Re: [Phys-l] data analysis : pendulum period versus length

[John Denker <jsd@av8n.com>]

On 05/12/2011 06:37 PM, Josh Gates wrote:
>   Also notice that in my data tabulated above, the scatter is
>   huge compared to the quantization error, so the details of
>   how the quantization was done don't matter.  This is how it
>   should be.

> > That's a great point that I hadn't thought of.  How about if we're
> > measuring a distance?  There's not any scatter there, really - I
> > put down the ruler and get a measurement. 

It depends on what you mean by "measurement":
  a) If a "measurement" consists of a multiple repeated
   observations, the set of observations (i.e. the "sample")
   will have some mean and some standard deviation.  The
   sample-mean and sample-stdev are properties of the whole
   set, not properties of any particular observation.
  b) If a "measurement" is a single observation, then by the
   time it gets written down it is just a number, with no 
   uncertainty.
  c) OTOH if you have a single observation, you might 
   imagine, based on theory, including a lifetime of
   experience with rulers, that this observation is
   drawn from some imaginary ensemble, i.e. the set of
   observations you /would/ have gotten if you had
   repeated the observation.  There is some uncertainty
   associated with this theoretical / imaginary ensemble,
   but as always, it is a property of the ensemble, not
   a property of any single observation.

OK?

> > I estimate the last
> > fraction of a mm or fraction of an inch, so there's some
> > uncertainty in that measurement, right?  Can I say that my
> > measurement is a distribution?
 
See above.

>  On the first observation, the
>   reading on the stopwatch was 1.235 and I am absolutely
>   certain about that. 

> > I'm absolutely certain about that reading, but it represents a range of possible values, right?

There is a crucial distinction between what the reading "is"
and what it "represents".  When it comes to numerals or
any other symbol, what it "is" is definite and cut-and-dried.
What it "represents" is much more indefinite, much more in 
the eye of the beholder.

To quantify the range of possible values, you need a lot
more than one reading.  You need an empirical ensemble of 
actual readings, and/or some theory that says what the 
theoretical ensemble of readings looks like.

This is one of the things that is pernicious about the
"random variable x" notation:  It uses the same symbol
to represent both the reading and the ensemble from which
the reading was drawn.

In a somewhat similar way, this is one of the things that
is pernicious about the "significant figures" idea.  It
uses a single numeral to represent both the reading and
the uncertainty of the ensemble from which the reading
was drawn.