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Bush__The_Essential_Physics_for_Medical_Imaging - Biomedical ...

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TABLE 20-3. FRACTIONAL ERRORS(PERCENT STANDARD DEVIATIONS)FOR SEVERAL NUMBERS OF COUNTS1001,00010,000100,00010.03.21.00.32Because the standard deviation is approximately the square-root of the mean, it isalso approximately the square-root of the single measurement:cr = -v.; [20-14]where x is the single measurement. <strong>The</strong> fractional standard deviation of a singlemeasurementmay also be estimated:. cr -v.; 1FractIonal error = - = -- = ~r [20-15]For example, a single measurement<strong>The</strong> estimated standard deviation is:x x vxof a radioactive source yields 1,256 counts.cr = V 1,256 cts = 35.4 ctsFractional error = 35.4 cts / 1,256 cts = 0.028 = 2.8 %Table 20-3 lists the estimated fractional errors <strong>for</strong> various numbers of counts.<strong>The</strong> fractional error decreases with the number of counts.Table 20-4 lists intervals about a measurement, called confidence intervals, <strong>for</strong> whichthe probability of containing the true mean is specified. <strong>The</strong>re is a 68.3% probabilitythan the true mean is within one standard deviation (l cr) of a measurement,a 95% probability that it is within 2 cr of a measurement, and a 99.7% probabilitythat it is within 3 cr of a measurement, assuming that the measurements follow aGaussian distribution.±0.674cr±1cr±1.64cr±1.96cr±2.58cr±3cr50.068.390.095.099.099.7Source: Adapted from Knoll, GF.Radiation detection and measurement, 2nd ed. New York: John Wiley,1989.

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