[Phys-L] Re: fall cleanup: sig figs

[jsd at AV8N.COM (John Denker)]

cliff wrote:

>  How can
> digits that carry no significance (meaning I have no idea what the real
> value in that place should be) guard anything?

There are several scenarios where this could happen.  One
of the simplest is as follows:

Executive summary:  guard digits don't make the noise on
the raw data any smaller ... but they *do* make the roundoff
errors smaller.

In more detail:
Suppose I have 10,000 raw numbers.  Each of them has 0.1 unit
of absolute uncertainty, uncorrelated and normally distributed.
Now I average them all together.  The average has only 0.001
unit of absolute uncertainty.

This is quite a bit trickier than non-experts would have
guessed, for the following reason:  Suppose we write each
raw number in the form Ai +- Bi, where Bi is the uncertainty.
Beware: uncertainty is not the same as roundoff error.  After
roundoff, we have something like Ai +- Ri +- Bi, where Ri is
the roundoff error.  It is easy to fall into situations where
even though the Bi are independent and normally distributed,
the Ri are all one-sided, and they accumulate like crazy.

Gaussian-distributed errors add in quadrature, while one-sided
errors just plain add, linearly.  Early in the game, the
randomness of the Bi smears out the one-sidedness of the Ri,
making the Ri much more random, but as soon as the signal-averaging
starts to take effect, the Ri become one-sided again, and
you're in big trouble.

There is a spreadsheet that implements a numerical example of
this, as discussed at
   http://www.av8n.com/physics/uncertainty.htm#sec-extracting

There are two sub-scenarios to consider.  As mentioned above,
and mentioned by Ludwik, you need to worry about this with
plain old raw physics data.

The other sub-scenario arises in computer programs, typically
when you are doing something like solving a differential
equation by time-stepping it.  You can improve your time
resolution by making the time-step smaller, but if you go
too far in this direction the roundoff monster will eat you.