Re: [Phys-L] Timing Statistic

[John Denker <jsd@av8n.com>]

On 11/6/18 12:19 PM, Paul Nord wrote:

>  The variance on the mean should properly be calculated as
> sigma/sqrt(n) for a selected set of random samples.  

Trivial point: That's obviously a typo.
Variance is σ^2.  The standard deviation of the mean is

	            σ_1
	σ[mean] = --------
	             √n

or the variance of the mean is

	               σ^2_1
	σ^2[mean] = -----------
	                 n

or more simply and more powerfully, the variances just add:

	σ^2[sum] =  ∑ σ^2_i

This typo has no effect on the rest of the discussion.

> But I don't think that is valid in this case because the samples are correlated.  

Yes, correlations certainly invalidate all such expressions.

> Vernier's
> software may even be doing some smoothing and approximation for angular
> positions that fall between the position encoder's discrete digitization
> steps.

That hypothesis is
  a) interesting, and
  b) easily testable.

Collect a lot more than 5 data points, and plot σ^2[sum]
as a function of n.  Either it's a straight line or it's
not.  It's important to plot the data.  There's a facetious
proverb that says:
	When all else fails,
	look at the data.

Generally speaking, digitization errors aka quantization
errors aka roundoff errors are not random and don't add
in quadrature.  You can round π to 3.14 as many times as
you like and the roundoff errors will not average out.

> We're collecting 20 data points per second.  The interaction takes about a
> quarter of a second.  Students take the 5 data points before and the 5
> points after the "collision" and look at the mean angular velocity for
> each.

I'm not convinced that's an optimal design.  It throws away
the data points /during/ the interaction.  Back when I was a
sorcerer's apprentice, I was taught not to throw away data.
Keep the data, and find a way to model it.

In more detail:
  a) Don't throw away data without a reeeeally good reason.
  b) Nobody in the introductory course will ever have that
   kind of good reason.  Not even close.  Angels fear to
   tread here.

Again:  Keep the data, and find a way to model it.

In this case, modeling an abrupt collision would be messy
... but since the experiment is still in the design phase,
why not simplify the interaction?  Rather than relying on
friction, consider using one axle supporting two disks,
each with its own bearing, with an eddy current coupling
between them.  Such a coupling is easy to model with
high accuracy.  It's also tunable, e.g. by varying the
spacing between disks and/or by swapping out the magnets.

Spin the two disks in the same direction, to measure the
coupling (i.e. loss of angular momentum) to the outside
world, due to presumably small friction in the bearings,
aerodynamics, et cetera.  Then spin them in opposite
directions; the coupling to the outside world should
be nearly the same, but now there is disk-to-disk coupling.

Tuning the interaction to be weaker and slower than an
abrupt collision should make you less vulnerable to
vagaries of the instrumentation.  Smoothing is less of
a factor, and you have more points to deal with.  One
of the big advantages of taking data with a computer
is that you can collect huge amounts of data.  To say
the same thing the other way, n=5 reminds me of a guy
walking down the street carrying a bicycle.  It's a
lot more fun if you hop on the thing and ride it.

Note that you can model smoothing and roundoff error.
  a) First, come up with a formula for what you think
   the true physical angular velocity is, as a function
   of time.
  b) Write down another formula, i.e. the rounded and
   smoothed version of formula (a).
  c) See how well the observed data fits formula (b).

Minor point:  I'd be tempted to report the result in terms
of χ^2 (chi squared) rather than Z-value.  It's essentially
the same thing conceptually, but there are better tools for
dealing with χ^2.

That's all the tirade I can provide at the moment.  I had
a 20-hour workday yesterday, wearing my Election Inspector
hat, and I'm still pretty zonked.