Re: Significant figures - a Modest Proposal

[John Denker <jsd@MONMOUTH.COM>]

At 07:32 PM 8/26/99 -0700, Leigh Palmer wrote:
> Has anyone got the impression I've had for years that teaching the
> topic of significant figures is counterproductive?

Exactly so!

In its usual form, fussing with sig figs does some harm and some good, but
more harm than good, IMHO.

> [we care about] the *uncertainty* in the result,
> *not* the "number of significant digits".

Exactly so!

> The big problem with rules for significant digits is that the number is
> necessarily quantized. Uncertainty is not quantized to factors of ten.

Right!  I don't even know what "XX significant figures" means.  The number
7.35 could mean 7.35 (+- 1 in the last place) or 7.35 (+- 5 in the last
place) or who-knows-what.

> I could construct many examples of the application of significant digit
> rules where the baby gets thrown out and the bathwater gets retained
> because of these rules, but I won't.

One can harvest examples from the article on Physical Principles and
Concepts in _Encyclopedia Britannica_.  It turns out that many important
physical quantities have been measured so carefully that their
*uncertainty* is known to two significant digits (if I may use the concept
to undermine itself).  Specifically, the charge on the electron is given as
        1.6021892(46) e-19 Coulombs

> .... the
> precision-losing practice of writing down intermediate results in a
> calculation and then reentering them from the keyboard, often after
> rounding and truncating. Surely this is an even worse sin.

Exactly so!  In the field of numerical analysis, it is mandatory to keep
_guard digits_, so that the inevitable accumulation of roundoff errors does
not degrade the final result.

Back in the days before calculators, it was unduly laborious to keep more
guard digits than necessary.  But these days, why not?  Calculating with
three guard digits is typically no more expensive than calculating with one
guard digit.

> the
> significant digit rules are not part of the laws of Nature, and they
> play no part in reporting results in the scientific literature.

That's a slight overstatement.  Talking about significant digits, flawed as
the concept might be, is part of the everyday language of real scientists
and engineers.  Someone who didn't know the lingo would be at a
disadvantage in the real world.


===============

Take-home messages:

*) Having too many digits is *way* better than having too few.

*) The notation we use today was not handed down to us on stone tablets.
It was constructed by hard-working clever people over the years.  It is
by-and-large very good, but not perfect.
  --) If you think the notation of vector calculus is confusing, you should
see what Maxwell had to go through to do electromagnetism *without* modern
notation.
  --) Even today, the notation for probabilities is so imperfect that it
interferes with many people's ability to reason about probabilities.

*) The professional ways to quote the uncertainty are explicit:
        7.35(7)   or   7.35 (1%)

*) There are many advantages and few disadvantages to keeping some guard
digits:
        7.355(70)  or   7.355 (1%)

*) Quoting something to N decimal places is a common but clumsy figure of
speech and shorthand for saying that the mean is the quoted number and the
standard deviation is a "few" counts in the Nth decimal place.  Quoting
7.35(7) as 7.35 arguably understates the standard deviation, but rounding
it off to 7.3 introduces a significant and needless error in the mean.