Re: [Phys-l] projectile motion lab

[Brian Whatcott <betwys1@sbcglobal.net>]

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!