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Re: [Phys-l] projectile motion lab



I don't know about authoritative, but the book required for the Advanced lab (UCSC) is, An Intro. to Error Analysis -- The Study of Uncertainties in Physical Measurements, John R. Taylor W/ the famous pic. of the train out the front of the GARE DE L'OUEST on the cover. 1982 (270 pp.)

Standard deviation, 84--87 and 13 sub topics.

Std. d. of mean 89--90
justification of, 127--130

bc, who remembers JD recommends Bevington

p.s. Taylor should appeal to the maths minded as it's not just a handbook, but includes proofs (tho. brief?).



Jack Uretsky wrote:

Hi all-
I just nodded in on this discussion. Can someone direct me to an authoritative sork on statistics that distinguishes between "standar deviation" and "standard deviation of the mean" ?
Regards,
Jack



On Tue, 10 Oct 2006, Brian Whatcott wrote:


At 04:25 PM 10/9/2006, Krishna, you wrote:


On 10/9/06, Folkerts, Timothy J <FolkertsT@bartonccc.edu> wrote:


I'm interested. What is the mean time and the standard deviation
using this data set? Also, I suspect that you should use the
standard deviation of the mean as your uncertainty.

I don't think the standard deviation of the mean (aka "standard error")
is what is needed here. The st. dev. of the mean tells you how well you
know the mean value, but that isn't what is important. You want the
effect of the variation of the individual trials, which is related to
the plain old standard deviation.

Now that you point this out, I'm not sure what to do.

Let's consider Justin's original experiment, where he has 50 measurments of
the time it takes for the marble to roll between two photogates. (Other
posters have pointed out the need to be careful with photogate measurements,
but I don't have much to say about that.) From those 50 measurements of
time T, we can calculate the mean <T> in the standard way. We can
calculate the standard deviation s in the standard way. From this, I can
further calculate the standard deviation of the mean as s/Sqrt(N) where N is
the number of measurements.

If I were going to report the "time it takes for the marble to roll between
the two photogates", I would report <T> +/- s/Sqrt(N). I hope that is
correct.

In propagating uncertainty through the calculation, should I use s as Tim
suggests, or should I use s/Sqrt(N) as I originally thought? Another way to
think of this: If I were going to do a Monte Carlo error analysis assuming
a Gaussian distribution, would the spread in my Gaussian be the standard
deviation or the standard deviation of the mean?


The standard deviation is related to

how close the center of the trials would be to the "true" value.


Here I think it should say "standard deviation _of the mean_ is related to
how close the center (which I understand as the mean) of the trials would be
to the "true" value." ? (seems like just a typo)

--
regards
-Krishna

Krishna Chowdary

I found this question interesting. From the follow on thread, I can see
that you are not offered a single incontrovertible answer.
Here is what I started to ponder....

If I want to report an experimental result with error bars it
is respectable to use the mean and standard error of the (sample) mean.
(I think!)

But you follow with a concern about uncertainty and propagating values
through a computation.
This seems to have the object of noting a computed result along with the
confidence you have in quoting the range of the answer.

When I put it this way, I see that the range I offer depends
upon the confidence with which I offer that range.
Hence, if I want to provide the highest confidence that my range
reflects the data, I would quote the widest range.
How would that wide range be computed?
To include the effect of rare outliers, I might want to multiply the
standard deviation of individual components of the computation
in the usual way to cover an increasing proportion of the set.

To take an obvious case: a computation consisting of A - B
where A is an established constant, and B is an experimental observable
might give hugely different results for A close to B
with the experimental extremes in B accounted for.
This does not even mention the possibility that the distributions
in question are skewed and non-Gaussian....


Sadly, this means that I conclude that the answer to Krishna's question is...
"It depends".

p.s. Supposing a Gaussian distribution for a Monte Carlo
seems circular to me: what am I missing?


Brian Whatcott Altus OK Eureka!


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