MARTIN H. LITTLE' Union College, Schenectady, New York
As E m m 2 has pointed out, it is not always possible
can be used for all values if the adjusting is downward. (Compare Method 5.) Method 5. Seifert4gives a method which makes use of the slide rule, but in his method the A and B scales are used, and the scales have to be renumbered from right to left. This method does not give a proportional increase (when adjusting upward), and favors the students with the lower marks, much more so than in Method 1, since larger increments are added to the lower marks than to the higher marks. Using the hypothetical values again, we find that the average 50 becomes 75, 67 becomes 83, 25 becomes 62. The following illustrates the nonproportional results:
to write examinations for which the class average will he close to 75 per cent, 75 usually being considered as the "ideal" average. The purpose of this article is to compare the various methods that may be used for adjusting to 75 the average of a set of marks, which average deviates considerably from 75. FIVE METHODS O F ADJUSTING GRADES
Method 1 . A method which is often practiced and the one which is perhaps the simplest, is merely to add or subtract the same number of percentage points from each student's original mark. This method tends to favor the student with the lower mark (when marks are adjusted upward). To illustrate with an extreme case: Suppose the class average in an examination is 50 per cent and the top mark in the class, 67 per cent. Adjustment would be made by adding 25 percentage points to the original mark. In the first instance the raverage mark ig three-fourths of the top mark. After adjustment the average mark is close to five-sixths of the top mark. Method 8. The method of using a slide rule is a h o s t as simple in procedure as the first. This method gives propol+onal increase (or decrease) to each mark. In the hypothetical case: 50 becomes 75; 67 becomes 100. Fifty is three-fourths of 67 and 75 is three-fourths of 100. Method 3. A simple graphical method described by Horton3 avoids a difficulty that may arise with the slide rule method. Let us reconsider the hypothetical case of a class with a test average of 50 per cent. If there are marks above 67 per cent the slide rule method will give values above 100. Horton's method favors (when adjusting is upward) those marks close to the average, i. e., as marks approach 100 or approach zero, they are affectedto a progressively lesser extent. Method 4. The method of Ehret2 involves the use of a specially constructed graph. In adjusting marks upward, marks below 75 are proportionally adjusted. Those above 75 will be decreasingly affected as the marks approach 100. It can be seen that Ehret's chart also overcomes the difficulty that may be encountered with the slide rule method. Ehret notes in passing that a simple slide rule setting can be made for values below the average, but not for those above. The slide rule 1 Present address: Albany College of Phermacy, Albany, New York. 2 EHRET, W. F., THISJOURNAL, 25,69C-l(1948). HOIITON, W.S., i b d , 26,286 (1949).
Adjusted
Originally
upward,
When adjustment is the possibility of obtabing an adjusted mark greater than is very improbable; however, when adjustment is downward, a very poor student could receive an adjusted mark of less than (See Table ~h~~~discusses if^^^,^ method further in letter to the Editor,s COMPARISON OF RESULTS
In Table 1 the results obtained by the various methods are tabulated. In column B the original marks obtained from a test taken by the author's class of 59 students we given. In column A the number of students receiving a particular mark is given. (Thus, four .students received a mark of 88 per cent.) The remain1% columns are numbered to correspond to the method used to arrive at the adjusted mark tabulated in these columns. At the bottom of each column the average, median, etc., are given as indicated. Methods 1 and 2 give similar results. Down to a mark of 55 per cent there is at most a difference of one percentage point. Method 1 gives values higher than obtained with Method 2 for marks above the average, but lower for marks below the average. The reverse situation exist's when marks %,readjusted upward. Both are the same in the vicinity of 75 per cent. Below 55 per cent a greater differenceexists. Needless to say the greater the number of percentage
267
6
SEIFERT, R. L., THISJOURNAL, 26.381-2 (1949). EHRET, W, F., aid.,26,394 (1949).
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JOURNAL OF CHEMICAL EDUCATION
TABLE 1 A
B
l
Z
S
4
6
increments added to marks above 75 ner cent are not as large as they should be; decreasing6 so as the marks approach 100 per cent. When adjustment is downward the increments subtracted are not as large as they should be for proportional results (cf. Method 2). Above 75 per cent these three methods are similar. [Methods 4 and 5 should be identical. (See Seifert.') Discrepancies may be attributed to errors in readmg from the graph used in Method 4.1 Below 75 per cent Methods 3 a n d 4 are similar, but marks obtained by Method 5 deviate greatly as descending values are tabulated. (Also note that below 75 per cent values for Methods 3 and 4 are similar to those for Method 2.) Similar results should be expected with Methods 3 and 4 since both methods have zero as the lowest mark, 100 as the highest mark, and make the greatest amount of adjustment to those marks close to the average. Any discrepancies may again be attributed to errors in reading from either graph. Further comparison is made in Table 2 where the 10 marks are those marks which were arbitrarily selected by Seifert. In this case marks are adjusted upward. '
Average for 59 students Median No. of A's (9&100) No. of B's N??9i) C's (7S79) No. of D's (61-69) No. of F's (below 60) Spread between highest and lowest mark
TABLE 2 81 82
75 76
75 76
76 76
76 76
Av.
The author hopes that this paper bas shown the advantages and disadvantages of each method so that each instructor may choose the method he feels satisfies his points that the original average mark must be changed purpose. The author prefers Method 2 which gives for adjustment to an average of 75 per cent the greater equitable results. When the difficulty arises of having the deviation of adjusted marks between methods 1 and marks adjusted to values greater than 100 per cent, the 2. (Compare results in Table 1 with those in Table 2.) followingmodification of Horton's method is suggested: However, it would seem that for averages that are ad- For marks above the average, instead of constructing justed within six percentage points, either method the graoh bv drawing a line from the average ooint (75. would be satisfactoly, if there are no adjusted marks 64 f i r the marks tabilated in Table 2) to t h i & n t (100; below 50 per cent. loo), draw the line to (100, X), where X is the value of When adiustment is uoward. Methods 3. 4. and 5 the highest mark of the particular set of marks which take care oithe difficultyAthat lhay arise with ~ e t h o d s are to be adjusted. (For the marks tabulated in Table 1 and 2, namely, having adjusted marks greater than 2, X is 87.) Call this method, Method M3. It can 100 per cent. However, in the process of eliminating readily be seen that the average of adjusted marks will this difficulty, the adjusted averages obtained with be closer to 75 when Method M3 is used than when Methods 3 and 4 tiill be less than 75 per cent, since the either Method 3 or Method 4 is used. 75
75
70
76
77
