Re: [Phys-l] Basic statistics

[John Denker <jsd@av8n.com>]

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.

> 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.

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

Oh, but they can.

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.

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.