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



A stab at a formal analysis:
Measurement i is of the form Xi = xo + Ei , where Ei is a random
variable with a symmetric distribution (ie mean of Ei approaches zero for
large N )

Therfor the mean value of Xi approaches Xo for large N .

Now consider the function Y = X^2. Applied to each measurement:
Yi =(Xo + Ei)^2 = Xo^2 + 2XoEi +Ei^2.
Considering the sum of the Yi ,and the properties of the Ei, we see that
the second term will approach sero, but that the third term will NOT
approach zero, In fact the third term approaches
N times the Variance of the Ei distribution.

TO take a more general approach, consider Y = f(x) and expand to second
order:

Yi = f(Xo) + f'(Xo) Ei + 0.5 f"(Xo) Ei^2

Now you can see that the mean of Yi will NOT approach f(Xo) UNLESS the
range of the errors is so small as to be negligible.
My conclusion is : Average first.


From: ludwik kowalski <kowalskil@MAIL.MONTCLAIR.EDU>
Reply-To: Forum for Physics Educators <PHYS-L@list1.ucc.nau.edu>
To: PHYS-L@LISTS.NAU.EDU
Subject: Re: Average earlier or average later?
Date: Fri, 02 Apr 1976 14:44:47 -0500

On Sep 9, 2005, at 11:43 PM, Bernard Cleyet wrote:

Ludwik!
pse post the source code.

PROGRAM TESTING
!***********************
randomize
let n=1000 ! how many samples from random distr
let meanx=2.2
let stdev=0.5
let sum=0
print "stdev=";stdev
for i=1 to n
call GET_X(mean,stdev,x) ! "measuring" (get x)
let y=x^4 ! calculating from x
let sum=sum+y
next i
let YY=sum/n ! mean from y
print " YY=";YY
end

SUB GET_X(meanx,stdev,x) ! "measuring one x" from
!********************** ! an imposed Gaussian distr
let sum=0
for i=1 to 12
let sum=sum+rnd
next i
let z=sum-6
let x=z*stdev+meanx
end sub


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:


cut


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 . . .
0.1 23.71 not a large systematic error . . .
0.25 25.32
0.50 30.25 even larger systematic error due . . .
0.75 42.39 very big systematic error . . .
1.00 60.40 worse
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