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Re: [Phys-l] formatting uncertainties



On 01/24/2008 07:23 AM, Spagna Jr., George wrote:
One of the "calculator challenges" is that students often don't know
how to use the calculator (or spreadsheet) efficiently. Writing down
intermediate answers, and then re-keying them for the next step of
the calculation will inevitably propagate truncation errors through
the operation.

I encourage students to use calculator memory to store intermediate
results for retrieval in subsequent stages.

This is excellent advice; see e.g.
http://www.av8n.com/physics/uncertainty.htm#hi-between-steps


I do, however, fall into the school of asking that students round the
final result appropriately,

"appropriately" is good. Now if we could just agree on what
is "appropriate" .......

with the understanding that those digits
in the answer which are to the right of the uncertainty are
effectively noise.

"Noise" is not a sufficiently precise term.
a) Is the claim that uncertain digits are uncertain?
b) Or is the claim that uncertain digits are insignificant
i.e. worthless?

The point is that (a) is true while (b) is false. Uncertainty
is not the same as insignificance!

Rounding to get rid of uncertain digits is a Bad Idea. Don't
do it. Don't permit (let alone require) your students to do
it.

The only reason a textbook exercise can get away with aggressive
rounding is because the whole exercise is academic in the worst
sense of the word; nobody really cares about the answer. They
only pretend to care. In contrast, for real-world data, where
folks actually care about the data, they would tar-and-feather
you for doing that to their data.

Did everybody notice that all of the fundamental constants on
the NIST site e.g.
http://physics.nist.gov/cgi-bin/cuu/Value?bg
are quoted with multiple uncertain digits? Haven't you ever
wondered if there might be a reason for that?

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

When I said there should be "no penalty" for carrying lots
of guard digits, I meant to say the teacher should never
impose an artificial penalty. This is a case where virtue
is its own reward, or rather vice is its own penalty. The
student who carries a ton of extra digits has already paid
the penalty of doing something slightly more laborious than
necessary. This very small penalty fits the very small crime.

I might help the student by hinting that it would have been
permissible to round off in this-or-that situation ...
while emphasizing that this is permissive, not obligatory
... and super-emphasizing that it would be a Bad Idea to
round to the point where roundoff error becomes a major
contribution to the overall uncertainty. The cost of
too few digits far exceeds the cost of too many.

Sorry for not saying this clearly the first time.

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

Back in the days before calculators, it might have
been necessary to round things right to the verge of
corrupting the data ... in order to get through the
calculation with any semblance of efficiency. But
that hasn't been true for 40 years. Wake up and
smell the silicon! Nowadays guard digits buy you
a lot of safety for a verrry small price.

Maybe in another hundred years or so the textbooks
will catch up with reality.