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[Phys-L] Re: infinite sig. figs.

After reading John Mallinckrodt's recent post I think we are near
agreement.

I'm sure that any disagreement is really mostly a result of thinking
about slightly different circumstances.

I agree that rounding takes place immediately prior to reporting. I
thought reporting was what we were talking about.

As did I. And my feeling is that the "rules of sig figs interpreted
a little loosely" are almost always sufficient to guide reporting in
situations like homework in introductory course work.

When I say, "interpreted a little loosely" I mean to say, that I am
not going to get my knickers in a knot over whether the result of
multiplying 1.07 and 8.96 should be reported as 9.59 or 9.6. The
rules interpreted strictly would argue for the former; I would argue
for the latter; I would accept either. I might even accept 9.587,
but I would certainly say something about it. I would NOT accept 10
or 9.58.

... The rub comes when the student *reports* a number that is rounded too
much, or not enough, and their question is how to determine what is too
much or not enough.

On homework I would use the "rules of sig figs interpreted a little
loosely" as explained above to determine how much to round. When
making primary measurements in lab, however, there is simply no way
to rationally determine an appropriate number of sig figs short of
having some idea about uncertainties.

Here's an example that comes up all the time in one of our labs. We
use the circular motion apparatus described in this procedure from
NJIT:

<http://physics.njit.edu/classes/physlab/laboratory111/lab114/lab114.htm>

A mass is attached to a spring and rotated in a horizontal circle.
Students adjust the rotation frequency so that the spring stretches
and the mass JUST makes contact with a pointer. They continue making
very small adjustments for a minute or so, timing the procedure with
a stopwatch while the apparatus counts rotations. In order to find
the rotation period they perform the obvious division. In order to
determine the uncertainty in the rotation period, they perform the
procedure several times. Here is a not so unusual set of data:

time (s) # of rotations
61.23 587
60.37 579
61.80 592
60.63 582

Students often worry about the fact that the numbers are so
different, varying in the second digit and decide, either on that
basis if not "just because", to calculate their periods accordingly
getting

time (s) # of rotations period (s) (or even) period (s)
61.23 587 0.104 0.10
60.37 580 0.104 0.10
61.80 595 0.104 0.10
60.63 581 0.104 0.10

The whole point of performing the procedure several times was to
obtain an idea of the uncertainty in the period and at this point
these students often are either completely confused or convinced that
the uncertainty is zero.

Obviously, however, they simply haven't kept enough digits to see
where and how much the values vary. Doing so yields

time (s) # of rotations period (s)
61.23 587 0.10431
60.37 580 0.10409
61.80 595 0.10387
60.63 582 0.10418

Now one see that the data is suggesting that the period is something
like (0.1041 +/- 0.0002)s and that the result is a number with
something like four sig figs.

... John said he would reject reports of 34.879 and 34.638 if the
uncertainties are 0.04 and 0.03. If that's the case then John would
reject many of the published numbers from NIST because they routinely
report one digit past the most significant uncertainty digit.

Not guilty as charged. I would most definitely reject the reports of
34.879 and 34.638 precisely because the uncertainties were given as
.04 and .03 and NOT, for instance, .043 and .029. NIST routinely
reports uncertainties to two sig figs. Students do too, but they are
almost always VERY wrong to do so. NIST routinely performs careful
enough experiments to be somewhat confident of their uncertainties to
two sig figs. That never, or almost never happens in the
introductory lab. Indeed, I would (and do) argue that uncertainties
in introductory labs are rarely known even to one sig fig. I am
often willing to interpret a quoted uncertainty of 0.05 as meaning
"an uncertainty somewhere between .1 and .02." It's quite a lot like
astrophysics. ;-) Just about the only time I am willing to consider
uncertainties reported to two sig figs is when the most significant
digit in the uncertainty is a 1 and the second one is a 4 or a 5.

Finally, to the following I can only say "Amen" (even if I wouldn't
want anyone to get the wrong idea.)

I spend a lot of time trying to get students to ask themselves what they
want to demonstrate and whether their data demonstrate that or something
different. If they are not sure, I want them to think about what new
data or new experiment they could do that would help them determine what
the data are saying. For example, sometimes when a student wonders if
the glider's loss of velocity is due to the glider going a little bit
uphill rather than from air friction, I ask if they can think of any new
data that might help answer that. Some will think of the idea of
keeping the track the way it is, but run the glider in the opposite
direction. Others have to be led to that idea by a series of questions
I ask them.

I sure would be happy to report that I have a lot of success turning
students into good experimentalists and good data analyzers, but I would
not be telling the truth. I do, of course, have some successes.

--
John "Slo" Mallinckrodt

Professor of Physics, Cal Poly Pomona
<http://www.csupomona.edu/~ajm>

and

Lead Guitarist, Out-Laws of Physics
<http://www.csupomona.edu/~hsleff/OoPs.html>
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