Re: [Phys-l] Basic statistics

[Ludwik Kowalski <kowalskil@mail.montclair.edu>]

On Nov 11, 2006, at 11:31 AM, Bernard Cleyet wrote:

> I think you two are "talking (writing) past each other".    One is a
> single distrib. tother several independent.  Furthermore, Wiki, claims
> that the Chi-square also does tend towards normalcy, but VERY slowly, 
> as
> the # of DOF increases.
>
> http://en.wikipedia.org/wiki/Chi-square_distribution
>
> > On Fri, 10 Nov 2006, Polvani, Donald G. wrote:
> >
> >
> >
> > > Jack Uretsky wrote:
> > >
> > >
> > >
> > > > This is a misconception. There are many distributions that become
> > > Gaussian in the large N limit, but not all.
> > >
> > >
> > > > The chi-squared distribution is an example of one that does not.
> > > However, for the SUM of independent, identically distributed random
> > > variables, each with mean mu and variance sigma^2, the central limit
> > > theorem tells us that the distribution (of the sum) tends to a 
> > > Gaussian
> > > as N approaches infinity.

Here is a way to illustrate the central limit theorem in action, at a 
very elementary level. General purpose computer languages, such as 
Basic, Fortran, etc. have an instruction to generate a number from a 
uniform distribution of x between 0 and 1. In True Basic, for example, 
the instruction is let x=rnd. Suppose ten students are sitting in front 
of computers which displays x, as soon as the return key is pressed. 
Each student gets one number and the sum of ten results is calculated. 
The teacher asks “what is the smallest possible sum?” Then s/he asks 
“what is the largest possible sum?”

After a short discussion ten sums are tabulated, one after another (on 
the blackboard). The teacher asks “how comes that not a single result 
is close to zero or close to 10? Why most of our sums happen to be 
close to 5?” The rest should be obvious.  To get a very large sum, all 
values of x must be close to 1; that is not likely to happen. Most 
often some of us get small x while others get large x and the sum 
hapens to be somewhere near the middle. Not surprisingly, distributions 
of sums (or mean values) are always bell-shaped. Actually, computers 
are not needed to conduct the activity. Just ask students to select a 
piece of paper from a box in which each piece has a numbers between 0 
and 1. Pieces of paper should be returned to the box as soon as numbers 
are recorded, unless the box contains at last 100 pieces with nubmers. 
How much more you want to do should depand on student’s age,  etc


Ludwik Kowalski
Let the perfect not be the enemy of the good.