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



On Sep 9, 2005, at 9:53 AM, ludwik kowalski wrote:

1) Suppose the true relation is strongly nonlinear, such as y=A*x^10.
2) Suppose the the distribution of experimentally measured x is
gaussian (due to random errors of measurements).
3) The distribution of the corresponding y values (calculated from
individual x) will not be Gaussian; it will be skewed.

For that reason averaging at the level of what is experimentally
measured seems to be preferable.

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 would randomize T around an assumed T=2.2, for example. That would
mean true Y=23.42. Instead of three measurements of T', as above, I
would simulate ten or twenty. Then I would "predict" the true Y by
using each method. My intuitive guess is that averaging at the level T
'would produce a better prediction than averaging at the level of Y'.
Perhaps somebody can do this quickly for us. Excel could also be used
for this purpose.
Ludwik Kowalski
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