[Phys-l] Curve fitting A salutary Moment?

[Brian Whatcott <betwys1@sbcglobal.net>]

Michael Edmiston wrote:
> > [Brian] I am visualizing taking an average of the differences
> > given by posn 1 - posn 0, posn2 - posn1
> > and so on ....
> >     
> > [Michael] Averaging the differences can't provide a smaller error because it provides exactly the same result.
> >
> > Explicitly write down the algebra for the average:
> >
> > [(P1-P0) + (P2-P1) + (P3-P2) + (P4-P3) + (P5-P4)] / 5
> >
> > Notice that all the positions between P0 and P5 appear in the sum twice; 
> > once as positive and once as negative.....You wasted your time obtaining data points P1 through P4, and you wasted 
> > your time doing the calculation using them.
> >
> > The result rests on the first and last points, and the error is totally 
> > dependent on how well you established the first and last points.
> >
> > Indeed, you can fabricate any outrageous data you want for the positions of 
> > node1 through node4 and you still get the correct result if the positions of node0 and node5 are correct...
> Michael D. Edmiston, Ph.D.


The slap of reality, if not administered too often, is always helpful.

It put me in mind of the recent PER thread, whereby I took my turn - 
with others -
at poking fun at John C's efforts to support PER initiatives. And I 
thought that I should take this opportunity to be more supportive of 
interactive style approaches....


  Thinking that Michael would (I imagine) never say this kind of thing 
to students.
"You wasted your time..." "Outrageous data..."
because while building empty self-esteem is worthy of disdain, lifting 
students who are addressing physics is probably a better way.
Like this.

1) display the fallacy of the mean difference of adjacent readings as 
Michael did.

2) ask for cures.
We have already seen one cure, the differencing of first and n/2 values, 
second and n/2 plus 1 values and so on.  
(This first cure was however dictated by the teacher - i.e. not 
participative...)
What other way might occur to students?

2a) Perhaps someone might propose differencing observation 1 and 2, 3 
and 4, 5 and 6, n-1 and n values

2b)Perhaps another might add the possibility of differencing 2 and 3, 4 
and 5, 6 and 7....
n-2 and n-1  values.

2c) It is possible one could encourage  a student to suggest a union, 
such as a weighted averaging  of these two averages,
one of n/2 differences, the other of n/2 - 1 differences to provide a 
grand average.

3) This would be the moment to mention a concrete experiment - say 
reading a scope with a graticule faceplate - or reading through a thick 
glass window and showing how the differences seem shorter at each end of 
the observations due to angle of view.

Here a plot showing the departure from a straight line - or in slightly 
more sophisticated terms - the tell-tale shape of the residuals from a 
fitted line - might vividly bring to life the virtues of curve fitting 
as well as some ways to average differences....

Brian W