Analytical Chem istry - DePauw University

66 Analytical Chemistry 2.0Problem 12 at the end of the chapter asksyou to show that this is true.ble in magnitude. The median, however, provides a more robust estimate ofcentral tendency because it is less sensitive to measurements with extremevalues. For example, introducing the transcription error discussed earlier forthe mean changes the median’s value from 3.107 g to 3.112 g.4A.2 Measures of SpreadIf the mean or median provides an estimate of a penny’s expected mass,then the spread of individual measurements provides an estimate of thedifference in mass among pennies or of the uncertainty in measuring masswith a balance. Although we often define spread relative to a specific measureof central tendency, its magnitude is independent of the central value.Changing all measurements in the same direction, by adding or subtractinga constant value, changes the mean or median, but does not change thespread. There are three common measures of spread: the range, the standarddeviation, and the variance.Ra n g eThe range, w, is the difference between a data set’s largest and smallestvalues.w = X largest – X smallestThe range provides information about the total variability in the data set,but does not provide any information about the distribution of individualvalues. The range for the data in Table 4.1 isSt a n d a r d Deviationw = 3.198 g – 3.056 g = 0.142 gThe standard deviation, s, describes the spread of a data set’s individualvalues about its mean, and is given ass =∑( X − X) 2iin −1where X i is one of n individual values in the data set, and X is the data set’smean value. Frequently, the relative standard deviation, s r , is reported.sr =The percent relative standard deviation, %s r , is s r × 100.Example 4.3What are the standard deviation, the relative standard deviation and thepercent relative standard deviation for the data in Table 4.1?sX4.1