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*From*: John Denker <jsd@av8n.com>*Date*: Wed, 24 Oct 2012 00:37:53 -0700

On 10/21/2012 02:23 PM, Folkerts, Timothy J wrote:

in many (all?) cases, the individual data point itself can be

considered to have been drawn from a random distribution. For

example, suppose I measure a voltage as 8.00 V using a specific

voltmeter. The manufacturer tells me that the accuracy of the meter

is 1 digit ± 0.5%. In effect, the manufacturer is telling me that if

I measured the SAME voltage with a set of their instruments, then the

readings of those meters would not all be the same and they should

range within 0.05 V of the "correct voltage". That single point is a

single instance of a value drawn from a random distribution. As

such, the error bars are telling us about the distribution of values

for that specific point, whereas the parameters of the curve itself

relate to a different random distribution.

I claim that there is not a "different random distribution" of the

kind described.

I've been thinking some more about this. I now have a concrete,

quantitative, objective, pedagogical example that illustrates

why it is a bad idea to associate error bars with the individual

reading. By "bad" I mean wrong in principle and demonstrably

problematic in practice.

Let's start with some "original" random distribution. In accordance

with the frequentist definition of probability, we make a humongous

number of readings, drawing them from this distribution. The result

is what physicists call an ensemble (and statisticians call a sample).

In the large-N limit, the frequency with which such-and-such event

occurs in the ensemble is the frequentist /definition/ of probability.

See e.g. Feynman volume I chapter 6.

If you plot the ensemble of points, it will have a certain spread.

It must be emphasized that the unadorned points -- without any alleged

error bars -- will exhibit spread. In fact they will exhibit exactly

the correct amount of spread. For instance, in the large-N limit, the

standard deviation of the ensemble of points will converge to the

standard deviation of the "original" theoretical ensemble. Other

statistical properties will converge as well.

My point is that if you attribute to each point some additional width,

perhaps by imagining that it wiggles around a little bit within its

error bars, you will get the wrong answer. It's wrong by a lot, as

you can see by glancing at this figure:

http://www.av8n.com/physics/probability-intro.htm#fig-frequentist-gaussian-recon-awry

The dashed red line is the right answer, while the black curve is what

you get by attributing error bars to the individual points. The black

curve does not converge to the right answer.

In contrast, if you treat the points as zero-size points, without any

error bars, the ensemble converges to the right answer. For details

see

http://www.av8n.com/physics/probability-intro.htm

especially

http://www.av8n.com/physics/probability-intro.htm#sec-convergence

That document is new and somewhat drafty. It will need to be rewritten

a few more times before it comes up to my standards. Questions and

suggestions are welcome.

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

I am quite aware that virtually everyone who ever took a high-school

chemistry course has been "taught" that every reading must have an

"associated" uncertainty.

That doesn't make it right. It's provably not right.

The uncertainty is a property of the distribution as a whole (aka the

ensemble as a whole) ... not of any individual point drawn from the

distribution.

**Follow-Ups**:**Re: [Phys-L] points don't have error bars (distributions do)***From:*"Folkerts, Timothy J" <FolkertsT@bartonccc.edu>

**References**:**[Phys-L] points don't have error bars (distributions do)***From:*John Denker <jsd@av8n.com>

**Re: [Phys-L] points don't have error bars (distributions do)***From:*"Folkerts, Timothy J" <FolkertsT@bartonccc.edu>

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