Counting Statistics
Gem-5 User’s Manual 273
In a special case when
and using the substituition
in Equation 14,
the two-tailed probability can be written in the following form:
Equation 15 One-Tailed Probability at x
−==
∫
∞−
−
−
σπ
2
x
erfc
2
1
1dte
1
)x(P
x
t
tailed1
2
(15)
One-tailed probability is 84.1% for <1σ and 97.7% for <2σ.
The probability of not false alarming on background data due to normal statistical
fluctuations, while successfully identifying a data value just above background for
Gaussian distributed data (that is, the likelihood that the value measured is the true or
correct value), is defined as the area under the normal distribution curve between
and
.
Note that as negative values of the count rate cannot exist, only the positive data
values are considered.
In the special case when
the probability for success is 1.0 for both one-tailed
and two-tailed distributions; otherwise the probability for success is less than one, but
it can be never negative.
Equation 16 Special Case When One-Tailed Probability Equals Two-Tailed
Probability
(16)
It can be shown that the standard deviation of Poisson or Gaussian distribution data is
the square root of number of counts.
The probabilities of success for the confidence level are based on a normal
(symmetric) distribution of data. The background count rates measured in situ are
asymmetric in nature or skewed; therefore the calculation of the probability will be in
error. Since the degree of asymmetry varies as the mean of the Poisson distribution, it
is not convenient to use asymmetric equations but rather to use a symmetric
approximation. Under these conditions, the two-tailed distribution “corrects” for
inaccuracies due to the asymmetric nature of the data better than for one-tailed
distribution. In addition, the two-tailed interval “corrects” for white noise (random
but not statistical) contributions, which may affect background determinations. At
higher count rates, the two-tailed distribution should not cause any problems.
Comparison of the one-tailed and two-tailed methods for False Alarm Rates (FAR)
was performed using computer simulations of random data from a Poisson
distribution with means between 5.0 and 8.0. The one-tailed method showed higher
than expected FAR. The same data using two-tailed intervals yielded the expected
result. The addition of white noise did not affect the observations.
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