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Re: [Phys-l] data analysis : pendulum period versus length



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From: John Denker [mailto:jsd@av8n.com]
To: Forum for Physics Educators [mailto:phys-l@carnot.physics.buffalo.edu]
Sent: Wed, 11 May 2011 16:44:35 -0400
Subject: [Phys-l] data analysis : pendulum period versus length

On 05/11/2011 04:29 AM, Josh Gates wrote:

> If I roll a pair of dice and observe five dots, then x_i=5 with no
> uncertainty. That works with dice, but how about the measurement of
> the period of a pendulum?

The dice and the pendulum are the same in principle. The
mathematics is slightly different for a discrete distribution
(dice spots) versus a continuous distribution (pendulum period)
... but only slightly. And you can simplify things by making
the pendulum measurement discrete: It might make a lot of
sense to measure the period with a digital stopwatch. The
data might look like:
x_1 = 1.235 (seconds)
x_2 = 1.244
x_3 = 1.253
x_4 = 1.238

Each observation is discrete. It is just as digital as the
number of spots on the dice. The same goes for measuring
the length: by the time you have written down the observed
length as a decimal number (or as a multiple of 1/16th of an
inch) it is a discrete observation. This is true whether or
not you think the underlying physical time and distance are
quantized.

Also notice that in my data tabulated above, the scatter is
huge compared to the quantization error, so the details of
how the quantization was done don't matter. This is how it
should be.That's a great point that I hadn't thought of. How about if we're measuring a distance? There's not any scatter there, really - I put down the ruler and get a measurement. I estimate the last fraction of a mm or fraction of an inch, so there's some uncertainty in that measurement, right? Can I say that my measurement is a distribution?

You see there is some scatter in the observations of the length,
and some scatter in the observations of the period. There is
also a model (shown in red) that tells us the theoretical
relationship of period versus length. The real question IMHO
is how well the data fits the model. More precisely, the
question is whether
a) the data fits the model /within the scatter/ which
means there is no sign of systematic error, or
b) there is evidence for some systematic error.
I love this. That's an entryway into comparisons of predictions and measurements with no math barrier to understanding, providing that we can get multiple measurements. I'll suggest this in the chem classes next year (where I'm lobbying to nix sig figs in favor of some thoughtful error analysis - which will _have_ to be simple and non-intensive on the math).



Note that the terminology is ambiguous here. Actually it's
worse than that, because _ambi-_ means two, and there are
more than two interpretations here:
-- The uncertainty of a single observation (which is zero).Is that always true? If I pace off 10 yards on the soccer field, I'm making a measurement, but it has considerable uncertainty (2-5%, in my experience). On second thought, that might be a crappy example. Hmmm...


-- The scatter, i.e. the empirical width of the /set/ of
observations.
-- The width of the underlying distribution from which the
observations were drawn. This may include contributions
from systematic error, which don't show up in the scatter.

As before, I emphasize that there is *no* uncertainty in
any single observation x_i. On the first observation, the
reading on the stopwatch was 1.235 and I am absolutely
certain about that. I'm absolutely certain about that reading, but it represents a range of possible values, right?

If at all possible, you are better off reformulating the
question so it doesn't look like a generalized comparison
between two distributions. In the pendulum example, the
reformulated question asks whether there is evidence for
systematic error. Best thing I've heard all week. Thanks!
jg




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