Nelson McKaig, Jr.
St. Petersburg Junior College st. Peterrburg, Florida
A
Grading Quantitative Analysis Laboratory Reports
beginning course in quantitative analysis usually includes the analysis of various pre-analyzed samples by the students. The objective is to develop skill in the use of basic chemical procedures and equipment. The student's skill can be measured best by obsewing the accuracy of his results, and these results should form the principal component of his laboratory grade. The purpose of this paper is to describe the development of a chart which provides a simple method of grading a student's analysis. I t seems to be applicable for several kinds of materials commonly used for gravimetric, volumetric, and at least one colorimetric procedure, and over a wide range of concentrations. Several methods have been suggested for grading such analyses. In most methods, no provision is made for the effect of concentration of the constituent, and some authors suggest separate scales for the various substances analyzed. Power (1) based his grading system on standard deviations of 6-8 parts per thousand (ppt) to represent an average grade, the exact value depending upon the substance analyzed. Blaedel, Jefferson, and Knight (3)have developed a point scale based on absolute deviations, in which an analysis with an error greater than 1.5% is given a credit of five points, for the experience gained in performing the experiment. More accurate analyses are given higher grades on a geometric scale, to a maximum of 10 points. Rosenstein and Smith (3) have developed a grading system adapted to computers, which permits comparison of a student's results with those obtained by the rest of the class. They base their system on several factors, including accuracy and precision, both of which are calculated as relative deviations. Daugherty and Robinson (4) use a graduated scale of relative deviations which is specific for each substance analyzed. Any of these systems can provide a fair and objective grade, if the samples used by the students are similar in concentration. But if the samples vary widely in concentration from one student to another, difficulties in grading may develop. A simple scale based on percentage of the entire sample (absolute error or absolute deviation) will favor students having samples in the lower concentration ranges. If the error is calculated on the basis of constituent (relative error or relative deviation), better grades can be obtained with samples in which the constituent is more concentrated. For example, it is usually easier to analyze a 5% ore with an accuracy of 0.10% than to get this level of accuracy with a 50% ore. But if this absolute value is converted to its equivalent relative value, 0.10% absolute error becomes Presented at the Florida Section Meet,ing-in-Miniature,American Chemical Society, Tampa, Florida, May, 1966.
20 ppt relative error in the case of the 5% ore, but is only 2 ppt in the case of the 50% ore. The answer to this problem is either to restrict the composition of the student samples to comparatively narrow limits, or to develop a system of grading which is applicable over a wide range of concentrations. The latter alternative provides more flexibility in conducting the course, and also reduces the difficulty of obtaining a supply of suitable samples. Pre-analyzed samples of copper ore have been obtained from suppliers with a range of 475 to over 40% Cu; iron ore samples have been received ranging from 13% to nearly 70% Fe.
-1.75~'
000
'
50
1.50
100
I
Concentration of constituent, log percent Figure 1.
Relation between the concentration of constituent in student
romplor and the abrolvte error of the analyses.
In an attempt to develop a satisfactory system for grading samples of such widely different concentration, astudy was made of the results of more than700 student analyses of pre-analyzed materials, made during the last 10 semesters. All analyses graded F were omitted, since in these the errors were very large and were mainly caused by carelessness, spilling, or failure to follow correct procedures. Preliminary calculations showed that about 9% of the results had been graded D. These accounted for almost 50% of the summation when the data were added to calculate averages, and over 50% when standard deviations were calculated. It was considered that inclusion of D-grade results placed too much emphasis on the poorest data, a t the expense of the good and average data. Hence only student analyses graded A, B, and C were used. The data were divided into groups consisting of lower and higher concentrations of the various materials analyzed, and the average percent concentration of constituent and percent of error was calculated for each group. All the averages were then converted to logarithmic values, and a graph was prepared. The result is Figure 1. The symbols for each different kind of Volume 44, Number 3,
March
1967 / 169
Comparison of Observed and Calculated Average Errors of Student Analyses, and the Average Deviation of Replicates of Student Results.
Averwe deviation from correct analysis
Number
NIgO in limestone; gmv. as oxinate
17 18 12 47
,843 3-6 615 .8-15
Mn in steel; Mn04- with Duboscq or B&L Spectronic 20
10 32 42
,752.0 -
.2(t.75
.2&2.0
,045 .(LS3 ,121 ,079 .024 ,029 ,028
27.8 17.0 13.8 16.9 45.6 29.7 31.9
analysis are connected by lines, with the chemical symbol for the constituent shown near the line. I n addition, there is a single point at concentration 4.94% (log = .694) which represents the average concentration and absolute error of 23 analyses for SiOl plus R203in limestone. These 23 samples were so similar in the total of these constituents that subdivision into high and low concentrations was meaningless. The SiOl and R203 were determined and reported separately by the students, but the results were combined in assigning a grade because the students could not purify either constituent from contamination with the other, due to lack of sufficient platinum crucibles. The most striking feature of Figure 1is that the data tend to form a single line, or narrow area, the axis of which can be calculated. This line, called the locus of absolute deviation in the figure, is described by the equation log d = 5 4 log C - 1.52
(1)
where d is the average deviation corresponding to a given concentration C of any constituent. The constant -1.52 is theintercept of the locus with the Y-axis, log 1.00. Similar graphs were prepared, using the same data but calculated as relative and as standard deviations. The graph for the relative deviations was practically a mirror image of Figure 1, with the vertical scale extending from over 45 ppt in the case of the low manganese concentrations to less than 4 ppt in the case of various samples containing over 50% of C1, CaO, NazO, or Fe. The equation for the locus of relative deviations was log d = -.47 log C
+ 1.48
(2)
Absolute deviation values, calculated by inserting a wide range of concentration values in eqn. (1) and then converting to relative deviations, were found to give numerical results almost identical to corresponding values calculated directly by eqn. (2). Similar results were obtained by using eqn. (2) to calculate absolute deviations. Therefore, either equation could be used
' Copies of the unabridged table, giving the data. for all the materials analyzed, and of the complete grading chart, a portion of which is shown in Figure 2, are available. The latter is mimeographed on both sides of a heavy sheet of 8.5 by 11 in. paper. Send requests to the Natural Science Department, St,. Petersburg Junior College, St. Petersburg, Florida 33710. 170
/
Journal of Chemkol Education
,040 ,071 ,098 ,070
24.1 14.4 10.9 14.6
005 012 @Xi ,009
3.7 2.6 2.9 2.3
,021 ,030 ,028
40.7 30.6 32.3
,003 -.001 -.000
4.9 -.9 -.4
.I2 .l i
.39 .21 ,024 ,033 .@31
78 34 43 45 45 35 36
for calculating the data on either basis. The graph for standard deviations was similar in appearance to Figure 1, the principal differences being a different slope of the locus and a greater vertical range among the points, caused by squaring the individual values and thus emphasizing the effect of the poorer analyses. The locus for standard deviations is shown in Figure 1. Its equation is log d = .46 log C
- 1.29
(3)
The difference in slope of the two loci shown in Figure 1 indicates that use of standard deviations as a basis for a grading system might result in varying grades for the same quality of work, depending upon the concentration of the constituent analyzed. For example, at the 50% constituent level, the absolute deviation shown by the graph is 0.259;; and the standard deviation is 0.319;& But at the 5% constituent level, the corresponding values are 0.072 and 0.105%, respectively. Thus at 50% constituent, the ratio of the two systems is 0.80, but a t 5% constituent, the ratio is 0.68. Student results are usually reported as percentage of the sample. The difference between the reporded analysis and the correct analysis is the absolute deviation. Therefore, comparison of a student result with a scale prepared from absolute deviation data should provide the simplest method of assigning a grade. Conversion of the absolute deviation to any other basis requires additional calculations. These require time, offer opportunity for error, and add nothing to the quality or accuracy of the data. Equation (1) should provide the simplest and best grading scale, if values calculated by it agree closely with the observed analytical data at all concentrations and for all the materials analyzed. To determine the validity of the derived equations, the observed average errors of the student analyses were compared with values calculated by eqns. (1) and (2). The table shows three of these comparisons,' covering a range of constituent from 0.20 to 69% of the sample, and illustrating a volumetric, a gravimetric, and a colorimetric procedure. The observed average values are shown in double column 4; the corresponding calculated values are in double column 5. The degree of agreement between the observed and the calculated values are shown in double column 6 headed "Difference." I n the table, the agreement is within 0.025% over this very wide range. In all the data used in pre-
paring Figure 1, t,he agreement in absolute deviation between the observed and calculated values was 0.0370 or less in all except four cases; and smaller than 3 ppt, of relative deviation in all hut two cases. These exceptions are believed to be random errors, except possibly for the errors of the MgO analyses. The MgO determinations, being at the end of an analytical sequence, contained the accumulated inaccuracies of t,he preceding determinations. This error was small, however, averaging less than 0.01% for the 47 samples studied. Little difference is noted in comparing the magnitude of the errors of either the gravimetric, volumetric, or colorimetric operations. All seem to conform closely to the deviations predicted by eqn. (1). This result is not, entirely unexpected. All tbe analytical methods used are capable of producing highly accurate results in the hands of a skilled analyst. Given good equipment (S), the inherent accuracy of any of the methods is many times better than the results likely to be obtained by unskilled students who are using the method, probably for the first time. Hence any inherent error of method or equipment can be neglected, for purposes of grading, and essentially all the error can be c.onsidered as student manipulative error, regardless of the analytical met,hod used. The table confirms the trend of Figure 1 in showing that the average deviation is affected more by concent,ration of const,ituent in the sample t,han it is by the nature of the material analyzed. I n the unabridged table, at coucentrations of about 50%) the average student errors in analyses for CaO, C1, Fe, and N a 0 all had about the same magnitude, namely 0.24 to 0.27%. At concentrations of about 25%, the average student errors in the analyses for Fe, Na,O, SO3,and Pz06were between 0.13 and 0.17%. The relationship is not so apparent when the data are expressed as relative deviations. An absolute deviation of 0.25% at 50y0 eoncentration is 5 ppt relative deviation; 0.15% absolute deviation at 25% constituent is G ppt relative deviation. This demonstrates the desirability of using relative, rather than absolute deviations, in grading student results in more concentrated samples, if a single-unit grading scale is used. But at lower concentrations of constituent, this advantage tends to disappear. The average absolute deviation of analyses for Cu, Mg0, and SiO? % 0 3 , when constituting about 5y0 of the sample, is 0.07 to 0.08%; this becomes 14 to 16 ppt relative deviat,ion. Since the above values are averages of student work, and in many cases the individual values were obtained by the same students, it appears that 0.25% deviation at the 50% constituent level, 0.15% absolute deviation a t the 2.5% level, and 0.075% absolute deviation at the 5% level can all be considered equivalent for grading purposes, for any of the materials cited. Although all the substances analyzed in this study seemed to give about the same analyt,ical errors at equal concentrations, the effect of concent,ration on the average error is quite large. For example, the maximum range of individual copper ore samples ext,ended from 3.90y0 Cu to 43.03y0 Cu, and the average of all the samples was 12.87%. Absolute errors calculated for these low, average, and high samples are 0.063, 0.12, and 0.23%, respectively. This is equivalent to 16 ppt, 9.3 ppt, and 5.4 ppt of relative
+
error. If the average were used as a basis of grading, on either the absolut,e or the relative deviation scale, the effectof the concentrat,ion factor on the student's grade is obvious. Ahout 80% of the student reports included in this study were based on the results of triplicate determinations; the others, having either lost a replicate or having one outside the limits of the Q-test ( 5 ) ,had an acceptable accuracy based on results of the surviving duplicates. The 0.95 confidence limits (5) of all the triplicate analyses were averaged, and the results are shown in the last double column of the table. Com-
Constituent, Figure 2.
percent
Section of a chart for grading quantitative anolyris experiments.
parison of these values with the differences shown in the preceding double column indicate that eqn. (1) should be satisfactorily precise for grading the preanalyzed samples listed. Any errors resulting from its use are small compared with the errors in the student's work. From the preceding discussion and the data of the table, it seems that eqn. (1) should be acceptable for grading pre-analyzed samples such as those discussed in this study. However, operational differences in various laboratories might influence the numerical values shown in the equation. The intercept constant, -1.52, was derived from average and better data. Therefore, it represents a higher value than a typical "average" grade. The slope constant, 0.54, represents the effect of concentration on the analytical errors. It permits calculation of the error at any concentration, after the scale has been established by selecting the value of the intercept constant. It would be laborious to calculate each student's grade by eqn. (1). It can be used to develop a table, or more conveniently a chart, for grading individual samples. I n Figure 2, analytical errors of 0.20, 0.40, 0.70, and 1.00% have been chosen as minimum values for grades of A, B, C, and D, respectively, a t 50Y0 constituent. Substituting the logarithms of these numbers into eqn. (I), the values of the intercept constant for each grade are -1.617, -1.316, -1.073 and -.91S, respectively. Using these values in eqn. (I), the error corresponding to a minimum grade of A, B, C, or D can be calculated for any concentration of coustituent. A chart covering the concentration range 0.25% to 100.00% has been prepared in a size suitable for grading student report^.^ In addition to indicating the deviaSee footnale 1.
Volume 44, Number
3,
Morch 1967
/
171
tions corresponding to grades of A, B, C, and D, provision has been made for percentile rating. This is based on a grade of 60% arbitrarily assigned for the minimum grade of D. Lines representing intermediate percentile grades, up to 96%, were then drawn on the chart, using the letter grade lines as guides. A section of this chart. is shown in Figure 2. As an example of its use, assume a student reports 6.90% for a sample containing 7.00% of a constituent. The error of his analysis is 0.10%. From Figure 2, the intercept of 0.10 on the vertical axis and 7.00 on the horizontal axis is located in the zone of B grades; more specifically, the grade can be estimated as 89%. Other scales can be prepared by choice of values for the deviations. The chart of which Figure 2 is a part,
172 / Journal o f Chemical Education
has been used for an elementary quantitative analysis course. It has also been used for grading freshman quantitative experiments. In the latter case, the numerical values on the Y-axis were made half as large, thus doubling the permissible error for a given grade. Literature Cited (1) POWER,F.W., J. CAEM.EDUC.,15, 339 (1938). W., J., JEFFERSON, J. R., AND KNIGHT,H. T.,J. (2) B L ~ D E L CAEM.EDUC.,29, 480 (1952). (3) ROSENSTEIN, R.D., AND SMITH,S. R., J. CUE>,.EDUC.,39, 620 (1962). (4) DAUGHERTY, K. E., AND ROBINSON, R. J., J. CHEM.EDUC., 41, 51 (1964). (5) DEAN,R. B., AND DIXON, W. J., Anal. Chem., 23,636 (1951).
