A STANDARD SCALE RALPH L. SEIFERT Carleton College, Northfield, Minnesota
IF
VARIOUS phases of a student's work are evaluated numerically and the numerical evaluations are averaged to obtain a &a1 grade for the course, then all the numerical evaluations should be reduced t o the same scale before they are averaged. In the case of written examinations the numerical grades on subjective or essay type examinations will in general range between 50% and 100Yo with a class average from 70% to 80%, if the instructor considers 75% to be indicative of average student performance. With objective examinations, graded to compensate for guessing, and with numerical problems the grades are more apt to range between 0% and 100yo with a class average near 50%. If a student consistently does the same quality of work relative to that of the other members of a class his h a 1 standing in the class will be approximately the same no matter what methods of evaluation are used nor how the evaluations are averaged. This; however, is not true for the student who is not consistent in the quality of his work. If the different sets of grades are not each reduced to a standard scale before they are averaged, the effective weighting factors for the various phases of the students' work will be different for each student and will influence the relative standing of all students! particularly those who are not consistent in the quahty .of their work. Various objective methods have been used to adjust grades in classes which are large enough to give significance to the class average. I n small classes objective methods must give way to subjective methods. The most equitable adjustment of grades to a consistent scale is obtained by the comput&ion of standard scores (z-scores)' or modifications thereof. This procedure is most accurate in very large classes. In any case it involves considerable computation on the part of the instructor and unless students have already been oriented to this method of scoring such orientation will be necessary. Any method of adjusting grades which will produce approximately the same results without
requiring much time or effort on the part of the instructor will be of value, particularly if the final scores fit into traditional schemes of grading. Many instructors who have to evaluate a student's work by a letter grade like to associate each letter grade with a definite numerical range, such as: A, 94-100; B, 85-93; C, 71-84; D, 60-70; E, less than 60. Dierent methods have been used to adjust student scores so that the adjusted grades would place each student in his proper letter classification. Ehret2 has recently described a device to adjust grades so that they are related to some arbitrary figure which has been chosen to represent average performance by a class. The principle underlying part of the method reported by Ehret is identical to the principle involved in the method which was devised by the author twelve years ago and has been used successfully by him during ten years of teaching. This latter method has the following advantages over Ehret's device. 1. It produces a better correlation between student grades obtained with different types of examinations and other written work. 2. I t requires only a slide rule with A and B scales, though a modification of Ehre.t's"device can be used. 3. It actually adjusts a set of class grades to any desired class average. The method used by Ehretz adjusts grades below class average according to the relation
where M is the average, or mean, raw score for the class, M,, is the, standard class average to which the grades are to be adjusted (taken as 75 by Ehret), S is the raw score for a given student, and Sad is the adjusted score. For raw scores above the average raw score the method used by Ehret utilizes the relation
' S ~ T EG. , M., "A Simplified Guide to Statistics for Students of Psychology and Education," Farrrtr and Rinehart, Inc., New York, 1938, p. 36. A
99
,
B
99
Adiuated soorea end selected standard average 97 96 95 94 93 92 91 90
98
. . . . . .
,
97
98
F
96
i1
95
, , ,
. . .
,
94 93 92 91 90 80 Rar. scores and average of raw soores
.
80
. 70
.
70
. 60
.
50
40 30 20 10 0
. . , . . 50 40 30 20 10 0
R e v k d Xumberinrof A and-B 6clln for Us. in Adjusting O r d e a
381
60
382
JOURNAL OF CHEMICAL EDUCATION
The adjustment in each case is readily made by the device described by Ehret. Unless the class range of raw scores is approximately the same on all examinations and in all phases of the students' work this method will not result in an equitable adjustment of grades suitable for combination to obtain an over-all average. A set of raw grades ranging from 0% to 100% will give adjusted grades with the same range even though the class average is adjusted to approximately 75%. A set of raw grades ranging from 50% to 100% will give adjusted grades with approvimately the same range. If equation (2) is used to adjust all the grades a more equitable adjustment results, for then all sets of grades will not only be reduced to the same class average but also to the same range of grades. The resulting adjustedgrades will be comparable to standard scores.' If 75% is selected to represent average performance by a class, then any set of grades with a range from n% 100)/2% will be adto 100% and an average of (n justed to the range 50% to 100% and an average of 75%. If 50% is selected as average performance the adjusted grades will be in the range 0% to 100%. If m% is selected as average performance the adjusted grades will be in the range 100% to (2m - loo)%. Of course it is only in very large homogeneous classes that the above relation will exist between the extreme grades and the average grade in accordance with a normal distribution curve. In most cases the approximation to this relation justifies the use of equation (2) to obtain an equitable adjustment of grades. Equation (2) can be used to adjust grades by proper use of the A and B scales of a slide rule. The scales must be read backwards starting with zero a t the right end. This can be done readily with a little practice. A better method is to shave the numbers from the A and B scales of an inexpensive wooden slide rule and renumber the scales in accordance with Figure 1. One scale (B in Figure 1) is selected to represent the raw scores. The other scale ( Ain Figure 1) represents the adjusted scores. If the averagsof the raw scores (on the raw score scale) is placed opposite the selected standard average, the proper adjusted grade will lie opposite each raw score. The two scales shown in Figure 1 are in position to correspond to a law score average of 57% which is being adjusted to a standard average of 75%. In this case a raw score of 83% is adjusted to a score of 90%. Any value may be selected as the standard average.
+
Since the above method for the adjustment of scores is based on equation (2), a device similar to that described by Ehret2 could be employed. However, all the lines on his device must be drawn through the same focus which he uses for grades above 75%. Such a device would still require visual interpolation or the use of a hundred radiating lines which might be confusing. The use of the A and B scales of a slide rule is much s.mpler. The method described herein has an additional advantage over that deecribed by Ehret. The average of the adjusted grades will equal the selected standard average. This will be true if either equation (1) or equation (2) is used exclusively for all grades. I t is not true of the adjusted grades obtained by the use of both equations. Thus the ten grades, 37, 45, 51, 54, 62,70,71,80,86, and 87, arbitrarily selected, give a raw score average of 64.3. If these grades are adjusted by the device described by Ehret for an assumed average class performance of 75%, the average of the adjusted grades is 71.6%. If the same grades are adjusted to a standard average of 75% by the method described herein the average of the adjusted grades is exactly 75%. In applying this method, as with most methods of scoring, some modification may occasionally have to be made because of the nature o f a given set of scores. This method assumes that the maximum possible raw score in each case is. 100 and that the examination or other work was of such a nature that the best students might be expected to obtain scores close to the maximum possible score. If these two conditions are not met in a given case all raw scores should first be multiplied by a factor which will raise the highest raw score to 100 or to some arbitrarily selected value near 100. The fmtor will be 160 (or the arbitrarily selected value) divided by the highest raw score. The selection of the factor to be used will be iuflwnced by the instructor's evaluation of the quality of work shown by the highest raw score. To this extent this method is not truly objective. The labor required for the adjustment of grades by this method is almost negligible compared to that required by more exact statistical methods, however the results obtained are similar. The raw scores in all phases of the students' work are reduced to the same class average and approximately the same spread of scores. This method has the further advantage of producing adjusted scores which fit easily into grading schemes that are traditional a t many schools.
