Kenneth E. Daugherty and Rex J. Robinson
University of Woshington Seattle
I
I
A Statistid Analysis of R ~ S U ~ S Undergraduate quantitative experiments
During the past seven years, data have been collected on student laboratory analyses in the introductory quantitative analysis course a t the University of Washington. The experiments are gravimetric chloride as AgCl, gravimetric sulfate as BaSOg, volumetric permanganimetric calcium, and volumetric iodometric chromium. The laboratory procedures for these determinations have been developed and used a t Washington for many years and are, in general, similar to the standard procedures for these determinations such as found in Pierce, Haenisch, and Sawyer,' "Quantitative Analysis." Percentage Distribution of Student Results. Error, parts per thousand
C1:emt.
Percentage distribution SOn--exot. Ca-exot. Cr-exut.
The student results of each of the above experiments have been analyzed statistically in order to determine the probability of accuracy on an analysis of an unknown sample. These experiments were selected for the study because of the high degree of accuracy in evaluation, since the samples were distributed to students as measured volumes of standard solutions: XaCl for chloride, K2SOI for sulfate, CaCOs dissolved in HCl for calcium and KzCrpOrfor chromium. The exact quantity for each student sample was easily calculated knowing the volume and concentration of the sample. Student results were graded on the basis of parts per thousand deviation from the calculated values. The parts per thousand error have been converted to per cent error and graphs plotted of frequency versus per cent error. The curves appear to be Gaussian, which is evidence that our statistical sample was large enough. The range of errors on the graphs of the four ex~erimentswere selected in order to portray the IPIERCE,W. C., HAENISOH, E. L., AND SAWYEB,D. T., "Quantitative Analysis," 4th ed., John Wiley and Sons, Inc., New York, 1958.
maximum swecp of the Gaussian curve, and in no way are meant to indicate satisfactory or unsatisfactory results on the determinations. This spread of results included all except those of high error. The chloride determination was the first laboratory experiment conducted by the students, followed by the sulfate, the calcium, and the chromium. The information portrayed in the curves may be utilized in establishing grade distributions and evaluating the magnitude and character of student errors in the four experiments. These errors may be either inherent errors of the determination or student errors. Mathematical errors were minimized since the laboratory notebooks were checked for mathematical accuracy by the teaching assistants before the results were accepted for grading. The table shows the percentage distribution of the student results based on parts per thousand. Student results past these percentage distributions were not tabulated and hence not shown on the Gaussian curves as the results were considered unacceptable.
PERCENT&=
ERROR
Figure 1. Gravimetris chloride or silver chloride. The dotted line is the arithmetic mean at -0.1 6%.
In the gravimetric chloride experiment 2570 individual determinations were tabulated. Of these, 2112 determinations, or 82'%, fell between +l.OO% of the theoretical value; 68% of the determinations were from -1.00% to O'%, and 32% from O'% to +1.00%. The standard deviation was 0.35%. Thus, the probability of a student getting a low result on the AgCl experiment is approximately two to one. The most likely negative errors are solubility of AgCI, loss in transferring precipitate, over washing the precipitate, Volume 41, Number I , January 1964
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and photochemical decomposition of AgCl in the crucible with resulting chlorine loss. These student results are remarkable in view of the fact that this determination was the first student analysis. I n the gravimetric sulfate experiment, there were 3124 individual determinations. Of these, 2137 determinations, or 68%, were between *4.00% of the theoretical value; 88% of the determinations were from -4.00% to 0% and 12% from 070 to +4.00%. Thus, of every eight determinations about seven were low. The standard deviation was 1.07%. This substantiates the reputation of the gravimetric sulfate as being one of the least accurate and most difficult gravimetric determinations. Solubility of barium sulfate, coprecipitation, transfer loss, and reduction of Bas04 during ignition of the filter paper are the most
PERCEMAGE
01 -IW
-075
-0.50
0
a25
PERCENTAGE
ERROR
-025
PERCENWGE
ERROR
Figure 2. Gmvimetric sulfate or barium r u l f d e . orithmetic mean a t -1.04%.
The dotted line i. t h e
0.50
075
ID0
Figvre 3. Volumetric permongimetric calcium. The dotted line is t h e arithmetic mean a t -0.047-
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Journal of Chemicol Education
significant of the negative errors. This experiment was the second determination in the laboratory schedule of work. I n the volumetric calcium experiment, there were 2024 individual determinations. Of these, 1475 determinations, or 73%, were between *1.00% of the theoretical value; 53% of the determinations were from -1.00% to 0% and 47% from 0% to +1.00%. A student has about an equal probability of obtaining either a high or a low result. The standard deviation was 0.47%. This determination was the third analysis of the course and the first volumetric determination. The KMnOl normality as obtained by the student was checked experimentally and graded by the teaching assistant; thus the calcium results were calculated using a correct KMnOanormality.
ERROR
Figure 4. Volumetric iodometric chromium. orithrnetic mervl o t 0.1 6%.
T h e dotted
line i s t h e
The last experiment examined was the volumetric chromium experiment with 1790 individual determinations. Of these, 1403 determinations, or 78Y0, were between *2.00% of the theoretical value. Twentyseven per cent of the experiments were from -2.00% to 0%, and 73% from 0% to +2.00%. Thus, of four student results, one would expect one low result and three high results. The standard deviation was 0.33%. The thiosulfate standardization was not graded separately; thus this determination is more susceptible to student error. The chromium experiment was the first iodometric determination. This statistical analysis presents an accurate picture of what one should expect from results of unknowns in an introductory quantitative analysis course. Given a group of students conducting one of these experiments, their results as a group can be predicted. It should enable the assignment of more realistic grades to student results on these four common experiments.
