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Re: [Phys-l] Significant figures -- again



On Mar 12, 2012, at 1:09 PM, I wrote in part:

2) He or she is demonstrating his or her numerical ignorance.

I do not think I am numerically ignorant.

On 03/12/2012 05:10 PM, John Mallinckrodt responded:

I couldn't agree more so I should probably have added that I'd be
utterly flabbergasted to hear you quote a length as 3.8675309 cm if
you only knew it to within a few mm.

Sometimes a picture helps explain things. Try this:
http://www.av8n.com/physics/img48/gaussian-roundoff.png
That is a graph of the original distribution
3.8675309 ± 0.1 solid blue line
and two other distributions derived from that by rounding off:
3.87 ± 0.1 dashed yellow line
3.9 ± 0.1 dotted red line

As I see it, the blue curve is the best. It is the most faithful
representation of the real, original distribution.

As I see it, the dashed yellow curve is not better, but it's not much
worse. Its Kullback-Leibler information divergence (relative to the
blue curve) is about 0.0003.

The dotted red curve is clearly worse. You can see at a glance that
it represents a different distribution. It's K-L information divergence
(relative to the blue curve) is more than 0.05.

To put it another way, are you really arguing that the blue curve
represents ignorance and illogic? Please tell us, in what way is
the blue curve worse? To what extent are you harmed by the blue
curve?

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

Remember the quote from G.B. Shaw:

Actress: What kind of woman do you think I am?!
Shaw: Madam, we've already established that.
Now we are haggling about the price.

I mention that because AFAICT we have already established that the
textbook rules about sig figs are grossly wrong and destructive.
Now we are haggling about how many guard digits to keep.

Even the sig-figs supporters are saying the textbook rules should
not be used for intermediate results, and even for "final" results
(whatever that might mean) one or two guard digits would be a good
idea.

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

Just today -- literally this afternoon -- I was working a chemistry
problem. I needed to evaluate the Moore-Penrose pseudo-inverse of
a non-square matrix. Rounding off to machine precision (at double
precision, roughly 15 decimal digits) was causing me grief. I was
able to work around it by complexifying the algorithm and fussing
with one of the parameters, but doing so cost me time and effort, and
made the product less robust and less user-friendly, including less
understandable to students and others who might be interested.

That and about 100 other experiences over the years, all leading by
different means to the same conclusion, are IMHO sufficient reasons
for me to not round anything off unless there is a clear objective
reason for doing so. Appeal to the authority of the textbook is not
a sufficient reason, especially when the textbook rules are obviously
wrong by a wide margin.

If I want to read about roundoff errors, I'm not going to rely on
some high-school physics text written by a highly skilled cartoonist.
I have other books, including books with "Numerical Methods" in the
title. I also have my own quantitative observations to draw on.

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

Also, FWIW: Students know that the sig figs rules are a crock. Even
at the high-school level, they can figure out that following the textbook
rules seriously degrades the accuracy of the result. So you're teaching
kids stuff they know can't possibly be true ... and that's a problem,
for them and for you.

They probably won't /say/ anything about it, because they know that any
sign of critical thinking during school hours will just get them into
trouble ... but that means you have two problems.