D. E. Jones' and F. E. Lylle Purdue University West Lofoyette, Indiana 47907
Computer Aided Grading of Quantitative Unknowns
In the teaching of quantitative analvsis, a continuina problem has been the-fair grading of unknowns. To ac; complish this, several questions must be satisfactorily answered. What level of accuracy and precision can be expected from the students for a particular experiment? How can a faulty procedure he detected? How can "bad" unknowns be recognized? And finally, how can the correct results be posted without promoting "dry lahing" in subsequent semesters? The computer aided grading discussed in the following paragraphs provides a means of approaching the answers to these questions. The scheme is to dispense random volumes of a stock solution containing only a roughly known concentration of the sought for constituent. These volumes are held hetween 20 and 30 ml to minimize both the dispensing error and any concentration dependent experimental error. All solutions are then diluted in an appropriate volumetric flask by the student. When all of the results have been returned, they are punched onto paper tape along with the volume given and an identification numher. This tape is then read by a laboratory computer where a linear regression is performed.2 If the students' results contain no determinate errors, the concentration is determined within the standard deviation of the data. Also, if the correlation coefficient from the regression analysis were close to unity, the instructor could he satisfied that the spread in the answers was of acceotahle limits. With such a situation, there would he no gasis for distinguishing among the students' abilities. and evervone would receive the same grade. In practice, however, a significant numher of students would have either poor precision or determinate errors. Both possibilities lead to very low values of the correlation coefficient and are easily detected. The method of operation then eliminates from the regression those .students considered errant, and the class is graded against the best of their peers for each separate experiment. It is quite arbitrary how student results are eliminated during the procedure. In the promam utilized. if the raw inputyields a correlation eoefiici&t below an operator set value, e.g., 0.95, results greater than an arbitrary number of sigmaunits from the calculated curve are dropped. The program then repeats the regression and computes a new coefficient until the correlation limit is either reached or exceeded. At this time, all of the students' values are compared to the standard deviation of the last run. The assignment of grades is straightforward with %2 o units 3.00 receiving SO%, etc. The receiving 100% credit, 2.01 computer output is so arranged that i t can be posted. The students can then check to see how well they have done and look for obvious mathematical errors like factors of two or ten. Any repeaters are given new unknowns, so that the data cannot be used to dry lab. Also, their answers are easily graded by using the parameters of the regressed curve and the volume dispensed.
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'Current address: Western Maryland College, Westminster, Md. 21157. aBwington, P. R., "Data Reduction and E m Analysis far the Physical Sciences," McGraw-Hill Book Company, New York,
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aBlaedel, W. J., Jefferson, J. H., and Knight, H. T., J. CHEM.
EDUC., 29,480 (1952).
In addition to the grading information that the regression coefficient and the standard deviation yield, the,slope and the intercept are also calculated and can he used as checks on the l&oratory procedure and the quality of the unknowns.3 The student standard deviation of each experiment can he recorded from class to class so that it is easy to detect when a given experiment is no longer under "control." The actual plotted data is helpful in many cases. For pre-analyzed solids, a cluster of points away from the regressed line could indicate a had unknown. A non-zero intercept could indicate a constant determinate error, and a slope different than that expected could indicate a proportionate error. Two exampies should suffice. In a Kjeldabl nitrogen determination utilizing solid commercially analyzed unknowns, a unity slope was expected since the regression involved percent against percent. In actualitv, the slope was -0.95. and the orohlem was traced to a ti'tration. The laborator; handout heglected to specify an indicator and the students simply suhstituted one used in an earlier experiment. unfortunately, the equivalence- and end-points were about 5% different. In a photometric determination of iron by its colored 1,lO-pheoanthroline derivative, a non-zero intercept and a
Computer Output for Nickel Gravimetrica
Ready Run Minimum corr. coef. for final calc. = ?.945
w o l i level for no. of sigmas = ?1.50 Slope = ,12462 Std. dev. = 2656443.02 Inter. = 1.45534 Std. dev. = ,677225 Corr. emf. = ,51803 Student std. dev. = .425993 Total data deleted this cycle 803
Slope = ,158014 Inter. = ,514371 Corr. coef. = .958352 Total deleted = 27 Used for last cam. coef. = 35 Student True 1 2 3 5 6 7 8 9 10
Reported
Sigmas
3.84658 4.35874 5.03846 4.57372 4.55159 4.5263 3.91297 5.03688 4.21173
Ready 'Underlined values are operator input. The numbers are not nearly as precise as the computer output might infer! Volume 50, Number 4, April 1973
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285
. 21.w
7.w
'%?spensed
f?%ume
2903
(ml?
Figure 1. The raw data after the first regression. (The graphical computei output Corresponding to the data of the table.)
zero slope were ohtained. The problem was traced to a stock iron solution excessively concentrated so that the measured absorbance reflected the ligand concentration. The entire routine, written in BASIC, is designed to handle 100 students and is run on a 2116A Hewlett Packard Computer with 8K of memory. An undocumented copy of the program is available on request. At the beginning of the routine, the computer prompts the operator for the desired correlation coefficient and the error level in sigma units to be used for dropping data. After each regression, the computer prints out the slope, the standard deviation of the slope, the intercept, the standard deviation of the intercept, the correlation coefficient, the student standard deviation, and those data dropped prior to the next cycle. Also, in order to have better operator control, a plot of the regressed line, f1 of the line, and the utilized data is output either on a calcomp plotter or a display scope. I£ the computer is unable to reach the desired level for the correlation coefficient by ignoring appropriate student data, an interrupt routine requests the operator to readjust one or both of the parameters. Whenever the desired correlation limit is reached, the computer will output the student identification number, the true value calculated from the regressed line, the reported value and the error in units of the student standard deviation. Also, a final graphic output is displayed, so that it can he ~ o s t e dalone with mades. An abbreviated example of a typical run is ihown & the table and Figures 1-3. w e have found emdricallv in our laboratories that about 25-30 students &e necessary to he reasonably sure of a reliable regression of the data. The scheme we have presented provides a quick, reasonable way to grade students in an analytical laboratory. The level of precision and accuracy for the experiment is quickly ohtained and the students' grades are dependent on the combined experiences of the students as well as that of the instructor. A faulty procedure can be easily spotted by examination of the slope and intercept of the line. Since new samples can be prepared semester by se-
286
/ Journal of Chemical Education
Figure 2 . The remaining data after an intermediate regression. (The graphical computer output correspnding to the data of the table.)
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2100
23.W
25.W
2703
29.W
Dispensed Ni Volume ( m l l Figure 3. All of the students' results plotted along with the final value of the slope and intercept. (The graphical computer output Corresponding to thedata of the tab1e.I
mester with little effort, the students can know the true value and compare their results with it without concern about dry lahing. This procedure has now been used for seven semesters for a variety of experiments, including, acid-base titrations, nickel-EDTA titrations, nickel-DMG orecioitations. oermaneanate-oxalate titrations and potentiometric tit;ations.'~he classes have been composed of maiors and non-majors and have averaged about 80 students in size. It should he noted that the use of the linear regression coefficient as a figure of merit does not have rigorous statistical foundations. However, experience indicates that this parameter used with discretion and in conjunction with the graphical output yields consistent, useahle results.
