A Graphic Grade-Scaling Method John E. el son' Bldg 54 Room 109, Loyola University Medical Center, Maywood, IL 60153 Pratibha Varma-Nelson St. Xavier College, Chicago, IL60655 Therese A. Kloempken Hines VA Hospital, Hines, IL 60141 Assignment of student grades can generate considerable controversy, especially if those involved fail to understand the grading process or if inequity is perceived. Some complaints can be avoided by using exams and exam questions that are pretested for validity and reliability ( I ) . However, standardized achievement tests are usually not available in all curriculum areas. After administration of a nonstandardized test, the instructor is faced with the task of translating a nuB 4 meric test score on one scale to a letter or number 2 40 grade on another scale in an equitable and politically expedient manner. When exam scores are low or maldistributed, or when a student has been excused 20 from an exam, this translation process may be problematic. Previous articles in this Journal have discussed various aspects of grading: numeric techniques for ado justing student scores (2, 3); statistical methods for .OI .I 1 5 10 20 30 so 7060 90 95 99 99.9 99.99 assigninggrades to absent students (4,5);the hazards Cumulative Frequency inherent to exponential scaling equations; as well as the advantages and disadvantages of the linear ing of test scores. Unfortunately, the numeric methods Figure 2. Construction of a scaling diagram for conventing raw scores. described are difficult to explain and justify to students. Using our technique, the instructor can choose to scale grades to a predetermined minimum and maximum score or to a fixed mean and standard deviation. Excessive skew Our Method or kurtosis in the grade distribution is readily apparent, and the trends and amounts by which scores are raised is For these reasons, we propose a graphic grading process obvious. A relative cumulative frequency graph of raw and that makes use of commercially available normal pmbabilscaled scores is generated that is easily understood by stuity graph paper (6).Our system encompasses many of the dents. Using it, they can quickly determine their pernumeric schemes discussed above. centile rank in the class. We demonstrate this system uslng typical exam scores (Figure 1). Minimum = 16
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1/
Maximum = 66 Pnint.; . ..... = RR . . Mean =40.6
SU. Deviation = 15.0
I
15
25
35
45
55
65
Raw Test Scores -
Journal of Chemical Education
Our Procedure Regin by sortmg the test scores from lowest to higbest. Calculate the oercentaeeofstudents who fall at or below each score (relative cumulative frequency). Plot the raw scores (y) versus cumulative frequency (x) on normal probability graph paper (Figure 2). The smoothness of the curve depends on how evenly the raw scores are distributed. If several students have obtained the same raw score, the raw curve will be "flat" at that point. If the data approximates a straight line, it can be considered to have a normal distribution (7.8).The standard deviation determines the slope of the line, and the mean falls at the center of the x axis. A straight line is 'fit" to the raw data by inspection or by drawing a straight line through the points describing the mean, plus and minus one standard devi-
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Figure 1. A relative cumulative frequency graph using typical exam scores. 462
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'Author to whom correspondence should be addressed
ation (Figure 2, LineR). Recall that for a normal distribution, the following three statements are true. The mean, minus one standard deviation, falls at 16%. The mean falls at 50%. The mean, plus one standard deviation, falls at 84%.
' 1
Scaling Scores A "scaling linen is then drawn on the graph. The instructor may choose to scale the grades to a fixed mean and standard deviation. or to a fixed minimum and maximum score.ARer seleding the dksired mean and standard deviation (or desired hieh and low scores1 a straight line is drawn through th&e points (Figure 2, Line S). To convert raw scores to scaled scores, pmceed as follows. Travel from the raw score on the v axis to the fit line (R), up to the scaling line (S), then back to they 55 65 75 85 95 105 axis, where the scaled grade is read (Figure 2, Lines A, Scaled Test Scores B, C respectively). Alternatively the raw data points, may be used. rather than the fit line (R), Figure 3. A frequencyhistogram of scores after scaling The following caveats are offered in regard to seleo predicted. In this way, the student's performance relative tion of the scaling line. If the scaling line is parallel to the to his or her classmates is preserved. Strictly speaking, the raw fit line, the scores are linearly scaled. If the slope of predictive validity of exams should be established before the scaling line is greater than the fit line, small differusing this method (5).However, this is rarely done. ences in the raw scores are magnified. Low scores are traWe will not attempt to defend the fairness of "normalizditionally raised more than high scores, lest you ''rob fmm ing" or predicting grades. Our only claim is that this the poor to give to the rich". method makes the process easier to visualize and underIt is readily apparent that the least controversy is enstand. The actual grade that a student receives is detercountered when dimcult exams with a wide distribution of mined by how the instructor draws the scaling line. Obviscores are scaled ( I ) , in contrast to less challenging exams, ously, the scaling line should be drawn so that students where the scores fall over a narrow range. A slight smoothreceive a grade that is commensurate with their perforing of the distribution occurs during the scaling process. mance. However, students with identical raw scores end up with the same scaled scores. Informing the Students If desired, the equations (3) describing the fit (R) and Minimal criteria for passing a course should be estabscaling (S)lines can be used to convert raw scores to scaled lished at the start of the term. Then if this or any other scores. The mean, standard deviation, score, and m u l a grade-scaling, predicting, or correcting method is to be tive frequency are represented by b, m, y, and norm (x), used, students should be explicitly informed of that policy. respectively. Equations 1and 2 describe the fit line. EquaReference should be made to the method used in the course tion 3 describes the scaling line. Equation 4 describes the outline where grading is discussed (9). scaling formula. ~~~~~
yR = mR * norm (xR)+ bR
~
(1)
Computer Information
(2)
It is possible that others have also discovered this system for scaling and predicting grades. However, we can find no mention of it in the literature. The data analysis shown here-includingline fitting, calculations, and printing of probability graphs and bar histograms-was carried out with KaleidaGraph, a commercially available (10) scientific graphing program, on a Macintosh I1 (11)personal computer with Apple Laserwriter printer.
Rearranging, we get (YR - bR) norm (xR)= -
m~
ys = ms
* norm (xs) + bs
(3)
Substituting norm (XR)for norm (xs), we get ys=[~)*(Y~-bd+bs
(4)
PredictingScores
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mance (4,5),the followingprocedure is us&. The stident's percentile rank on a previous exam is established. If severa1 exams have been completed, the mean percentile rank can be determined. Using the student's previous rank and the sealing line for the exam missed, the student's score is
Literature Cited 1. Gay, L R. Edu~oliaMIReezmh? C o m p L s ~ i o s ~ r & i o s I ynndAppIimtiion, sis 2nd ed.: Cherl~eE. M a d Columbua.OH, 1981: Chapter 5. 2. M ~ ~ O J. Y ,T J. them E&C 1890,67,414-415. 3. Becker. C.E. J. Chem.Edvc 1881.68.309.
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7. ~uringt0n.u.8.; MUY,D. c.~o~lba~~r~mba6iiityond~t~tiatica ~ i t nh b h , znd d.; McCraw Hill: New Ymk, 1970: pp 141-143. 8. Spiegel M. R. Sehnumb ofPmbobilityondSfntisa~; M-w.Hill: 19,6 236. 11s146. 11. APPIS computer. cupertino. CA.
Volume 69 Number 6 June 1992
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