Re: [Phys-l] Symbol for uncertainty

[James McLean <mclean@geneseo.edu>]

On 5/11/11 4:17 AM, John Denker wrote:
> ...
> Therefore we say u is a functional of X and denote it u[X] ...
> using square brackets to emphasize that it is a functional (not a mere
> function).  This is consistent with writing m[X] to represent the mean
> of the distribution X.
OK, now I feel better about the u[X] notation for uncertainty.  But it 
still strikes me as a little clumsy.

> 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))    ?

I don't think so.  The m[omega] version isn't any better.  When forced 
to be explicit, I use the variable with an overbar for the distribution 
mean, but even that is too much overhead (ooo, a pun) to maintain long term.

> 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.

> The one thing that is not a viable option is to use x as a shorthand
> for the distribution X.  I am quite aware that there are entire books
> on the subject of "random variables" that use this notation, but it
> is truly a terrible notation.  It is just begging to be misunderstood.
> It is doubly-especially unsuitable for use in an introductory course.
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?

We have already claimed capitalization as a meaningful element in 
assigning algebraic symbols for physical quantities.  Perhaps we can use 
adornment?
   distribution of period measurements: T-tilde
   measurements of period:              T_1, T_2, T_3, ...
   mean of the distribution:            T
   uncertainty of the distribution:     u[T-tilde] or u_T-tilde or ...
Is there any pre-existing standard?
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?

Cheers,
-- James