Re: [Phys-l] Curve fitting versus averaging [was question on averaging]

[John Denker <jsd@av8n.com>]

On 02/19/2009 07:45 AM, Edmiston, Mike wrote:

> The typical student just wants to find the differences between these
> positions and average the differences to find the average
> half-wavelength.  However, doing this actually only utilizes the first
> and last position because all the intermediate positions drop out during
> the averaging process.  This means you are wasting your time to find all
> those intermediate resonance points.
>
> This strikes the students as odd.  Surely having all that extra data is
> good for something.  Actually all that data is indeed worthless if you
> simply take differences and average the differences.  But it seems
> curve-fitting might be a way to utilize all the information.  Plot the
> resonance positions against "n" or an integer index number, and fit a
> linear regression line to the positions, and use the slope as the length
> of one-half-wavelength.  It seems this would be using all the points and
> would give a better estimate of the half-wavelength than just taking the
> difference between the first and last positions and divding by the
> number of half-waves between those two positions.
>  
> Am I looking at this correctly?

Yes.






=================

Continuing down that road:  

1) This "odd" feature of the average is called a _telescoping sum_.  
Tell students they should always be on the lookout for telescoping
sums, because that is super-easy way of doing sums.  (They should 
have learned this in math class.)

2) Averaging is linear.  Almost any reasonable criterion for "goodness 
of fit" is nonlinear.  Logic and judgment are nonlinear.

On rare occasions the thing you care about is a linear function of
the data -- for instance arrival time depending on the average
velocity -- but you can't depend on this in general.

>  Plot the resonance positions against "n" or .......

3) Yes, plot them.  Don't just plug into some formula.  Plot the
data and the fitted function, and *look* at the plot.  

If the residuals are small, plot them also, on their own zoomed-in 
scale.

By doing this, you might discover that your initial choice of
fitting function is slightly imperfect (or completely disastrous).

Examples where students might expect one thing and get another
include:

 *)   Suppose we are looking at the resonances of a string (as
opposed to a tube) as a function of length.  Due to stiffness
in the string, the frequency as a function of wavenumber will not 
be a simple proportional relationship.

 *)   Frequency versus wavenumber for waves in a waveguide,
not too far above cutoff.

 *)   Frequency versus wavenumber for water waves, plane waves
in a ripple tank.  If you're expecting this to be linear (as it 
is for sound and for light waves in the same geometry) then 
you're in for a surprise.

 *)   Frequency versus n for waves with cylindrical or spherical
symmetry (as opposed to plane waves).

 *)   Force versus length of a nonideal spring.

 *)   Resistance versus temperature for an Allen-Bradley
carbon-comp resistor.  You might expect Arrhenius- or Richardson-law 
behavior i.e. exp(-1/T), but in fact it goes more like exp(-1/sqrt T).
I don't know why, but it does.

 *)  Real data (including friction) versus theory (neglecting
friction) in simple kinematics experiments, and innumerable other
situations as well.

 *)  Et cetera....................


===================

Pedagogical judgment is required here.  You probably want to _start_
with systems where the irregularities and nonidealities are small,
but students need to learn to handle irregularities sooner or later.
The real world is highly nonideal.

Curve fitting is the traditional and reasonable way of seeing the
relationship between the regularities and the irregularities.