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Re: [Phys-l] Significant figures -- again



Once upon a time, a teacher had an ensemble of 2x2 matrices.
He assigned a team of students to calculate the eigenvalues
and produce a histogram thereof.

One student, Bob, worked all day calculating the matrix elements.
Another student, Carol, worked all night calculating the eigen-
values and plotting the histogram.

Bob found that all the matrices were of the form

[ 1.0012346 0.9987654 ]
M = [ ] [1a]
[ 0.9987654 1.0012346 ]

where each matrix element had an uncertainty of ±0.12 percent.
As another way of saying the same thing:

[ 1.0012346 0.9987654 ]
[ ±0.0012 ±0.0012 ]
M = [ ] [1b]
[ 0.9987654 1.0012346 ]
[ ±0.0012 ±0.0012 ]

For several reasons, Bob rounded off each matrix, in accordance
with the "textbook" sig figs rules:

[ 1.001 0.999 ]
M' = [ ] [2]
[ 0.999 1.001 ]

Bob's reasons included:
a) He felt obliged to communicate the uncertainty to Carol.
Writing down a large number of digits would "imply" (via
the sig-figs rules) a very small uncertainty, which in this
case would be quite wrong. It would be downright dishonest.
b) Equation [2] "looks nicer" than equation [1].
c) He knew the teacher would call him "numerically ignorant"
and would take off points if he quoted a bunch of trailing
digits, i.e. uncertain, irreproducible digits.

All in all, it was "obvious" to Bob that equation [2] was the
right way to express things.

That night, Carol calculated the eigenvalues for each matrix.
For example, the eigenvalues for matrix [2] are
ev1 = 0.002 [3]
ev2 = 2
as you can readily verify.

However, I happen to know (based on where the matrices come from)
that the actual distribution of eigenvalues is
ev1 = 0.002469(3) [4]
ev2 = 2.0000(24)

Carol's result for the small eigenvalue (ev1) is off by 190 sigma.

If Bob had broken the sig figs rules and kept 4 decimal places
(rather than 3), the small eigenvalue would have been off by
"only" 23 sigma.

If Bob had kept 5 decimal places, the small eigenvalue would
have been off by "only" 3 sigma.

If Bob had kept 6 decimal places, the small eigenvalue would
have been off by about 0.28 sigma ... a goodly fraction of an
error bar. This is arguably almost acceptable, but still a
bit larger than I would like to see.

Now you know why I quoted the matrix elements in equation [1]
using seven decimal places ... which is four digits more than
would be allowed by the sig figs rules.

DISCUSSION ==============

1) Situations like this come up all the time in the real world,
in many different situations, ranging from astronomy to zoology.

2) Indeed, this example is based on a real-world situation that
is much more extreme than what you see here. It took some effort
to simplify the example to the point where you could get by with
*only* seven decimal places.

3) The underlying issue can be summarized, roughly speaking, by
talking about "small differences between large numbers". More
specifically, if you have a small difference between large numbers
and the fluctuations are uncorrelated, it is a "noise amplifier".
In contrast, if you have a small difference between large numbers
and the fluctuations are highly correlated, there is relatively
little intrinsic noise, but you have a *roundoff error amplifier*.

I happen to know (based on where the matrices come from) that
the uncertainties in equation [1] are highly correlated.

4) In this example, we actually know the uncertainty. However,
we should also keep in mind that there are very many real-world
situations where you need to write down a number when the uncertainty
is NOT KNOWN ... and may not be know until weeks or months later,
if at all.

I have asked about this several times, and haven't heard anything
resembling a meaningful answer from the sig-figs advocates.

My advice is: If you have a number that ought to be written down,
write it down. Just write it down already. You can worry about
the uncertainty later, if necessary. Write down plenty of guard
digits. The number of digits you write down does not imply
anything about the uncertainty,precision, tolerance, significance,
or anything else.

5) Contrary to Bob's belief mentioned in reason (a) above, you
are not obliged to attach an implicit (or explicit) uncertainty
to numbers you write down. If you have an ensemble of numbers,
you /might/ be able to summarize it in terms of a mean and a
standard deviation, but you might not ... and even if you are
able to summarize it, you are not obliged to. The ensemble
speaks for itself, better than any summary ever could. In our
example, Carol can calculate the ensemble of eigenvalues just
fine without knowing the uncertainty of this-or-that matrix
element.

6) It really makes me cringe when students get points taken away
and get called "numerically ignorant" for doing exactly the right
thing, i.e. keeping plenty of guard digits.

7) You can adapt this example for use at the high-school level.
It would not be appropriate to talk about matrices, but the task
of finding the eigenvalues of a 2x2 matrix boils down to finding
the roots of a quadratic equation ... which is something students
learn in Algebra I.

Every student who has ever plugged uncertain numbers into the
quadratic formula knows that the sig figs rules are a complete
crock. What I wonder is, How come the teachers don't seem to
know this?


On 03/12/2012 09:04 PM, John Mallinckrodt wrote:

All duly noted and understood. And yet I'd still be flabbergasted.

Hmm. It seems a bit odd to "understand" several independent
lines of reasoning all supporting the same conclusion ... and
then reject the conclusion.