Correction: Chi-Square Test and Counting Statistics

[Judd Sills <sillsj@GAT.COM>]

With respect to my posting on this subject late
yesterday, Ed Leithold (I apologize if I misspelled
your name) left me a voice mail this morning to point
out an omission that I made in the equations
which I would like to correct here.  In case people 
want the entire posting for future reference (as I
often do), and would prefer it to not have any
errors in it (this is a no brainer for me), I will
re post the note with the corrections.  The corrections
are that I forgot to show the raising of the difference
from the mean term to the power of two in the Chi
square formula and the standard deviation formula.
I apologize for the error...I was challenged by
trying to find a way to write the equation without
a word processor so that everyone could follow it.
My thanks to Ed for his good catch, and my thanks to
everyone who has expressed interest in this topic.

Happy weekend to all.
=========================================================
The Chi-square test is a means to test the variability
of a counter-scaler system to ensure that the results are
consistent with the random nature of radioactive decay which
can be predicted according to a probability.  Simply
stated, when multiple counts of a radioactive sample
(or natural background) are made, the results should
approximate a Poisson distribution.  The Chi-square test
is a method to compare the variability of the results to
the predicted variability from a true Poisson distribution.

If the system results are too consistent (little variability)
then it probably isn't performing properly, and similarly,
if the variability is excessive, then it also probably isn't
performing properly.  The Chi-square test should be performed
for both background and a test source on a regular basis (e.g.,
monthly) and whenever the system has been altered by maintenance
(repair or adjustments) or calibration.  It is performed by collecting a 
series of counts (source or background, according to which Chi-square is
being performed).  The more observations that are collected, the
smaller the standard deviation of the observations would be.  I recommend
that you perform 20 observations (n).  The Chi-square is calculated
as the sum of the squares of the differences of the observed count
rates from the mean count rates divided by the mean count rate itself.
                     2
                    X  = {Sum(i to n) [xi - xbar]^2}/xbar
where xbar is the sample mean for the n counts.

For 20 observations, following the 99th percentile, the Chi-square
range should fall between 8.91 and 32.9.   Other acceptance criteria are
needed if a different number of observations (counts) are taken.

After collecting this data and meeting the acceptable Chi-square
result, control charts (graphs) should be developed to test daily
background and source checks for acceptability.  Determine the standard
deviation of the observations:
     SD = Square Root(sum(i to n)[xi - xbar]^2/{n - 1})
where n is the number of observations (e.g., 20 for my recommendation)
On the Control Chart, draw a line representing the mean (xbar), two
SD (xbar +/- 2SD), and three SD (xbar +/- 3SD).  On a daily basis
plot the observation (background or source) and compare it to these
bounds.  If the daily observations are between the two 2SD lines, the result 
is acceptable.  If the daily observation is between the 2SD and 3SD line,
then a recount should be performed (if it is between the 2 SD lines,
then this is acceptable).  If the daily observation is beyond the 3SD
lines, the system should be taken out of service for repair/evaluation.

I recommend that you use the same sources for daily testing that
you used for the Chi-square testing.  Your source should be representative
of your typical sample decay energy.  It is also recommended that you
compare your empirical counter efficiency with what your calibration
lab has reported for the isotope that you are using for your test source
as another check of system performance.

This was a long winded answer, but hopefully explained the reasons for
the test, and how to perform it and use it properly.


Judd M. Sills, CHP           |   Office: (619)455-2049
General Atomics, Room 01-166C|      Fax: (619)455-3181
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