Error bars

[LUDWIK KOWALSKI <KOWALSKIL@alpha.montclair.edu>]

As posted on  Fri, 24 Oct 1997

> > The goal of my activity was to show that the acceleration of free fall
> > does not depend on speed, for example. Fluctuations of the order of 50%
> > and more (big error bars) defeat the purpose. The shutter speed was 1/1000.

> It doesn't matter that large error bars defeat the purpose. If they exist 
> they should be acknowledged by showing them. In the cases where they are 
> small that should inspire confidence in the results.

Hwo can deny this? The trouble is, as outlined by Brian, in that the 
evaluation of errors is not at all trivial in this case. It would be trivial 
if we could ASSUME that the TRUE value of g is very close to 9.8 m/s^2 and
that the TRUE nature of the process is that g is practically independant
of y or v. In that case the seven values of differences (between what is
measured and what it should be) could be used to calculate the variance.
The error bars equal to SQRT(variance) would be used on a graph of a(t).

But we do not know what the true value is and we can not assume that the
acceleration is constant (without defeating the purpose of the experiment).
So how do decide about the error bar if we know very little about what
happens in the camcorder. 

Consider a simplier situation. We have no idea how the length of a rod
depends on the temperature and we decide to find out. We take a yard
stick and find that L=20.0", 19.2" and 22.1" (inches) at three different
temperatures. The division lines on the yardstick are 0.25" apart and
fractional digits, in the above results, are our "best estimates". How 
can we reach an agreement on the bars of errors? One way is to use the
instrument many times at each temperature, use the mean value instead
of the true value and to calculate standard deviations. What is the 
equivalence of this for the measurements of y with a camcorder? It is 
not easy to impose nearly identical conditions without synchronizing
the camcorder (at the microsecond's level) with the relasing mechanism.

Back to the easier rod example. Suppose that reproducibile conditions
can not be created for some reason. We would probably agree that a
typical error bar in a quick measurement is less than 0.2". And we would 
not argue that it must be larger than 0.02" (naked eye, parallax, etc.).
But should it be 0.1? Why not 0.12"? Why not 0.05? 

Or, in the case of our camcorder --> ten pixels? five pixels? two pixels?
Intimate knowledge of the whole recording process (from tiny pixels in the
focal plane to the pixels on my monitor) is necessary to decide about the 
error bars in y. And even then the decision is arbitrary. Once the standard
error in y is decided upon we have a less complicated problem of the so-
called "propagation of errors" (from y to v to a). Keep in mind that each
value of a is calculated from a difference of differences in y.

                                                     Ludwik Kowalski
P.S.
There is a very good article in The New York Times today (strating on the
front page, October 25, 1997) entitled "Internet's Value In U.S. Schools
Still in Question". Interesting statistical data and a lot of problems to
think about. Amy Harmon is the author.