[Phys-L] Re: Average earlier or average later?

[ludwik kowalski <kowalskil@MAIL.MONTCLAIR.EDU>]

On Sep 9, 2005, at 11:42 AM, Pamela L. Gay wrote:

> Okay I'm game for a simulation. Played with excel for 30 seconds and
> for
>      N = 1000
>      <X_N> = 2.200
>      stdev(X_N) = 0.374
>
> I found
>      Y=  Sum over N (X_N)^4  / N = 27.499
>      Y = <X_N>^4 = 23.423
>
> On Sep 9, 2005, at 11:23 AM, ludwik kowalski wrote:
> > A numerical illustration:
> > 1) Y=T^4 (measuring temperature to calculate the rate of cooling by
> > radiation, Y)
> > 2) Three values of T' were measured: 1, 2 and 3. The mean T' is 2 and
> > we conclude that the true Y is close to 16.
> > 3) Without calculating mean T' we calculate individual Y' from
> > individual T' as: 1, 16 and 81. And we conclude that the true Y is
> > close 36.
> >
> > It is the propagation of error issue. We know that Y=T^4 is highly
> > reliable. And our object is definitely loosing heat by radiation at
> > some unknown rate, Y. We measured T three times and made two
> > conclusions based on different methods. My desktop computer is
> > dead. If
> > it were not dead I would simulate the situation by randomizing T with
> > True Basic. That would definitely tell me which of the two methods is
> > better. That is what computer simulations are good for.

I do not know how wide was the distribution of T in Pamela's
simulation. It turns out, as intuitively expected, that the discrepancy
between the true Y and the predicted Y grows rapidly when the
distribution of T becomes wider. This is shown below. I am talking
about averaging at the level of Y; averaging at the level of T produces
a  nearly perfect prediction at any standard deviation. Like Pamela, I
used 1000 simulated measurements for each sigma and the mean T=2.2. My
distributions of T were Gaussian.

stdev        predicted Y    comment
0.0001     23.426           nearly perfect because all T are
practically identical
0.1            23.71             not a large systematic error when the
distribution of T is narrow (+/- 4.5%)
0.25         25.32              stdev is 22.7% of the mean T. It
produces a larger systematic error
0.50         30.25             even larger systematic error due to the
wrong way of averaging.
0.75         42.39             Very big systematic error (factor of 1.8)
1.00         60.40             Factor ~2.6

Ludwik (happy to find an old True Basic on the cloned drive)
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