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



On 03/13/2012 06:41 AM, Folkerts, Timothy J wrote:

I question the value of
http://www.av8n.com/physics/img48/gaussian-roundoff.png for
illustrating "3.8675309 ± 0.1".

My concern goes back to the data that would have generated the graph.
Certainly if the EXACT curve is a Gaussian with a mean of exactly
3.8675309 and a standard deviation of exactly 0.1, then plotting the
exact curve will give the best fit.

But in "real life" we would never know the curve ahead of time. To
get such a curve experimentally, you would measure many individual
values. Then you would estimate the center. And the standard
deviation. And the uncertainty of the center. And the uncertainly
of the standard deviation. And whether the data is truly Gaussian.

Getting such a smooth curve experimentally would take a HUGE number
of data points -- let's say 1010. Then the uncertainly of the mean
would be around 0.1 / (1010)0.5 = 0.000001. The error in the
standard deviation is also of order N-0.5. So you are really
illustrating something more like "A distribution with a mean of
3.8675309(10) and a standard deviation of 0.100000(1)"

If you generate only 100 or 1,000 points randomly from the
appropriate distribution, I suspect you would have a hard time
telling which fit the data better.

Sorry, that argument is invalid and the conclusion is wrong.

The argument is invalid because it is circular, or worse. It
assumes I know a number such as 3.8675325 "exactly" ... but I
just got through saying that the uncertainty is 0.1.

I reckon the only reason anybody would think I knew it exactly
is because of the many digits. However, I have said over and
over that when I write a number, the number of digits doesn't
imply anything about the uncertainty, significance, tolerance,
or anything else. Please believe me when I say that. You can't
convince me that sig figs are a good idea if you start by assuming
that I am already using sig figs notation ... especially when I
have repeatedly and vehemently denied it!

So, you may ask, where did all those trailing digits come from.
Obviously, they did not come from an "infinite" number of
measurements.

Here's a plausible scenario.

Suppose, based on ten measurements, I determined x to be
drawn from a certain distribution, namely 3 ± 0.1 (and no,
I am *not* going to write that as 3.0 ± 0.1).

Next, I measured a new variable, y1, and determined it to be
equal to x + 334/385. I don't have to expand that in decimal
notation, but if I do, it comes out to 3.8675325 ± 0.1.

Before you rush to round that off, I should tell you that I also
measured y2 and found it to be equal to x + 335/385. If I expand
that in decimal notation, it comes out to be 3.8701299 ± 0.1.
Neither y1 nor y2 is known to any great accuracy, but the difference
y2-y1 is known much more accurately. Indeed y2-y1 is equal to
1/385 which comes out to approximately 0.002597402597403(/) ...
so in fact rounding y1 and y2 to "only" seven places loses some
accuracy. Whether or not this loss is /significant/ depends on
how the numbers will be used, which we have not yet discussed.

Situations like this -- where there is correlated uncertainty --
come up all the time, in every field from astronomy to zoology.

I'm not kidding about that. The last time I was doing zoology
fieldwork, we were keeping lat/lon coordinates in a geospatial
information system (GIS). We recorded y1 and y2 with IIRC ten
or eleven decimal digits ... even though neither y1 nor y2 could
possibly be known exactly, because of uncertainty in the datum
if nothing else. Continental drift and all that. However, the
displacement between points (i.e. y2-y1) was known more accurately
than the datum was known, and was relevant.

So, folks should think twice before you call us "numerically
ignorant" for keeping lots of digits. That shoe is on the other
foot, and provably so.

On 03/13/2012 08:37 AM, Jeffrey Schnick wrote:
Harry Meiners taught us to write down as many more digits as we
wanted and put a bar over the first one that was just an estimate
(typically but optionally extending the bar over all the digits less
significant that that one too). One could have plenty of guard
digits while still conveying a crude estimate of the uncertainty.
The method eliminates all the bad things about the significant
figures method and still allows one to convey a rough estimate of the
uncertainty. I don't know why it (or something similar--an
underscore is easier to type than an overbar) isn't more widespread.

Although that solves several of the worst problems with the sig
figs approach, it doesn't solve them all.

In particular, it still means you can only express the uncertainty
to the nearest power of ten. Often the uncertainty is known much
more accurately than that. As a specific example NIST reports the
charge on the electron as
1.602176565(35) x 10^-19 C
http://physics.nist.gov/cgi-bin/cuu/Value?e

Using a bar, you'd be hard pressed to communicate the uncertainty
so as to distinguish between 35 counts in the last place versus 15
or 25 or 45 or 55.

Perhaps most importantly, it still leaves us with no way to handle
the situation where you need to write down a number, but you simply
DO NOT KNOW the uncertainty.