Re: error analysis

[kyle forinash <kforinas@IUS.EDU>]

A further point about error analysis which I rarely see presented to students.

Most of us are familiar with the old linear expansion apparatus?
Where you heat up a metal rod, measure the expansion and calculate
the coefficient of expansion. Two of the measurements made are delta
L (change in length of the rod) and L (total length of the rod).

Rhetorical question (which I ask my students): Why is it you can get
away with measuring L with a meter stick but delta L has to be
measured with a micrometer? My point is: No measurement ever made in
science is perfect but some measurements are more important than
others.

Students ought to be able to do an error analysis which answers questions like:

1. Of all the measurements made in this experiment, improving which
measurement(s) (if any) would lead to the biggest improvement in the
accuracy of the answer?
2.  Given this particular equipment with its inherent limitations
(which all instruments have), what is the smallest error you could
possibly get in your answer? The biggest (if all instruments are at
their maximum error)?
3. Is there any way to change the procedure to get a more accurate answer?
4. Can any error in one place be offset by another error somewhere
else? (For example starting a calorimeter experiment below room temp
and ending above room temp by the same amount to cancle heat
exchange.)

my 2 cents

kyle

> --Boundary_(ID_cMjr0TIr97iSwlxAUp0xdg)
> Date: Thu, 8 Mar 2001 13:08:35 -0500
> From: "John S. Denker" <jsd@MONMOUTH.COM>
> Subject: Re: error analysis (was: middle school science...)
> MIME-version: 1.0
> Content-type: text/plain; charset=us-ascii; format=flowed
>
> Hugh wrote:
>
> >  The other quibble has to do with the idea of "expected results." I
> >  have a running battle with our chemistry teachers who seem to have
> >  the idea that their students can do error analysis in their labs by
> >  simply calculating the difference between the answers they obtain and
> >  the "correct" answer--that is, the answer in the text. This is to
> >  error analysis as "paint by numbers" is to art.
>
> And then at 10:24 AM 3/8/01 -0600, RAUBER, JOEL offered a small difference
> of opinion:
>
> I believe that when measuring a known quantity, where there exists a
> nominally correct value, that calculating per cent error is a legitimate
> part of the error analysis.  It provides a good quick reality check on what
> you are measuring and can often quickly reveal calculational mistakes.
>
> I hasten to add that this isn't and should never be the complete error
> analysis!
>
> Let me take the thought one step farther:
>
> Consider the situation where we are measuring the properties of, say, a
> real spring.  Not some fairy-tale ideal spring, but a real spring.  It will
> exhibit some nonlinear force-versus-extension relationship.
>
> Now suppose that we do a really good job of measuring this
> relationship.  The data is reproducible to some huge number of significant
> digits.  For all practical purposes, the data is exact.
>
> Next, suppose we want to model this data.  Modelling is an important
> scientific activity.  We can model the data using a straight line.  We can
> also model it using an Nth-order polynomial.  No matter what we do, there
> will always be some "error".  This is an error in the model, not in the
> observed data.  It will lead to errors in whatever predictions we make with
> the model.
>
> Proper error analysis will tell us *bounds* on the errors of the predictions.
>
> Is this an example of "if it doesn't work, it's physics"?  No!  An inexact
> prediction is often tremendously valuable.  An approximate prediction is a
> lot better than no prediction.
>
> I mention this because far too many intro-level science books seem to
> describe a fairy-tale axiomatic world where the theorists are always right
> and the experimentalists are always wrong.  Phooey!
>
> It is very important to realize that error analysis is NOT limited to
> hunting for errors in the data.  In the above example, the data is
> essentially exact.  The spring is not "at fault" for not adhering to
> Hooke's so-called law.
>
> A huge part of real-world physics (and indeed a huge part of real life in
> general) depends on making approximations, which includes finding and using
> phenomenological relationships.  The thing that sets the big leagues apart
> from the bush leagues is the ability to make _controlled_ approximations.
--
-----------------------------------------------------
  kyle forinash , PhD
  kforinas@ius.edu 812-941-2390
  Natural Science Division
  Indiana University Southeast
  New Albany, IN    47150
  http://Physics.ius.edu/
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