Re: [Phys-l] Basic statistics

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

On Nov 9, 2006, at 4:18 PM, John Denker wrote:

> On 11/09/2006 12:56 PM, Ludwik Kowalski wrote:
>
>
> > ... we assume that true values do not fluctuate.
>
> Fluctuations are not the issue.  Everybody is assuming there
> is an underlying distribution from which the samples are drawn,
> and that this distribution is unchanging.

1) What is a distribution of gravitational acceleration, or of the mass 
of a brick? They are negligible in comparison with fluctuations 
associated with limited precision of instruments we use, including our 
own eyes, reaction time, etc. That is what I had in mind.

> > Distributions are
> > assumed to be due to random errors, which mean they are Gaussian.
>
> In general random does not imply Gaussian.  But for present
> purposes let's restrict the discussion to the case where
> the underlying distribution happens to be Gaussian.

2) I agree, its not hard to generate a distribution of any shape by 
using random numbers. And an "underlying" distribution of wavelengths 
of photons (random for each photon) may be very non-Gaussian. 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. 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.

> > The
> > question "1.2 or 0.4 ?"  was posed in that context. Both answers 
> > cannot
> > possibly be correct.
>
> Oh, but they can.

I am not convinced.

> The notation A ± B is ambiguous.  There are two different
> probability distributions in play.  If you have a cluster
> of N observations (N=9 in this case), you can ask about
> the statistics that govern drawing one more observation,
> or you can ask about the statistics that govern drawing
> one more entire cluster.

3) I was asking about the uncertainty associated with a mean value, 
<x>, derived from 9 measurements. Should it be s=1.2 or should it be 
E=s/sqr(N)=0.4, in order to be right 68% of the time. In that context 
only 0.4 is correct. But I agree that a different question could have 
been asked. "What error bar should be assigned to a prediction that the 
next measurement result, x, would be the same as the <x> obtained from 
the previous 9 results?"  In that case 1.2 would be the correct answer.

> Different questions lead to different answers.  In
> either case the answer is denoted A ± B, so you can't
> easily tell which question goes with a given answer.
>
> This is a perennial source of confusion.  I misunderstood
> a question on this topic, on this list, just a couple of
> weeks ago.
>
> I fleshed out and cleaned up the explanation and put it
> up at
>   http://www.av8n.com/physics/uncertainty.htm#sec-samples
>
> This includes a possibly-helpful diagram
>   http://www.av8n.com/physics/uncertainty.htm#fig-sample-mean
> showing the relationships among the quantities of interest.

4) I will read your piece a little later today. And I might ask some 
questions. All this is far from being trivial.

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