Analytical Chem istry - DePauw University

Chapter 4 Evaluating Analytical Data65X=∑inXiwhere X i is the i th measurement, and n is the size of the data set.Example 4.1What is the mean for the data in Table 4.1?So l u t i o nTo calculate the mean we add together the results for all measurements3.080 + 3.094 + 3.107 + 3.056 + 3.112 + 3.174 + 3.198 = 21.821 gand divide by the number of measurements21.821 gX = = 3.117 g7The mean is the most common estimator of central tendency. It is nota robust estimator, however, because an extreme value—one much largeror much smaller than the remainder of the data—strongly influences themean’s value. 1 For example, if we mistakenly record the third penny’s massas 31.07 g instead of 3.107 g, the mean changes from 3.117 g to 7.112 g!An estimator is robust if its value is notaffected too much by an unusually largeor unusually small measurement.Me d i a nThe median, X , is the middle value when we order our data from thesmallest to the largest value. When the data set includes an odd number ofentries, the median is the middle value. For an even number of entries, themedian is the average of the n/2 and the (n/2) + 1 values, where n is thesize of the data set.Example 4.2What is the median for the data in Table 4.1?When n = 5, the median is the third valuein the ordered data set; for n = 6, the medianis the average of the third and fourthmembers of the ordered data set.So l u t i o nTo determine the median we order the measurements from the smallest tothe largest value3.056 3.080 3.094 3.107 3.112 3.174 3.198Because there are seven measurements, the median is the fourth value inthe ordered data set; thus, the median is 3.107 g.As shown by Examples 4.1 and 4.2, the mean and the median providesimilar estimates of central tendency when all measurements are compara-1 Rousseeuw, P. J. J. Chemom. 1991, 5, 1–20.