## Science & Research

# Volume III - 4.8 Statistics Applied to Radioactivity

| DOCUMENT NO.:
| VERSION NO.:1.4 | ||

| EFFECTIVE DATE: 10/01/2003 | REVISED: 1-31-13 |

- 4.8.1 Introduction
- 4.8.2 Sample Counting
- 4.8.3 Standard Deviation and Confidence Levels
- 4.8.4 Counting Rate and Activity

ORA laboratories may be involved in the identification and quantitative measurement of radionuclides in foods, drugs, and the environment. Instrumentation varies from simple counters to solid state detectors that measure both discrete energy levels and the quantity of radiation in these samples. The correct application of statistical principles is important for arriving at the correct analytical result that will support regulatory decisions.

### 4.8.1 Introduction

Statistics is directly and intimately involved in measurements of radioactivity. Whereas most measurements made in the ORA laboratory are based on variables which vary continuously, radioactivity measurements are based on the counting of discrete, random events. In this case, the normal distribution probability function is replaced by the Poisson distribution, and the associated statistical parameters (mean, standard deviation) are therefore expressed differently.

### 4.8.2 Sample Counting

Radioactive decay is a random process that is described quantitatively in statistical terms. Therefore repeatedly counting radioactive transformations in a sample under identical conditions will not necessarily result in identical values. The result of counting sample radiations is

*number of sample counts = Ns*

The standard deviation of the sample counts, based on Poisson statistics, is

*standard deviation of sample counts =* σs = √*Ns*.

Noise originating in the background, also a random process, simultaneously generates counts that are indistinguishable from those originating in the sample, and therefore the total or gross counts observed from counting a sample include background counts,

*gross sample counts = Ng = Ns + Nb*

where *N _{s} = sample counts*, and

*N*.

_{b}= background countsIt follows that the counts due to sample radioactivity are obtained by subtracting the background noise count from the sample gross counts

*Ns= Ng - Nb*

The counting rate due to sample radioactivity is

*Rs*= *Ns* / *ts*

where *ts* = sample counting interval.

The sample counting rate can also be expressed as

*Rs* = *Rg* - *Rb* = ( *Ng* / *tg* ) - ( *Nb* / *tb* )

where *Rg = gross sample count rate*,

*Rb = background counting rate* .

*tg = gross sample count interval*, and .

*tb = gbackground counting interval*.

### 4.8.3 Standard Deviation and Confidence Levels

The standard deviation is a measure of the dispersion of values of a random variable about the mean value. For a large number of measurements, 68 percent would be expected to lie within plus and minus one standard deviation of the mean of the measurements; 96 percent would occur within plus or minus two standard deviations.

The standard deviation of the sample counting rate, σ_{R}s is given by

where * σ _{R}g* = standard deviation of the gross sample counting rate, and

*σ*= standard deviation of the background counting rate.

_{R}bThe sample rate plus or minus one standard deviation is reported as

If a measured value is reported within the limits of one standard deviation, there is a 68 percent certainty, or 68 percent confidence level, that the true value of the measured quantity is between the given limits. In other words, there is a 68 percent certainty that the real value lies within the limits. If the value is reported at the 96 percent confidence level, the true value is within plus or minus two standard deviations of the reported value. Several confidence levels are tabulated below:

Confidence Level (%) | Number of Standard Deviations (σ_{s}) |

90 | 1.645 |

95 | 1.960 |

96 | 2.0 |

99 | 2.58 |

*Example*. A sample counted for 100 seconds yields 2300 gross counts. The background measured under identical conditions yields 100 counts in 10 seconds. Calculate the sample counting rate (counts per second) and the standard deviation of the sample counting rate. Report the results at the 96% confidence level.

### 4.8.4 Counting Rate and Activity

The sample counting rate is proportional to sample activity, and may be converted to radioactivity units using correction factors. These may include detector efficiency in units of counts per disintegration, chemical recovery fraction, fractional radiation yield, and others. The sample activity may be obtained from the counting rate as follows:

where ε = *efficiency detector*, and

r = *chemical recovery,* and

Y = *yield radiation*.

*Example*. A Sr-89 sample, counted using a detector having a 50% beta-particle detection efficiency for Sr-89 (0.5 counts per Sr-89 disintegration which emits one beta particle per disintegration), yields 500 gross counts in 10 seconds. The background count was 100 counts in 60 seconds. The chemical recovery of strontium was 86%. Report the approximate activity in the sample at the 68% confidence level.

where 1 Bq (Becquerel) = 1 disintegration/s.