Successive counts of a long-lived radioisotope will yield different counts, even though the sample geometry, equipment settings, etc. remain unchanged. The differences are, of course, due to the random nature of the decay process.
If we were to take a very large number of counts, the distribution of the recorded data would be quite accurately described by the NORMAL DISTRIBUTION, or GAUSSIAN curve. This is expressed as:
Where:
Note: The non-symmetrical POISSON distribution more accurately models radioactive decay, but is far more difficult to work with, and the GAUSSIAN gives results which are just about as accurate.
If we define the STANDARD DEVIATION as the square root of the mean:
Then the GAUSSIAN may be expressed as:
Looking at the areas under the Gaussian curve in terms of sigma:
Remember, when working with probability curves, the ordinate of the curve is not the whole story: it is the area under the curve that is significant.
The area under the curve, bounded by two coordinate values n1 and n2, is the probability of attaining a value of n between n1 and n2. The value under the entire curve is 1 (n must be somewhere!). Thus we see that:
In practice we often have the opportunity to take only one count of a sample. In this case, the single count becomes the only estimate of the mean, and the standard deviation is given by the following expression, from before:
Counts should always be reported as (A ± a), and the confidence level specified. In this notation, [A] is the counts measured, and [a] defines the range of values having the given confidence level. As a convention, no specification of confidence level implies the 1band (68.3% confidence that the true mean lies within the range [A-a] to [A+a]).
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Often we must manipulate data, as when correcting for background. The manipulation must properly reflect the uncertainties associated with the measured values. The uncertainties propagated by basic mathematic manipulations are:
Occasionally one or more measurements may exhibit considerable variance with the mean. If the observation is erroneous, a large error may be created. Some criteria is required to evaluate the data: Chauvenet's Criteria is used. It states that an observation should be discarded if the probability of its occurrence is less than, or equal to:
For the number of observations (j), Chauvenet showed that the value of the following ratio:
For any observation of the set must not exceed the value shown on the curve.
Examination of the data may show some points which should be discarded (since they do not meet the criteria). Then, after discarding the point, the data should be re-analyzed using the (j-1) valid remaining points. The process continues until all points pass the test.
When radioactive samples are measured, it is important to determine if any component of the events recorded are the result of radiation originating from a location other than the sample. This is easily accomplished by repeating the measurement without the sample in (or near) the detector. Subtraction of this background contribution from the sample reading yields the component due to the sample.
Since there are uncertainties associated with both measurements it is important to evaluate that uncertainty properly. The basic mathematical treatment is indicated in the previous section (subtraction.)
There is one last detail that needs consideration however: we often take counts of differing duration. The problem then becomes one of evaluating the entire set of data on a consistent basis. This is easily accomplished by normalizing each data sample to one minute, and reporting the results in counts per minute. That normalization is obvious for the base reading, but the treatment of the uncertainty deserves some care.
The rationale is best presented by the following example:
Note that you have gained certainty by sampling more events. We have gone from having 68.3% confidence that the real mean is between 90 and 110 to achieving the same confidence that the mean is between 96.7 and 103.3.
Note that THE STD. DEV. IS NOT given by
