Analytical Chem istry - DePauw University

94 Analytical Chemistry 2.0Figure 4.10 The blue bars showa histogram for the data in Table4.13. The height of a bar correspondsto the percentage of pennieswithin the mass intervals shown inTable 4.14. Superimposed on thehistogram is a normal distributioncurve assuming that m and s 2 forthe population are equivalent toX and s 2 for the sample. The totalarea of the histogram’s bars and thearea under the normal distributioncurve are equal.You might reasonably ask whether thisaspect of the central limit theorem is importantas it is unlikely that we will complete10 000 analyses, each of which isthe average of 10 individual trials. This isdeceiving. When we acquire a sample foranalysis—a sample of soil, for example—it consists of many individual particles,each of which is an individual sample ofthe soil. Our analysis of the gross sample,therefore, is the mean for this large numberof individual soil particles. Because ofthis, the central limit theorem is relevant.2.95 3.00 3.05 3.10 3.15 3.20 3.25Mass of Pennies (g)is normally distributed. It is easy to imagine that the histogram will moreclosely approximate a normal distribution if we include additional penniesin our sample.We will not offer a formal proof that the sample of pennies in Table 4.13and the population of all circulating U. S. pennies are normally distributed.The evidence we have seen, however, strongly suggests that this is true. Althoughwe can not claim that the results for all analytical experiments arenormally distributed, in most cases the data we collect in the laboratoryare, in fact, drawn from a normally distributed population. According tothe central limit theorem, when a system is subject to a variety of indeterminateerrors, the results approximate a normal distribution. 6 As thenumber of sources of indeterminate error increases, the results more closelyapproximate a normal distribution. The central limit theorem holds trueeven if the individual sources of indeterminate error are not normally distributed.The chief limitation to the central limit theorem is that the sourcesof indeterminate error must be independent and of similar magnitude sothat no one source of error dominates the final distribution.An additional feature of the central limit theorem is that a distributionof means for samples drawn from a population with any distributionwill closely approximate a normal distribution if the size of the samples islarge enough. Figure 4.11 shows the distribution for two samples of 10 000drawn from a uniform distribution in which every value between 0 and 1occurs with an equal frequency. For samples of size n = 1, the resulting distributionclosely approximates the population’s uniform distribution. Thedistribution of the means for samples of size n = 10, however, closely approximatesa normal distribution.6 Mark, H.; Workman, J. Spectroscopy 1988, 3, 44–48.