Counting Statistics

Subgroup I: Radiation Physics and Detection

In nuclear medicine, statistics has two main uses

1. describing the amount of uncertainty associated with a measurement

2. determining whether or not equipment is functioning normally

sample mean

measure of central tendency

variance

quantifies the amount of internal fluctuation in the data set

standard deviation

a measure of the dispersion around the mean

fractional standard deviation (coefficient deviation)

measure of the precision, usually expressed as a percent

frequency distributions

1. Binomial

2. Poisson

3. Gaussian (normal)

Binomial distribution

• applies to data of a yes or no variety

• depends on number of successes and number of failures

Poisson distribution

• applies to data sets that are very large

• used to approximate the binomial distribution

• only depends on the mean

Gaussian (normal) distribution

• applies when the mean value >20

• used as an approximation of Poisson distribution

radiation measurements

• only have a single number N to represent the population distribution

• we assume that the measurement is close to the mean

radiation measurement of standard deviation

SD = √N

radiation measurement of fractional standard deviation

fractional SD = k/√N

(k = 1 for 68% confidence level)
(k = 2 for 95% confidence level)
(k = 3 for 99% confidence level)

radiation measurement percent error

percent error = FSD * 100 = 100k/√N

(k = 1 for 68% confidence level)
(k = 2 for 95% confidence level)
(k = 3 for 99% confidence level)

What is the standard deviation and percent error for N = 9025 counts?

s = 95
percent error = 1.05%

How many counts are required to achieve a 2% error at the 95% confidence level?

10,000 counts

percent error is a measure of

precision

When the counts increase, the precision ___ and percent error ___

increases, decreases

For a given count rate, the percent error can be decreased by

counting longer

Due to propagation of errors, we must make sure that

uncertainties are calculated, so that the final results can be expressed more accurately

Chi-square test

• statistically testing equipment for proper operation

• determines how well the measured counts ‘fit’ the Poisson distribution

• deviation from the assumed distribution implies faulty equipment performance

Chi-square test ideal probability

0.5

Chi-square probability (p-value) acceptable range

0.1 - 0.9

Chi-square p-value < 0.1

the data is too random

Chi-square p-value > 0.9

the data is not random enough

If the p-value for a Chi-square test is outside the 0.1-0.9 acceptable range

the equipment should not be used