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Re: [Phys-l] Symbol for uncertainty



On 05/13/2011 07:18 PM, James McLean wrote:

Another option is to write X as a subscript, i.e. m_X ± u_X.

Notationally very consistent, which is nice.

:-)

But suppose that in an elementary lab I hang a spring, and measure the
elongation x that occurs when I hang a mass m1 (due to gravitational
acceleration g). I then wish to predict the natural angular frequency
when a different mass m2 hangs on the spring. All quantities have
associated uncertainties. Is anyone really going to write the equation
m_omega = sqrt((m_m1*m_g)/(m_m2*m_x)) ?

We don't write that and we shouldn't write that, because it is
(almost certainly) wrong physics.

Again, the underlying conceptual point is that we need to distinguish
a _number_ from the _probability distribution_ from which that number
was supposedly drawn.

An equation that holds for each observed value of x, m, and omega might
very well not hold for the estimated mean values m_x, m_m, and m_omega.

Do we need to notate the fact that in lab, we don't actually work with
the distribution mean, but instead an estimate of the distribution mean?

Yes.

Suppose we have an unknown Gaussian distribution X characterized by
its equally unknown mean m[X] and standard deviation sigma[X].
We then draw from it the observed value x1. In the absence of
other information, x1 is our best estimate of m[X], and indeed our
only estimate ... but it is not equal to m[X] ... and it may not even
be a good estimate.

As before, suppose we measure the pendulum period using a digital
stopwatch. The observed reading is x1 = 1.234 seconds. That is
the reading, with no uncertainty. I am absolutely certain that
x1 = 1.234 seconds.

As a separate matter, you may believe that this reading must have
been drawn from a distribution X with some mean and some standard
deviation. You may believe that the standard deviation of the
distribution X is on the order of 0.01 seconds.

In this situation, it would be quite abusive to write x1 = 1.234 ± 0.01
because the first part of the RHS represents the actual factual observation
x1, while the second part of the RHS represents something to do with the
distribution X from which you believe x1 must have been drawn. This is
like mixing apples and fertilizer. It may be that fertilizer is useful
for growing apples, but it is not ordinarily a good idea to mix the
apples with the fertilizer.

For this theoretical reason, and for about a dozen practical reasons,
I recommend writing simply x1 = 1.234 ... and then *separately*
writing down whatever we know (or believe) about the distribution X.

The +- notation is no help at all.

To put another way, I am NOT asking about how people like to notate
uncertainty of a specific measurement.

You miss the point. The ± notation works just fine for non-specific
algebraic quantities.
-- We agree it is well established to write {1.234 ± 0.010} to
describe a specific distribution
-- It also works just fine to write {Fred ± Wilma} to describe
some some not-yet-specific distribution. The name of the
variable that describes the uncertainty is not "±"; the name
of the variable is Wilma. You can also express the same thing
as NormDist(, Fred, Wilma) or in many other ways.

Another option is simply to write A ± B and then explain that A is
the nominal value of the distribution X while B is the corresponding
uncertainty.

Nearly useless for teaching the mathematics of uncertainty propagation.
You really need a notational system that preserves a reminder of the
referenced quantity.

That could go either way. The best-practices policy for this is well
understood in the computer programming business:
-- Short names like A and B work fine when they have limited
scope. They go out-of-scope before you forget what they mean.
-- Slightly longer and and more meaningful names can be used when
the scope is slightly longer. If B is not a sufficient reminder
of A, then Wilma may serve as a better "reminder" of Fred.
-- Names with larger scope, especially global scope, need much
more explicit names, e.g.
MyModel::Earth::Orbit::radius__m::mean;
MyModel::Earth::Orbit::radius__m::stdev;

Note the suffix "__m" means "measured in meters".

Note that class declarations and namespace declarations can be used
to keep explicitness from becoming onerous in ordinary applications.
That is, if the "referenced quantity" is the thing mentioned above,
we might write
{
using MyModel::Earth::Orbit::radius__m;
mean = 1.49597870691e11;
stdev = 0.00000000006e11;
y = NormDist(x, mean, stdev);
}

For more on this style of naming, see
http://www.av8n.com/computer/htm/documenting.htm

Real-world problems are almost always sufficiently complex that no
standard notation suffices. Also, as the saying goes:
The nice thing about standards is that
there are so many to choose from.
Use the standards as a source of good ideas and as a starting point,
and then improvise as necessary.

Names should usually be kept short. Therefore the /meaning/ will
never be self-evident. Therefore you should write a _legend_ and
spend a few words, or a few sentences, or a few paragraphs explaining
what each name means.

On the other hand, using capitalization to distinguish between the
variable and the distribution is dead on arrival for physics. Do you
suggest that multiple measurements of a pendulum period T be represented
by t_1, t_2, t_3?

You are making conflicting demands:
-- You want the notation to be concise, without "overhead".
-- You want to keep standard symbols including capitalization.
-- You want to unambiguously represent several different things.
-- You want related things to have similar representations.

Those are noble goals, but they cannot all be achieved at the same
time. It is provably impossible. A certain amount of "overhead"
is the price that must be paid to keep things unambiguous.

The appropriate amount of overhead is partly a matter of convenience
and a matter of taste, and will vary from situation to situation.
De gustibus non disputandum.

Note that given a set of observations x1, x2, x2, et cetera, you
could easily have more than one distribution you need to talk about,
including perhaps an a_priori distribution on x, an a_posteriori
distribution on x, and who-knows-what else. So the idea of
automatically deriving the name of "the" distribution from the
name of the variable might work sometimes, but it definitely
won't work in general.

for teaching the mathematics of uncertainty propagation.

I'm not sure whether this is central to the discussion, or merely
tangential ... but here goes: Step-by-step propagation of uncertainty
is almost always a bad idea. It almost never gets the right answer
except in trivial cases. It has been obsolete for at least 65 years.
Monte Carlo is easier and better. Much, much easier and much, much
better.
http://www.av8n.com/physics/uncertainty.htm#sec-crank3