Biomedical Engineering Reference
In-Depth Information
11
Statistics
11.1
The Statistical World of Atoms and Radiation
As we have seen throughout this topic, the mathematical descriptions used to make
quantitative predictions about nature on the atomic scale do so on a statistical ba-
sis. How much energy will a 1-MeV proton lose in its next collision with an atomic
electron? Will a 400-keV photon penetrate a 2-mm lead shield without interact-
ing? How many disintegrations will occur during the next minute with a given
radioactive source? These questions can be answered precisely, but only in statisti-
cal terms. Whether we measure a number of counts to infer the activity of a source
or the number of electrons produced in a proportional counter to infer the energy
of a photon, there is an irreducible uncertainty due to statistical fluctuations in the
physical processes that occur. Repetition of the measurement results in a spread of
values. How certain, then, is a measurement?
In this chapter we formalize some statistical concepts in order to place confi-
dence limits on the measured values of quantities. The uncertainties we address
here are the inherent ones due to quantum physics. Other sources of error or un-
certainty, such as the precision with which the position of a pointer on a dial can
be read, are not considered here. We begin by analyzing the determination of the
activity of a radioactive sample by counting.
11.2
Radioactive Disintegration—Exponential Decay
To determine the activity of a long-lived radionuclide in a sample, the sample can
be counted for a specified length of time. Knowledge of the counter efficiency—
the ratio of the number of counts and the number of disintegrations—then yields
the sample activity. If the counting experiment is repeated many times, the num-
bers of counts observed in a fixed length of time will be found to be distributed
about their mean value, which represents the best estimate of the true activity. The
spread of the distribution about its mean is a measure of the uncertainty of the
determination.
