Re: [Phys-L] statistics of measurements question

[John Denker <jsd@av8n.com>]

On 02/26/2013 03:05 PM, Aburr@aol.com wrote:
> I  have some statistical questions regarding means and standard deviation 
> (sd) on  which I would appreciate answers and comments. A common way of 
> thinking about  simple measurements such as time (t) for any operation is to 
> consider the  population to be all possible t measurements from which you take a 
> random sample  using the mean and sd of the sample to infer the mean and sd 
> of the population.  All this predicated on the population being a gaussian 
> distribution. Yet it is  clear that the population is not gaussian (if for 
> no other reason than that  there can be no negative times.) Does one just 
> ignore the very small tail of the  gaussian here?

That's one of those simple questions with a complicated answer ... 
or perhaps ten different answers, depending on as-yet unspecified
details.  Let me say a few things that /might/ be helpful:

1) First and foremost, keep in mind that a very great deal of what
goes on in science consists of building /models/ to explain whatever 
is going on.

I mention that because "standard deviation" might be relevant to
some part of the model ... but standard deviation is not a model 
unto itself!  You need a lot more than that.


2) At a much more prosaic level:  Not all models are Gaussian.  You 
could, for instance, have a model that says the data has a /flat/
distribution, /evenly/ distributed over some interval from A to B.

Note that the standard deviation is perfectly well defined for
distributions other than Gaussian distributions.  It is an easy
exercise to calculate the standard deviation for the aforementioned
flat distribution.  It is related to |A-B| but it is not quite the
same thing.


3) In *some* cases you can have a Gaussian model that predicts that 
it is vanishingly unlikely that the observed time will be negative, 
in which case using the model does not violate the constraint and/or
the intuition that the times cannot be negative.


4) At the opposite extreme, I can easily imagine situations where 
the observed times *can* be negative ... for instance if two 
independent observers are timestamping the events and the elapsed 
time is calculated by subtracting the somewhat-noisy timestamps.


5) It may be that the start-timestamp and end-timestamp exhibit
nontrivial /correlations/ ... in which case the whole concept of
standard deviation goes out the window.  The standard deviation
is the square root of the variance ... but when there are N
variables there is in general an N by N covariance matrix.
Neglecting the off-diagonal elements is a verrry bad idea.


6) Last but not least, it is fairly well known that the standard
deviation of the /sample/ is perfectly well defined, but is a
terrible estimator of the standard deviation of the underlying
/population/.  For starters, it is a biased estimator, and even
if we take steps to remove the bias it is still quite a noisy
estimator.

=================

Many of these issues are discussed at
  http://www.av8n.com/physics/uncertainty.htm
and
  http://www.av8n.com/physics/probability-intro.htm

=================

If that doesn't answer the question, please re-ask the question.
Often there are answers to specific practical problems even
when the generic philosophical problem is unanswerable.