Re: [Phys-l] Student Misconceptions

[John Denker <jsd@av8n.com>]

In the context of:

> > If you drive somewhere with an average speed of 60 mph and return the
> > same way with an average speed of 30 mph, then is your average speed
> > for the entire trip less than, equal to, or more than 45 mph?    [1]

On 10/03/2011 05:00 PM, Bailey, Steven wrote:

> I might be missing something (which I often do)  but this is a conceptual question.  

Actually there are /two/ conceptual questions on the table,
because Robert Cohen asked a follow-on question:

> > Average over time or over space?

It's partly a trick question, because in the context of
question [1] the conventional assumption is that we are
interested in the average with respect to time.  This is
conventional, and there are -- in this context -- reasonable
physical grounds for focusing attention on the average over
time.  Any other type of average would be -- in this context --
eccentric, maybe even perverse.

However, in a wide range of seemingly-similar contexts, it
would be quite necessary to ask what kind of average we are
talking about.  One excellent example that Prof. Cohen has
already mentioned is 
  -- the force, averaged over time, versus
  -- the force, averaged over distance

Both of those have clear physical significance.  One has to
do with momentum, while the other has to do with energy.

Another example that comes up all the time concerns waves, 
namely the average over frequency versus the average over
period (or wavelength).  This matters, even for things as
simple as finding the peak of the power spectrum.

The conceptual point is simple:
  All averages are weighted averages.

Unless it is ultra-super-obvious what weighting you have 
chosen, you ought to spell it out ... and you should ask 
yourself how sensitive your results are to the choice.

In statistics, this shows up in Simpson's paradox:
 http://plato.stanford.edu/entries/paradox-simpson/

The example that started this thread (i.e. average speed) is
not the ideal example, but even slight changes make it much
more interesting, such as averaging the inverse speed w.r.t
distance, as previously mentioned.

Other examples abound.  The concept remains important.  All
averages are weighted averages, and you need to be careful.