In the Laboratory
Effect of Sample Size on Sampling Error
W
An Experiment for Introductory Analytical Chemistry Joseph E. Vitt and Royce C. Engstrom Department of Chemistry, University of South Dakota, Vermillion, SD 57069
All instructors of analytical chemistry would probably agree that sampling is a critically important step in any analytical procedure (1, 2). Many, if not most, would probably also agree that sampling receives less coverage in the analytical curriculum than its importance should warrant. However, it is difficult to effectively communicate the importance of sampling to the students in a typical introductory analytical class, because it is still common practice to provide students with a prepared sample in the laboratory. To quote an article in this Journal by J. A. Hern, “the process of sampling frequently consists of picking up a container at the prep-room window (3).” The importance of sampling continues to be addressed in current textbooks (4–6 ), although with far less detailed coverage than in previous generations of textbooks (7 ). Often, students receive a mixed message because we say one thing in the classroom (sampling is as important as measurement or data analysis), but that message is invalidated by our actions in the laboratory. Thankfully, several innovative laboratory exercises have been published lately that incorporate sampling as an important component of the experiment. For example, the article quoted earlier (3) reported an experiment in forensic analysis that involved a field trip to a police firing range to collect samples by wiping the hands of officers with cotton swabs moistened with nitric acid. The samples were subsequently tested using atomic absorption for the presence of antimony in the gunshot residue. More recently, a novel approach to teaching instrumental analysis focused on the analysis of lead by several different methods and incorporated sampling as a major component of the experiment (8). An innovative approach to the introductory analytical lab has been reported that involves monitoring several analytes in a marine aquarium using traditional wet chemical as well as instrumental methods of analysis (9). The aquarium approach was effective in increasing student interest through the use of a real-world sample whose chemistry is interesting and can readily be connected to the environment. Classroom activities have also been reported that illustrate sampling error (10–12), and the mathematics of how sample size affects sampling error has been dealt with in detail (13–14). A very early laboratory experiment reported in this Journal demonstrated the importance of a homogeneous sample by presenting the students with solid samples that were intentionally heterogeneous (15). We have developed an experiment for the introductory analytical laboratory that introduces the basic concepts of sampling and statistical analysis of data. This experiment is a variation of an experiment and lecture demonstration developed by Kratochvil et al. (11, 16 ). Many laboratory and classroom W
Supplementary materials for this article are available on JCE Online at http://jchemed.chem.wisc.edu/Journal/issues/1999/ Jan/abs99.html.
exercises covering the basics of statistical analysis of a data set have been reported in this Journal (17–23). This exercise could be used in place of or in addition to exercises dealing with statistics. We believe it is advantageous to introduce the basics of sampling and statistical treatment of data at the beginning of the introductory analytical laboratory. Indeed, experiments using a real-world sample that involve the students in the entire analytical process will be much more effective if the students first understand the individual steps in an analysis, including sampling and statistical treatment of results. This experiment is one of several done near the beginning of our analytical course, before executing experiments that incorporate more open-ended questions. The experimental procedure is quite simple, and the only materials needed in addition to those normally found in a laboratory are several hundred small (≈ 5 mm) beads, available from craft stores. Using beakers of increasing size, each student draws three samples of beads from a population that consists of a mixture of approximately 50% each of two colors of the beads. The number of beads of a specified color and the total number of beads are counted, and the percentage of the chosen color is calculated for all three samples. Each student reports the results on a class data sheet, which is then photocopied and provided to all students. The students then calculate the mean, standard deviation, and relative standard deviation (RSD) for each sample size. The RSD predicted by the binomial distribution (13) (eq 1) is then calculated for each sample size, and the students are asked to compare the observed and theoretical results and comment on the relationship between sample size and sampling error, as indicated by the RSD. In eq 1, n is the number of beads and p is the fraction of beads of the chosen color.
theoretical RSD =
1–p np
(1)
An example of class results for this experiment is shown in Table 1. Remarkable agreement was found between the theoretical values of RSD and the observed values of RSD for a class of 22 students. The results of this experiment have proven reliable for the past 12 years, although for a particularly small class (12 students) the agreement was not nearly Table 1. Values of RSD for Various Sample Sizes from a Class of 22 Students Sample Size Beaker (mL)
RSD
Mean No. of Beads
Observed
Theoretical
10
48
0.155
0.162
20
108
0.097
0.106
50
261
0.069
0.070
JChemEd.chem.wisc.edu • Vol. 76 No. 1 January 1999 • Journal of Chemical Education
99
Research: Science and Education
as close. For smaller classes we suggest that the number of samples be increased by having the students repeat the experiment so the total number of trials is at least 20. Most importantly, the results show the predicted decrease in sampling error as the sample size increases. In discussions with the students, the important connection to real samples can be made by discussing grinding and sieving of solid samples. We believe the connection to a real sampling situation can be made more readily when the students acquire the samples and generate the resulting data in the laboratory, as opposed to being given a data set with which to work. Variations on this experiment could be introduced relatively easily. For example, the effect of particle size on sampling error could be demonstrated by using a mixture of beads of two sizes, or one component (color) of the sample could be present at a trace level. Acknowledgment This experiment originated in part through discussions between one of the authors (RCE) and Walter Blaedel, University of Wisconsin. Literature Cited 1. Kratochvil, B.; Taylor, J. K. Anal. Chem. 1981, 53, 924A–938A. 2. Kratochvil, B. J. Chem. Educ. 1991, 68, 838–839.
100
3. Hern, J. A. J. Chem. Educ. 1988, 65, 1096. 4. Harris, D. C. Quantitative Chemical Analysis, 4th ed.; Freeman: New York, 1995; p 2. 5. Skoog, D. A.; West, D. M.; Holler, F. J. Analytical Chemistry— An Introduction; 6th ed.; Saunders: Philadelphia, 1994; pp 3–4. 6. Christian, G. D. Analytical Chemistry, 4th ed.; Wiley: New York, 1986; pp 49–56. 7. Laitinen, H. A.; Harris, W. E. Chemical Analysis, 2nd ed.; McGraw Hill: New York, 1975; pp 565–582. 8. Fitch, A.; Wang, Y.; Mellican, S.; Macha S. Anal. Chem. 1996, 68, 727A–731A. 9. Hughes, K. D. Anal. Chem. 1993, 65, 883A–889A. 10. Bauer, C. F. J. Chem. Educ. 1985, 62, 253. 11. Kratochvil, B.; Reid, R. S. J. Chem. Educ. 1985, 62, 252. 12. Clement, R. E. Anal. Chem. 1992, 64, 1076A–1081A. 13. Harris, W. E.; Kratochvil, B. Anal. Chem. 1974, 46, 313–315. 14. Cohen, R. D. J. Chem. Educ. 1992, 69, 200–203. 15. Herrington, B. L. J. Chem. Educ. 1937, 14, 544. 16. Kratochvil, B.; Reid, R. S.; Harris, W. E. J. Chem. Educ. 1980, 57, 518–520. 17. Salzsieder, J. C. J. Chem. Educ. 1995, 72, 623. 18. Stone, C. A.; Mumaw, L. D. J. Chem. Educ. 1995, 72, 518–524. 19. Spencer, R. D. J. Chem. Educ. 1984, 61, 555–563. 20. Paselk, R. A. J. Chem. Educ. 1985, 62, 536. 21. Gordus, A. A. J. Chem. Educ. 1987, 64, 376–377. 22. O’Reilly, J. E. J. Chem. Educ. 1986, 63, 894–896. 23. Richardson, T. H. J. Chem. Educ. 1991, 68, 310–311.
Journal of Chemical Education • Vol. 76 No. 1 January 1999 • JChemEd.chem.wisc.edu
