be skewed to the right. This also corresponds to
Pareto principle - that 80% of the wealth is concentrated among 20% of
the population

One possible explanation - is the poverty trap: poor people remain
stuck without education or other means to get out of the trap -
because they get a subsistence wage.

On Sat, Aug 31, 2013 at 1:12 AM, Steve Samuels <sjsamuels@gmail.com> wrote:
>
> 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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-- 
Dr. Yuval Arbel
School of Business
Carmel Academic Center
4 Shaar Palmer Street,
Haifa 33031, Israel
e-mail1: yuval.arbel@carmel.ac.il
e-mail2: yuval.arbel@gmail.com
You can access my latest paper on SSRN at:  http://ssrn.com/abstract=2263398
You can access previous papers on SSRN at: http://ssrn.com/author=1313670
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