Chapter 11
Statistics of Counting
11.1 Introduction
Practically every book on radiochemistry deals with the statistics of numbers and it
will be beyond the scope of this book to deal with this subject in much detail. However, we shall discuss the importance of statistical calculations in the detection and
measurement of radioactive isotopes by taking a few examples. The measurement of
radioactivity is always expressed by the number of counts per unit time. Counting is
done for a longer duration and the activity is calculated by dividing the total count
recorded by the time for which the counting was done. Sometimes the activity is
recorded for a very short duration, especially while counting short-lived isotopes.
The activities of the sample recorded by the counter are also subtracted for the background count rate to get the true count rate. Sometimes activities are measured several
times to get an average count rate. It may also be required to add two or more count
rates together, subtract, divide, or multiply one count rate from another count rate etc.,
to arrive at required results. For all these operations, a question arises as to how much
count should be recorded to get the minimum error in counting the activity for a sample. Should a single count for a longer duration (e.g., 5–10 min to get a large number
of count (e.g., 10,000) and divide it by time (i.e., 10,000/10) to get average count rate
(i.e., 100 cpm) be taken or should one take a large number of counts of small digits
(e.g., 100, 101, 99, 110, 106, 108. . .) for a shorter duration (e.g., 1–2 min) and take the
mean of all counts as most accurate reading? To know which of the two methods are
more accurate? if the count rate for background is low (i.e., of the order of 10–20 cpm);
should the background be counted for one minute or for 1 minutes × 10 times and take
an average count rate or should one count for 10 min duration to get one large count
and then divide by time of counting for calculating the count rate for background
activity? These questions emerge when the activity of a sample is to be reported
accurately. Normally, one tends to believe that the average of the count recorded
5–10 times (with each measurement of low duration) is an accurate count. But there
is a serious drawback, which can be perceived by examining the following example.
A family of five persons wished to cross a river on foot. The leader of the family
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. Sharon and M. Sharon, Nuclear Chemistry,
https://doi.org/10.1007/978-3-030-62018-9_11
171
Statistics of Counting
11.1 Introduction
Practically every book on radiochemistry deals with the statistics of numbers and it
will be beyond the scope of this book to deal with this subject in much detail. However, we shall discuss the importance of statistical calculations in the detection and
measurement of radioactive isotopes by taking a few examples. The measurement of
radioactivity is always expressed by the number of counts per unit time. Counting is
done for a longer duration and the activity is calculated by dividing the total count
recorded by the time for which the counting was done. Sometimes the activity is
recorded for a very short duration, especially while counting short-lived isotopes.
The activities of the sample recorded by the counter are also subtracted for the background count rate to get the true count rate. Sometimes activities are measured several
times to get an average count rate. It may also be required to add two or more count
rates together, subtract, divide, or multiply one count rate from another count rate etc.,
to arrive at required results. For all these operations, a question arises as to how much
count should be recorded to get the minimum error in counting the activity for a sample. Should a single count for a longer duration (e.g., 5–10 min to get a large number
of count (e.g., 10,000) and divide it by time (i.e., 10,000/10) to get average count rate
(i.e., 100 cpm) be taken or should one take a large number of counts of small digits
(e.g., 100, 101, 99, 110, 106, 108. . .) for a shorter duration (e.g., 1–2 min) and take the
mean of all counts as most accurate reading? To know which of the two methods are
more accurate? if the count rate for background is low (i.e., of the order of 10–20 cpm);
should the background be counted for one minute or for 1 minutes × 10 times and take
an average count rate or should one count for 10 min duration to get one large count
and then divide by time of counting for calculating the count rate for background
activity? These questions emerge when the activity of a sample is to be reported
accurately. Normally, one tends to believe that the average of the count recorded
5–10 times (with each measurement of low duration) is an accurate count. But there
is a serious drawback, which can be perceived by examining the following example.
A family of five persons wished to cross a river on foot. The leader of the family
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. Sharon and M. Sharon, Nuclear Chemistry,
https://doi.org/10.1007/978-3-030-62018-9_11
171