Re: [Phys-l] Basic statistics

[John Denker <jsd@av8n.com>]

On 11/09/2006 05:15 PM, Ludwik Kowalski wrote:

> I agree, its not hard to generate a distribution of any shape by 
> using random numbers. 

Yes, it can be done artificially ... but there are also lots
of natural, non-artificial distributions that are distinctly
non-Gaussian.

> And an "underlying" distribution of wavelengths 
> of photons (random for each photon) may be very non-Gaussian. 

Yes.


> I know that a distribution of <x> approaches a Gaussian shape, even 
> when distributions of x(i), in each sample, are not Gaussian, when the 
> number of samples approaches infinity.
            ^^^^^^^

Remark on terminology: This substantiates what I wrote at
  http://www.av8n.com/physics/uncertainty.htm#sec-stat-terms
that non-statisticians commonly use "sample" to refer to a single
observation.  In constrast, statisticians think of "sample" as a
/set/ of observations and would have said something like "sample
size approaches infinity".

Statistics, like other fields, has its own terminology and jargon,
which can be very confusing to outsiders.


> But what 
> is the general precondition of randomness of of x(i) in a our 
> laboratory samples? I do not remember how to answer this quesion. 

I doubt the question is answerable in general.

Here is an amusing example:  You are looking through my logbook.
You discover that I have used a cheap analog ohmmeter to measure
a resistor R1 and also the series combination R1+R2.  The readings
are:
   R1    : 100 ohms
   R1+R2 : 103 ohms
   all readings accurate ± 2%

The resistors are long gone, having been mixed with other resistors
and sold to an anonymous buyer ... so you must rely on the logbook,
and rely on your wits.  Let's stipulate that the uncertainty on the
two readings is Gaussian to a good approximation.  Your task is to
infer the value of R2.

Question 1:  Would you say that R2 = 3 ± 2 ohms?
             Or would R2 = 3 ± 3 ohms be a better answer?
Question 2:  Do you think the uncertainty is Gaussian?
Question 3:  Roughly what is the probability that R2 has a negative
             resistance?

   FWIW, to simplify the math in Q3, you may refer to
     http://www.av8n.com/physics/img48/gaussian-disk.png