Re: Measurement and Error

puppet_sock_at_hotmail.com
Date: 10/06/04 

Date: 6 Oct 2004 07:07:51 -0700

jhelfand@umd.edu (Joe) wrote in message news:<e62610ea.0410052227.1ffdc657@posting.google.com>...
> Someone told me whenever you make a measurement, you need to say what
> the uncertainty is. (Do you agree?!) If I mesaure with a ruler,
> what's the uncertainty?

Ok, first you need to go back and complain to your 8th grade
science teacher for not teaching you this. I recall many
tedious hours spent measuring the football field using first
a 1 cm ruler, then a 1 m ruler, then a 30 meter tape measure.

All measurements of real quantities (as opposed to digital
quantities such as "how many lions are there in my bathtub?")
involve uncertainty. Without some indication of the accuracy,
the measurement is less useful than it could be, possibly even
totally meaningless. (Digital quantities involve uncertainty
as to identifying or missing counts. "Is that a lion? Are
there any lions hiding under the soap? Is that lion actually
in the tub or do we count that as in the soap dish?")

The typical uncertainty in something like a ruler is half the
smallest division on the scale. However, there can easily be
more uncertainty than that. If you were trying, say, to measure
the height of water up the beach, then you'd be dealing with
waves, rough beach shape, etc. So you might be measuring with
a ruler with 1 mm divisions, but you might only be accurate to
a few cm if there were waves moving the water up and down that
much. Or to a few 10's of cm is the waves were bigger, or a few
meters if the waves were bigger. And so on.

Uncertainty is a fundamental of measurement and science. You have
to know how accurate a measurement is. If the measurement is
reported as 1.554243 m, then the default uncertainty is plus or
minus 1 in the last digit, or +/- 0.000001 m. Those who have
pocket calculators will often report some calculation as though
it had accuracy to the last digit. "One seventh of my 4.5m board
is 0.64285714285714285714285714285714 m." Clearly, if I'm using
a typical wood saw, I'm not really getting those last digits,
probably no more than an accuracy of +/- 1mm. The same is true
of all scientific measurements, and all calculations based on them.

It is fairly frequent to meet silly mistakes in uncertainty reporting.
Pocket calculators are just one. Recently I was reading a report about
a nuclear reactor. The dimension of a part was reported as 4.5 feet.
But we've all gone metric, so that had to be in meters. This was
dutifully reported to 5 significant digits, 1.4 etc. meters.

Uncertainty can arise through standard things like reading a scale.
It can arise through physical process that jumps around, like waves
messing up a water level measurement. It can arise through
repeatability issues, such as manufacturing with impurities, flaws,
variations in temperature, etc. And, at very fundamental levels,
it can arise from quantum mechanics.

Uncertainty means we have only a finitely accurate view of reality.
This means that, in the very typical case, there are many theories
that cannot be resolved with that accuracy. If two (or more) theories
manage to get inside the error bars on all measurements, then there
is no physical reason to prefer one over the other. There are usually
reasons of simplicity to prefer one. Or reasons of prior art, or
existing mathematical work, etc. Thus, at this time, the standard
theory of gravity is general relativity. There are other theories.
But so far, GR has worked every time we've tested it, and most of
those other theories are more complicated.
Socks

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