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5.3 Determination of Count Rates and Their Errors

Example 3. Consider the following series of measurements of the counts per minute from a detector viewing a 22Na source,

2201 2145 2222 2160 2300

What is the decay rate and its uncertainty?

Since radioactive decay is described by a Poisson distribution, we use the estimators for this distribution to find

muhat = xbar = 2205.6 and

sigma (muhat) = sqrt(xbar / n) = sqrt(2205.6 / 5) = 21.

The count rate is thus

Count Rate = (2206 ± 21) counts/mm.

It is interesting to see what would happen if instead of counting five one-minute periods we had counted the total 5 minutes without stopping. We would have then observed a total of 11028 counts. This constitutes a sample of n = 1. The mean count rate for 5 minutes is thus 11208 and the error on this, sigma = sqrt 11208 = 106. To find the counts per minute, we divide by 5 (see the next section) to obtain 2206 ± 21, which is identical to what was found before. Note that the error taken was the square root of the count rate in 5 minutes. A common error to be avoided is to first calculate the rate per minute and then take the square root of this number.

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