Re: error analysis

["John S. Denker" <jsd@MONMOUTH.COM>]

At 06:29 PM 3/8/01 -0400, Tim O'Donnell wrote:
> What types of error analysis would you like to see a high
> school teacher work with students. Other than percent
> differences and significant figures  which I do try to
> reinforce.

Good question.

A few suggestions:

  1) Despite the name I unwisely gave to this thread, it would be better
to speak of "uncertainty" rather than "error".  This is how the pros
express it.  See e.g.
  http://physics.nist.gov/cuu/Uncertainty/basic.html

  2) The notion of "significant figures" is highly over-rated.  The key
thing is to know the uncertainty, and to express it clearly.  Truncating a
result to a certain number of digits is a horribly clumsy way of attempting
this.  Better ways exist.  For instance, the Britannica reports the charge
of the electron as
   1.6021892(46) e-19 Coulombs

More discussion of this can be found in the phys-l archives at:
  http://mailgate.nau.edu/cgi-bin/wa?A2=ind9908&L=phys-l&P=R59015

3) For intermediate calculations there is much benefit, and very little
cost, in keeping a few _guard digits_.

As an exercise, consider the following numbers, which are accurate to 1
decimal place, and highly uncertain in the 2nd decimal place.
        1.12
        1.07
        1.14
        1.13
        1.11
        1.12
        1.13
        1.09
        1.13
        1.12

a) have the kids add the numbers and THEN round the answer to 1 decimal place.
b) Then have them round to 1 decimal place BEFORE adding, then add.

The answers differ quite significantly.  If I were trying to do serious
science, I would wish for a _third_ decimal place in those numbers, just to
make sure the "insignificant" digits didn't add up to something significant.

As the list of numbers becomes longer and/or the number of steps in the
calculation becomes greater, the more important it is to have guard digits.

This is discussed at
  http://mailgate.nau.edu/cgi-bin/wa?A2=ind0001&L=phys-l&P=R29416
and
  http://mailgate.nau.edu/cgi-bin/wa?A2=ind9908&L=phys-l&P=R61877

4) Percent differences can be tricky.  Suppose I go to Home Depot to buy
two pieces of pipe.  They're both about 10 feet long.  Somebody tells me
that they differ in length by 3 inches, plus or minus one percent.  One
percent of what?  One percent of the difference, or one percent of the pipe?

This is not a subtle problem, and it does not require a subtle
solution.  The rule is simple:  say whatever you have to say in order to
clearly communicate what the uncertainty is.

5) As mentioned earlier in this thread, sometimes the analysis is overly
focused on uncertainty in measurement.  The whole point of measurements is
to be used, in conjunction with a theoretical model, to make predictions
about what happens next.  So the key concept is uncertainty in
_prediction_.  Sometimes the uncertainty is due to the measurements, and
sometimes it is due to imperfections in the model.

6) Remember that uncertainty analysis is an advanced topic.  Sometimes in
order to teach this topic, people cook up examples with lots of
uncertainty.  It is important to do this WHEN THE TIME COMES.  But first it
is important for the students to get comfortable with the idea that you can
do physics experiments with lots and lots of accuracy.

Example:  Even a low-cost triple-beam balance can weigh things to one part
in 1000, and you can predict that the mass of three objects together equals
the sum of the separate masses, to high accuracy.  (Check the calibration
of your balance before assigning this experiment.)

Another example:  You can use one alarm clock to predict when another alarm
clock will go off, easily to one part in 100,000 or so.  In today's class,
set up one to go off in tomorrow's class, ten seconds ahead of the other
one.  Then the class can have a countdown
10...9...8...7...6...5...4...3...2...1 and the other should go off.

Remark:  This demo is psychologically more effective if the two clocks
don't LOOK alike.

These days even low-cost travel-clocks are based on quartz oscillators, so
you should be able to do pretty well, especially if they are kept at the
same temperature overnight.  Even if they don't keep perfect time, after
you've run the experiment a few times you'll know the appropriate
calibration constant, and you can make pretty impressive predictions.

7a) Again:  Uncertainty analysis is an advanced topic.  Any really
meaningful discussion of uncertainty requires notions of probability that
high-schoolers usually don't have (unless you just taught it to them).

7b) Almost any really meaningful work with uncertainty analysis requires
lots of data, which calls for computer skills that, once again,
high-schoolers usually don't have.  Maybe you could do it without a
computer, but most students would be horribly bored by that.

7c) The previous paragraph said "almost any" ... and here's a possibly
constructive suggestion:  You need an experiment that is simple enough that
you can do it umpteen times.  Maybe each person in the class does it once
or twice.  Then you can HISTOGRAM the results and see how big the spread
is.  The general idea of using histograms won't solve all the world's
problems, but it is a sound idea of lasting value, and it makes a good
foundation for the more advanced concepts.

At 10:36 AM 3/9/01 -0700, Daniel L. MacIsaac wrote:
>  I would NOT recommend teaching measurement analysis to
> non-engineers or scientists; they are learning physics for different reasons.

That seems going a bit far.  I would hope that future doctors, lawyers,
butchers, and bakers would benefit from having some notion of what's
reproducible and what's not.