Introduction to Health Physics: Fourth Edition - Ruang Baca FMIPA UB

490 CHAPTER 9
plus and minus one standard deviation of the mean, 96% between plus and minus
two standard deviations, and so on. The normal distribution, given by Eq. (9.33),
contains the standard deviation, σ , as one of the parameters. The Poisson distribution
(Eq. [9.36]), contains only one parameter, the mean. The standard deviation of the
Poisson distribution is equal to the square root of the mean number of observations
made during a given measurement interval:
σ (Poisson) = √ n. (9.38)
The “size” of the sample when we are counting events such as radioactive decay or
other nuclear events is the arbitrary length of time over which we are making the
measurement. In Eq. (9.38), n is equal to the total number of events that occurred
during that time of observation and is thus the average rate for that time interval.
Thus, if we observe 10,000 counts during a 10-minute counting interval, the standard
deviation of the observation is √ 10, 000 = 100 counts per 10 minutes. The
measurement represents a mean value of 10,000 counts for the 10-minute measurement
interval. One of the main virtues of the Poisson distribution is that, for practical
purposes, when n ≥ 20, it is indistinguishable from a normal distribution of the same
mean and of standard deviation equal to the square root of the mean. Under these
conditions, all statistical tests that are based on a normal distribution, such as the
t-test, the chi-square criterion, and the variance ratio test (which is called the F-test)
may also be used for Poisson distributions. Although all these tests are applicable
to radioactivity measurements, a full discussion of them is beyond the scope of this
book. Details and applications of these tests may be found in the Suggested Readings
at the end of this chapter.
In the example cited above, where 10,000 counts were recorded during 10 minutes
of counting, the mean counting rate, in counts per minute, and the standard
deviation of the mean rate is given by
r ± σr = n
t ±
√
n
. (9.39)
t
Since
σr =
we have
√
n
t =
n 1
×
t t =
r × 1
t ,
r ± σr = r ±
r
, (9.40)
t
which gives 1000 ± 10 cpm. If the activity had been measured over a 1-minute interval
and had given 1000 counts, we would have 1000 ± √ 1000 or 1000 ± 32 cpm.
Precision is a gauge of the reproducibility of a measurement. Thus, we are more
likely to observe 1000 cpm during a 10-minute counting time than during a 1-minute
counting period. Numerically, precision is given by the coefficient of variation, CV,
which is defined as the ratio of the standard deviation to the mean:
CV = σ
, (9.41a)
mean