Re: [Phys-L] teaching error analysis in high school

[John Denker <jsd@av8n.com>]

On 09/30/2012 12:19 PM, Jeff Bigler wrote:
> Agreed that the variables need to be uncorrelated, but as you say in
> your discussion, the same needs to be true for C3T.  I'm explicitly
> having them assume that the variables are uncorrelated because most of
> my students don't have the statistics background to deal with anything
> else. 

Let's talk about that.  We agree that Crank Three Times™ does not
by itself solve all the world's problems ... but it is in several
ways better and in no ways worse than the step-by-step "analytic" 
propagation techniques suggested at
  http://phys.columbia.edu/~tutorial/propagation/tut_e_4_1.html
and elsewhere.

For starters, consider a multi-step calculation, even something as
simple as the quadratic formula (in its usual form).  Even if the 
coefficients of the original polynomial are uncorrelated, intermediate 
steps in the calculation will suffer from nasty correlations, especially
in cases when there is one large root and one small root.

 a) In favorable sub-cases, Crank Three Times™ will give you the right
  answer.

 b) In other sub-cases, Crank Three Times™ will warn you that there is
  a problem.

 c) In these situations, step-by-step "analytic" propagation is likely
  to fail _without warning_.  This is very much worse than what Crank
  Three Times™ would have done.

In all cases, Crank Three Times™ lays the intellectual foundation for an
even more powerful technique, namely Monte Carlo.  So it can be considered 
a step in the right direction.  This stands in contrast with the step-by-
step "analytic" approach, which is a step in the wrong direction, and 
must be completely unlearned before progress can be made.  Unlearning is 
always difficult.

Given that the students don't understand about correlations, they won't
really understand /why/ Crank Three Times™ is better ... but you can
_tell_ them it's better.  You can tell them that later, when the subject
of correlations comes up, they will understand why it is better.

Note that the quadratic example could well be taken as a warning to never 
use the lame "textbook" form of the quadratic formula ... but again the
point remains:  Students would need to completely unlearn the step-by-
step "analytic" approach before they could understand /why/ the better
numerical methods are better.
  http://www.av8n.com/physics/uncertainty.htm#eq-smart-quadratic

Bottom line:  AFAICT, there is nothing to gain and much to lose by
teaching the step-by-step "analytic" propagation idea.