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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.
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.