5. R. Smith, R. Schor, and P. C. Donohue
University of Connecticut Storrs
11
Computer Grading of Small Numbers of Laboratory Unknowns
Rapid grading of commercia~yanalyzed laboratory unlaiowns by computing machine methods has been described previously.' In many cases, for general laboratory use or for examination purposes, instmctors prepare their own unlmowns and must then determine a "correct" value as a basis of grading. The determination of the most probable analysis prompted the investigation of the methods discussed in this paper. McNevina has reported the results of a laboratory examination where the "correct" value of the stock solution was determined by analysis of 58 student determinations. He reported that eight values were discarded as being "out of range." It is not possible to rigorously justify any rejection criterion for a small number of laboratory unknowns. Many rules for treatment of observations are available, however, and we have examined several of these to nbtain an objective method of determining the correct value as a basis for grading. In particular, we have considered methods that could be coupled to the machine grading technique that is used in our laboratory.
5% of the values will fall outside =t2u from the mean. (In a Student's distribution4for more than 10 cases this is also approximately true.) We have decided to reject any values which deviate by more than + 2u from the mean. The formulas for the mean and standard deviation of a Set of N measurements, (xi)where i = 1,. . . . . . . . . .N are :
Table 1 .
Data Reported by 16 Students for an Iron Unknown"
Reported result (mg/25 ml sample)
No. of cases
Deviation from uncorrected mean
Determination of the Median
Dean and Dixon3 have indicated that for a small number of results the median has certain advantages over the average when gross errors are suspected. The use of the median requires that student data be sorted in ascending order so that the middle value can be selected. This procedure is fairly rapid and can easily be adapted to an IBM grading program by use of an IBM sorter. The use of the card sorter requires that each datum be punched on a separate card. Determinotion of the "Correct Value" by a Rejection Procedure
Theoretically an average value of a series of independent observations will yield the best estimate of the true value. In practice, however, when dealing with a small number of student reports the mean can be affect,ed by a few extreme results. In such a case the mean should not be used as the "correct value" for grading. This difficulty is avoided by using the median as discussed above. Examination of a large number of laboratory results shows that the distribution of the students' determin* tion of an analysis approximates a Gaussian if extreme values are rejected. For a Gaussian distribution only ROSENSTEIN, R. D., AND SMITH,S. R., J. CBEM.EDUO.,39, 620 (1962). A., J. CHEM.EDUC.,38, 144 (1961). MCNEVIN, WILLIAM , 636 'DEAN,R. B., AND DIXON,W. J., Anal. C h e m i s t ~ 23, (1951).
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Median, 184.2; uncorrected mean, 183.0; Mean after 2.9 rejection, 184.2.
v =
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4.99.
Each student reported up to three values.
Table 1 lists a series of values reported by students for the analysis of an iron solution which is representative of a semester's laboratory results. Rejecting values which deviate by more than 2u from the mean (i.e., values less 'THOMPSON, WILLIAM R., Annals qf Math Statistics, 6, 214 (1935).
than 173.0 and greater than 193.0) the mean was recomputed as 184.2, which is identical to the median. The corrected mean and the median are generally in close agreement. A FORTRAN program for carrying out this procedure which can be coupled to the program previously described' has been written and is available in our library. Conclusion
Our results in the case indicated in Table 1 and in a number of other cases show that the "correct value" for grading obtained by a 2u rejection criterion is in substantial agreement with that obtained by using the median, and lends itself more readily to FORTRAN programming. The precise choice of the cut-off point is de-
pendent on the judgment of the grader. Our experience shows that changing the rejection criterion to 1.5~ or 2 . 5 ~will not significantly change the final results. We conclude that using either the median or the corrected mean as the correct value will give comparable results. Both methods are readily adaptable to machine scoring. We believe that either method, moreover, is more reliable than the use of the uncorrected mean since 'iunreasonable" results reported by the students are rejected. We wish to thank Professor J. L. C. Lof, Director of the University of Connecticut Computational Center, for his cooperation during this work, and the National Science Foundation for support of the computing facilities a t this University.
Volume 42, Number 4, April 1965
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