Bush__The_Essential_Physics_for_Medical_Imaging - Biomedical ...

For example, a counr of 853 is obrained. Derermine the interval about thiscount in which there is a 95% probability of finding the true mean.First, the standard deviation is estimated:a = Y853cts = 29.2 cts853 cts ± 1.96 a = 853 cts ± 1.96 (29.2 cts) = 853 cts ± 57.2 ctsSo, the 95% confidence interval ranges from 796 to 910 counts.In nuclear medicine, calculations are frequently performed using numbers thatincorporate random error. It is often necessary to estimate the uncertainty in theresults of these calculations. Although the standard deviation of a count may be estimatedby simply taking its square root, it is incorrect to calculate the standard deviationof the result of a calculation by taking its square root. Instead, the standarddeviations of the actual counts must first be calculated and entered into propagationof error equations to obtain the standard deviation of the result.Multiplication or Division of a Number with Error by a Numberwithout ErrorIt is often necessary to multiply or divide a number containing random error by anumber that does not contain random error. For example, to calculate a count rate,a count (which incorporates random error) is divided by a counting time (whichdoes involve significant random error). If a number x has a standard deviation a andis multiplied by a number c without random error, the standard deviation of theproduct cx is ca. If a number x has a standard deviation a and is divided by a numberc without random error, the standard deviation of the quotient x/c is alc.For example, a 5-minute count of a radioactive sample yields 952 counts. Thecount rate is 952 cts/5 min = 190 cts/min. The standard deviation and percent standarddeviation of the count are:a = Y952cts = 30.8 ctsFractional error = 30.8 cts/952 cts = 0.032 = 3.2%a =30.8 cts I 5 min = 6.16 cts/minFractional error = 6.16 cts/190 cts = 3.2%Notice that the percent standard deviation is not affected when a numbermultiplied or divided by a number without random error.isIt is often necessary to add or subtract numbers with random error. For example, abackground count may be subtracted from a count of a radioactive sample.