Physics and Radiobiology of Nuclear Medicine

standard deviation on either side of the mean, that is, within the range
±s; 95% of all measurements fall within the range ± 2 s; and 99%
fall within the range ± 3 s. Also the Gaussian curve shows that half of
the measurements are below the mean value, and the other half are
above it.
The standard deviations in radioactive measurements indicate the
statistical fluctuations of radioactive decay. For practical reasons, only
single counts are obtained on radioactive samples instead of multiple
repeat counts to determine the mean value. In this situation, if a single
countn of a radioactive sample is large, thenn can be estimated as close
to ; that is, =nand s=. It can then be said that there is a 68%
chance that the true value of the count falls within n ±sor that the
countn falls within one standard deviation of the true value (Fig. 4.1). This
is called the 68% confidence level. That is, one is 68% confident that the
countn is within one standard deviation of the true value. Similarly, 95%
and 99% confidence levels can be set at two standard deviations (2s) and
three standard deviations (3s), respectively, of any single radioactive
count.
Another useful quantity in the statistical analysis of the counting data is
the percent standard deviation, which is given as

(4.2)

Equation (4.2) indicates that asn increases, the %sdecreases, and hence,
precision of the measurement increases. Thus, the precision of a count of a
radioactive sample can be increased by accumulating a large number of
counts in a single measurement. For example, for a count of 10,000, %sis
1%, whereas for 1,000,000, %sis 0.1%.

Problem 4.1
How many counts should be collected for a radioactive sample to have a
2% error at a 95% comfidence level?

Answer
95% comfidence level is 2s, that is,

Therefore,

n=10,000 counts

n= 100

2

200

=

n

2

2 100 2 100

%=

×

=

s ×

n

n

n

2 n

%s

s

=× = =

n

n

n n

100

100 100

n n n

n

n n

36 4. Statistics of Radiation Counting