David P. Dingledy and Lawrence A. Patrie State University College Fredonia, N e w York 14063
Statistical Evaluation of Laboratory Experiments
It has been demonstrated that a statistical treatment of results reported by introductory laboratory course chemistry students leads to correct stoichiometry despite the considerable spread in these res u l t ~ . ' . ~The statistical averages and measures of dispersion obtained in this treatment are therefore useful in showing the value of laboratory exercises to the student and in evaluating student performance. Student results may be entered directly iuto the statistical calculations, or may be first corrected for obvious weighing errors or gross errors in arithmetic. I n the evaluation of the first exercise described below, that of tho synthesis of copper sulfide, the data are partially corrected because the results were obtained from the reports of six laboratory instructors, some of whom required that student reports be corrected for such errors before submission for grading. In the evaluation of the second exercise none of the reports were corrected. I n the evaluation of the third exercise, the synthesis of nickel sulfide, all data were corrected by us. It appears from a comparison of the final results that the extra step of correction is not necessary, probably because the statistical treatment minimizes the effect of values in the sets that are markedly outside the normal distributions. The standard deviation, s, is determined from the average, j., of all n rcsults in a set 12-1
and the standard error (8.E.)
The table lists values obtained in general chemistry laboratory exericses at Fredonia in which the student was expected to determine the stoichiometry of a sulfide prepared by heating a metal in an excess of sulfur.
' DINGLI:DY, D., A N D B ~ R N R DW. , M., J. CHI:M.EDUC., 44, 242 (1967). DAYIIIS, OWENL. (Edzlor), "Statistical Methods in Research and Production," Oliver and Boyd, London, 1949.
The expression in this table of average plus or minus twice t,he standard error is based on a normal distribution of results. For 30 or morc values, plus or minus twice the standard error represents a probability of nineteen to one that t,he true universe (population) average is between these limits. For less than 30 values an appropriate t value is obt,ained, as in the case of the lead sulfide, from tables of the percentage points of the t-distribution to correspond to the actual number of values giving this probability (573.3 Each of the above calculations may be performed in a few minutes if an electronic desk calculator such as the SCM Cogito is available. I t is interesting to note that none of the 129 nickel sulfide or the 9 lead sulfide results and only one of the 266 copper sulfide results were the expected value. The expected result was contained within the standard error limits in each case, however. Several applications may be made of the above statistics. One application is fundamental to the notion of laboratory exercises as teaching devices. These exercises should impress upon the student that application of the chemical laws of conservation of mass and of constant proportions in the laboratory should result in satisfactory stoichiometry. Yet it is apparent that the individual results will not demonstrate this stoichiometry and indeed differ enough from the expected result that the student doubts his own capabilities and the validity of the laws t,hat are supposed to be demonstrated. If, however, instead of leaving the student to evaluate his individual result, the lab instructor pcrforms the relatively simple statistical calculation for the entire section, the student then observes that correct stoichiometry is obtained. A second related application is one of grading. Having the standard deviation at hand supplies the instructor with a useful and realistic grading scale Difwnee of Value From Average (In Slandard Dmialion Units) Greater than 1.6s Less than 1.68, greater than 1.08 Less than 1.0s, greater than 0.49 Less than 0.4s, greater than 0.1s Less than 0.1s
Fraclion of Such Values In in Normal Distribulia
Grade E I1
yh
C
40 20 10
B A
Statistical Evaluation of the Stoichiometry o f Sulfides Preoared in General Chemistry Laboratory
Sulfide Prepared
Number of Results
Average Result P (yoMetal In Sulfide)
Copper sulfide
266
Lead sulfide Nickel sulfide
Standard Deviation,
Standard Error S. E . =
Average S. E.
Expected Sulfide
0.34
78.3 f 0 . 7
1.06
86.5
0.9
73.6 1 1.8
Cur&, Digenite PbS, Galena. Ni41, Ileadewoodite
s
3 1 6
78.3
5.5
9
86.5
3.2
129
73.6
10.1
3 2
zt
2.2
Expected % lMetal In Sulfide
Volume 46, Number 2, February 1969
78.1 86.6
73.3
/
1 19
Such a grading scheme allows for the variability in error inherent in various exercises. For example, standard deviation for the lead sulfide exercise is 3.2, but for the nickel sulfide exercise it is 10.1, and the grading of the latter exercise then allows larger numerical error limits in order to be comparable in terms of evaluation of student ability with the former exercise. The standard deviation also furnishes a useful tool in the choice and modification of exercises. Demonstration of a relatively low standard deviation will in gen-
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eral favor the use of the exercise because it indicates that there is better control over the experimental variables in the procedure used. A large standard deviation should result in modification of laboratory procedure toward possible removal, or at least minimizing, of these variables. We wish to thank Dr. Robert Grennell of the Fredonia Education Department for his interest in and assistance with the statistical calculations.
