[Phys-l] formatting uncertainties (was: New gravitational constant)

[John Denker <jsd@av8n.com>]

In the context of  
  6.67255 ± 0.001          [1]

On 01/22/2008 11:42 PM, Savinainen Antti wrote:
> ... So it is taught even in high school that you are not
> supposed to report a number like that 

OK, we all agree that students *are* taught in high school
not to report numbers like that.

My question is, _why_ are they taught that?  I don't see 
anything wrong with such an expression, and indeed there 
are situations where such an expression is entirely 
appropriate, as discussed below.

  Note that for present purposes, it is irrelevant whether
  expression [1] is the correct value of "G" or anything
  else.  That is, the semantics of expression [1] are
  irrelevant.  The topic of discussion is whether expression
  [1] is _syntactically_ acceptable or not.

The conceptual foundation for any understanding of uncertainty
rests on the mathematics of probability.  An expression such
as [1] represents an probability distribution, not a simple
number.

So my question can be restated in more quantitative terms:
Supposing that we have a distribution with mean of 6.67255
and standard deviation of 0.001, why should anyone be told
they are not supposed to report it as having that mean and 
that standard deviation?

Subsidiary questions include
 -- Is there a rule that says "anything that can be rounded
  should be rounded"???   Where did that rule come from?
  Is there any reason to believe that is a wise rule?

 -- Is there any reason to believe that in this context
  6.67255 can safely be rounded?  How do you know?  How 
  sure are you of that?


On 01/22/2008 09:09 AM, Todd Pedlar wrote in part:

> > .... would you care to comment on
> > the significance or utility of quoting those last two decimal
> > places when the uncertainty is in the 3rd place after the decimal?

My comment is simply that the distribution .67255 ± 0.001 
is a different from the distribution .6726 ± 0.001 and even
more different from .673 ± 0.001.

When I plot the distributions in the usual way, the difference
between .67255 ± 0.001 and .6726 ± 0.001 is readily perceptible
to the naked eye.  If somebody is unable to perceive the difference,
that's his problem, not mine.

Whether the difference between .67255 ± 0.001 and .673 ± 0.001 
is important in practice depends on as-yet-unspecified details 
of the situation.  Practical significance is *not* the same as
measurement uncertainty.  In some situations it is perfectly
practical to round off pi to one decimal place, even though
the value of pi is known to considerably more decimal places
than that.

Conversely and more importantly, it is quite possible for something 
to have practical significance even when it is quite small compared 
to the uncertainty.  Signal averaging is an elementary example.  
Also there are guys at JPL whose job it is to receive signals that 
are many,  many dB below the noise.  Other examples abound.
  http://www.av8n.com/physics/uncertainty.htm#sec-significance

 -- The cost of carrying extra digits is usually zero
 -- The cost of over-eager rounding is often catastrophic.

There are other reasons to eschew rounding.  As it turns out,
the constant G is the veritable poster child for correlated
uncertainties, as discussed at
  http://www.av8n.com/physics/uncertainty.htm#sec-correlated

  Consider the following example: Newton’s constant of universal 
  gravitation, G, is known to about 100 ppm. The mass of the 
  earth, M, is also known to about 100 ppm. So far so good. 
  The tricky thing is that the product GM is known to about 
  2 parts per billion.

  Now suppose you have a value of G and a value of M that are 
  consistent in the sense that when multiplied together, they 
  give the correct value of GM, accurate to 2 ppb. If you round 
  off G and/or M to four or five digits, on the theory that 
  that matches the uncertainty of those quantities, you will 
  very seriously degrate the accuracy of the product GM.

All in all, tt seems to me we should be encouraging students 
(and everyone else) to carry plenty of guard digits.

I would never complain about an answer of the form 6.67255 ± 0.001.
If a student (or anybody else) works out an answer that comes
to 6.67255 ± 0.001 they should report it as 6.67255 ± 0.001
unless they are sure it is safe to round off ... and figuring
out what's safe and what's not is usually very hard, well
beyond the scope of an introductory course.

For details on all of this, including guidelines for how to
do things properly, see
  http://www.av8n.com/physics/uncertainty.htm