Re: statistical fluctuations

[Ludwik Kowalski <kowalskil@MAIL.MONTCLAIR.EDU>]

On Friday, Sep 5, 2003, referring to my illustration
(see below) Dan Crowe wrote:

> The standard deviation is 9.2, which is the square
> root of (47 + 37). The signal is the difference between
> two measured values (47 - 37=10).  By Poisson statistics,
> the variance of each measured value equals the
> measured value (47 and 37, respectively).  The standard
> deviation of each measured value is the square root of
> the variance, or the square root of the measured value
> (square root of 47 and square root of 37, respectively).
> The variance of the difference between the measured
> values is the sum of the variances of the measured values
> (47 + 37 = 84).  Therefore, the standard deviation of the
> signal is the square root of the sum of the measured
> values (square root of 84 = 9.2).
> > In my illustration one chip recorded 47 tracks
> > (signal plus background) while another recorded
> > 37 (background). I subtract and obtain signal
> > equal to 10. But I can not say that signal=10
> > because both 47 and 37 would fluctuate from
> > chip to chip. The probability that the true signal
> > is 10 is very very small in this example.
> >
> > Now consider the Oriani's method. (His numbers
> > were actually much larger, I made them small to
> > focus on fluctuations. Suppose that  using his
> > method (one chip) I observe 37 before the
> > experiment and 47 after the experiment. I conclude
> > that the signal is 10. Yes, I do not expect that the next
> > experiment will yield the same signal but the result
> > will always be positive or zero. Is it correct to say
> > that the standard deviation for the signal should
> > be sqrt(10), in this illustration? I am not certain.
> > Ludwik Kowalski

In other words a person using two separate chips: one
to measure (signal+noise) and another to measure the
(noise only) should report that the signal is 10 +/- 9.2.
This is not a very convincing argument that the signal
is real. Most likely it is an illusion due to statistical
fluctuations affecting each independent measurement.

But now consider a person using only one chip, as
Oriani did. After the first development the (background
only) is 37, after the second development the (background
+ signal) is 47. The second person, if I am correct, should
report that the signal is 10 +/- 3.2. This is a much more
convincing argument that the signal is real; it differs
from the null result by about three standard deviations.

In both cases I am defining the +/- bar of error as one
standard deviation. Am I correct in saying that the second
method is much better that the first when a conclusion has
to be made on the basis of very limited number of counts,
as in the invented illustration?
Ludwik Kowalski