Sampling Error Lecture Demonstration Christopher F. Bauer University of New Hampshire, Durham. NH 03824
Samnline . .. is a subiect which is undertaueht in the analvtical chemistry curriculum. Since a likely reason for this omission is lack of time the concepts must be presented as efficiently as possible. ~emonstraiionscan fulfill this requirement, ~articularlvwhen student participation is involved (1-5).This k i c l e describes a demon&atio&hich illustrates the concept of samnline error and its corollaries. '!'he demonstration is part of a lecture on sampling statistin and stratixies. At the heginning of the lecture two paper bags, A and B, representing materials to be analyzed, are passed around the room. Bag A contains 10 large cork stoppers (#22), one of which is painted yellow and the rest black; B contains 100 small corks (#3), 10 of which are yellow. (These cork sizes were chosen to differ in volume bv rouehlv one order of maenitude.) The students are instruked "analyze" both materials for the number nercent of vellow corks bv withdrawing a sample that comprises 10%of thk available ma& (1 cork from A, 10 corks from B). They then record their results. Besides describing the task verbally, the instructor calls attention to the following rules that are needed to avoid bias in the results: (1) Shake hag. (2) Material A: Select one cork; Material B: Select ten corks. (3) No peeking into the bag or a t the results of others. (4) Record composition as "% yellow." (5)Replace corks. The instructor receives the results and plots a histogram on an overhead for the class. Meanwhile, two students open the bags and determine the "true" yellow compositions. The figure shows a typical result for a class size of ten. True compositions (10% yellow) are also written on the histograms. Although the composition and the analytical method are identical for both materials, the error is much greater for material A. Clearly, the sampling process alone can introduce sienificant error. " The discussion that follows covers the questions of when analvsts should be concerned about sampling error and how ll the problem can he avoided. ~ a r n ~ l i n ~ e r r & Cl.,h articles from automobill exh'akt. Assuming that the PC& and soil particles have a uniform narticle diameter of 50 um. the variation to he expected in the P b content of a 0.1-g sample of soil simply because of the samdina nrocess is about 71%. If enoueh of this soil were dissolvedsuc~thata 10 ppm solution resilted and a 0.1-mL (same mass as for solid sample) aliquot were taken, the expected variation in the P b content of this aliquot would be about 2 X 10-4%! Thenumber of definingunitsin the 0.1-g particulate sample is about 1.5 X 106 particles; in the 0.1-g solution sample about 1.5 X 10Z1molecules. Hence, one need not he concerned with the sampling error involved in removing aliquots from solutions. However, particulate materials are another story. A good discussion of this subject has been presented by Harris and Kratochvil(6), whose equations were used for the above estimates. Through discussion students are led to the conclusions that reducing particle size and in-
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% YELLOW Histogramof observations for 10 students where the sample represene 10% of the available mass for each material.
creasing aliquot size increase the number of units and thus decrease the error. One further point can be illustrated from the histograms. Calculation of the mean analytical results for each material yields 20% yellow for A and 14%for B compared to the true 10%comnosition-clearly there is less bias in the mean composition bf severalreplic& samples than in the composition of most single samples. This demonstration represents a case where there is zeroanalytical error (presumabfy everyone can distinguish yellow from black). In such a case i t is better to analyze many samples once than to analyze one sample several times because sampling error dominates the overall error. On the other hand. if the analvtical imnrecision were laree. for instance if one shade of black were deing distinguishei from another shade in a darkened room. havine renlicate samnles would be insignificant in reducinithe toral eiror. Plotting the histograms and the ensuing discussion require about 15 min. If this amount of time cannot be liberated from the lecture schedule. the demonstration mieht work out as an experiment to be performed during dead times in the laboratory, such as during check-in. Two laboratory experiments illustrating the importance of sampling error have been described (7,8). Obviously, corks are not the only substrate on which this demonstration could he based, and larger classes allow more flexibility in the illustrations that can be performed. ~orexample,conver~ence of the mean ofthesampling distribution can be explicitly demonstrated using populations of 10.20.40.80. and 160 narticioants. ~, ~. ~,~ .~ Sampling is an essentiai part 01analytical methodology and needs a thoroueh and thouehtful uresentation. The narticipatory demoGtration deskbed'herein provides a timeefficient means for meeting this need. Literature Cited
Volume 62 Number 3 March 1985
253
