Academic Forum 22 2004-05 What’s My Grade? Michael Lloyd, Ph.D. Professor of Mathematics and Computer Science Abstract The distribution of students' course grades for a variety of classes will be examined and a probabilistic prediction of their course grades based on their current performance will be derived. Besides being statistically interesting, the data and techniques could be used pedagogically. Introduction “What’s my grade?” is a common question instructors are asked by their students before they take the final. In this paper, I will investigate some predictors for a student’s course grade. The following table lists the courses that are studied in this paper and the corresponding level. Course Level College Algebra 1000 Plane Trigonometry 1000 Precalculus 1000 Statistical Methods 2000 Probability/Statistics 3000 These courses were selected because I teach them regularly and thus have sufficiently large sample sizes. Also, only data from the summer 2001 semester through the fall 2003 semester were used because it was in 2001 that I started giving short homework assignments to all of the aforementioned classes instead of quizzes. Withdrawals were ignored because I do not keep partial grade information for such students; also, there were very few incompletes, so these were also omitted. Dependence on Semester The following table show the results of three 1-way ANOVAs on the course grade where 4 was assigned to an A, 3 for a B, etc. Statistical Methods and Probability/Statistics are only taught one semester per year so they could not be included in these ANOVAs. 60 Academic Forum 22 There may only be a significant difference in the average course grade between semesters for Precalculus. Grades are probably higher in the spring because most students in that course are fresh out of high school in the fall. The students who fail Precalculus in the fall are more serious about studying when they retake that course in the spring. These ANOVAs support the decision that there is no semester dependence and thus lumping all the data together for each class is justifiable. 2004-05 Semester Significance Fall, College Spring, 0.47 Algebra Summer Fall, Plane Spring, 0.46 Trigonometry Summer Fall < 0.06 Precalculus Spring Distribution of Grades Plane Trigonometry 30 20 10 Percent Before investigating how to predict students’ grades, I thought it was a good idea to examine the distribution of course grades. The distribution for the lower level classes Plane Trigonometry, College Algebra, and Precalculus are similar. Statistical Methods stands out as having the property that lower course grades are earned with lessening probability. There are also many high course grades in Probability/Statistics. 0 A 40 30 30 20 20 10 10 Percent Percent D F B C D F Precalculus College Algebra 0 B C Grade 40 A B C D 0 A F Grade Grade 61 Academic Forum 22 2004-05 Statistical Methods Probability/Statistics 40 50 40 30 30 20 20 10 Percent Percent 10 0 A B C 0 D A Grade The following table gives 95% confidence intervals for passing each course, where passing is defined to be an A, B or C. Note that the pass rate for College Algebra is significantly less than both Statistical Methods and Probability/Statistics. Also, the Trigonometry pass rate is significantly less than Statistical Methods. B C D F Grade Number students Probability 63 ± 9% 115 College Algebra = (54, 71)% 67 ± 11% 75 Trigonometry = (56, 77)% 76 ± 8% 111 Precalculus = (68, 84)% 89 ± 10% 38 Statistical Methods = (80, 99)% 83 ± 11% 46 Probability/Statistics = (72, 94)% Course Average Prediction Based on the First Exam 62 Trigonometry 90 80 70 60 50 Course Average The following are scatter plots and regression equations of (exam 1, course average). (Students with a missing exam 1 score were omitted.) All significance levels were 0.000 except Probability/Statistics with a level of 0.062. Note that the coefficient of determination R2 almost decreases as course level increases. Thus, the first exam is more indicative of a student’s ultimate course grade in the lower-level courses. The variables are C = course average, E1 = exam 1 average, and n = sample size. Note that the exams are all out of 80 points. 40 30 20 10 30 40 50 60 70 80 Exam 1 C = 12.1 + 0.743E1, R2=0.35, n=72 90 Academic Forum 22 2004-05 Precalculus 90 70 80 60 70 50 60 40 50 30 40 Course Average Course Average College Algebra 80 20 10 0 10 20 30 40 50 60 70 80 90 30 20 10 20 Exam 1 30 40 50 60 70 80 90 Exam 1 C = 23.3 + 0.608E1, R2=0.31, n=111 2 C = 2.31 + 3.64E1, R =0.68, n=112 Statistical Methods Probability/Statistics 90 80 70 80 60 50 Course Average Course Average 70 60 50 50 60 70 80 40 30 20 10 Exam 1 20 30 40 50 60 70 80 90 Exam 1 2 C = 50.9 + 0.185E1, R2=0.05, n=46 C = 6.1 + 0.926E1, R =0.34, n=38 The following tables give the probability of passing or failing the course conditioned on passing or failing the first exam. A perfect correlation would correspond to the 1 0 . probability matrix 0 1 LM OP N Q Plane Trigonometry Pass first Exam Fail first Exam Precalculus College Algebra Pass CourseFail Course Pass first Exam 0.84 0.16 Fail first Exam 0.20 0.80 Statistical Methods Pass CourseFail Course Pass first Exam 0.92 0.08 Fail first Exam 0.00 1.00 63 Pass first Exam Fail first Exam Pass Fail Course Course 0.78 0.22 0.41 0.59 Pass Fail Course Course 0.89 0.11 0.38 0.62 Probability/ Pass Fail Statistics Course Course Pass first 0.91 0.09 Academic Forum 22 2004-05 Note that only 1 individual failed his or her first Statistical Methods exam. The following tables give the probability of passing or failing the course conditioned on the earning an A or B on first exam. For College Algebra and Plane Trigonometry, the first row of the probability matrix is closer to [1,0] than when conditioning on the first exam. Exam Fail first Exam 0.64 Pass Course A or B on first Exam 0.95 C or worse on first 0.39 Exam Fail Course 0.05 Pass Course Fail Course 0.90 0.10 0.58 0.42 Pass Course Fail Course 0.89 0.11 0.79 0.21 Plane Trigonometry Precalculus College Algebra Pass CourseFail Course A or B on first Exam 0.95 0.05 C or worse on first Exam 0.44 0.56 A or B on first Exam C or worse on first Exam Probability/ Statistics Statistical Methods Pass CourseFail Course A or B on first Exam 0.92 0.08 C or worse on first Exam 0.83 0.17 The following tables give the probability of passing or failing the course conditioned on earning an A on first exam. In every course except Probability/Statistics, an A on the first exam implies passing the course. In fact for Probability/Statistics, passing the first exam is almost independent of passing the course. College Algebra Pass Course Fail Course Exam 1 = A 1.00 0.00 Exam 1 < A 0.57 0.43 0.36 A or B on first Exam C or worse on first Exam Plane Trigonometry Exam 1 = A Exam 1 < A Pass Course 1.00 0.58 0.61 Fail Course 0.00 0.42 Precalculus Pass Course Fail Course Exam 1 = A 1.00 0.00 Exam 1 < A 0.70 0.30 Statistical Methods Pass Course Fail Course Exam 1 = A 1.00 0.00 Exam 1 < A 0.88 0.12 64 Probability/ Statistics Exam 1 = A Exam 1 < A Pass Course 0.83 0.82 Fail Course 0.17 0.18 Academic Forum 22 2004-05 Course Average Predictions Based on the First Three Exams Near the end of the course, students become increasingly concerned with what their course grade will be. The current homework average was not included because I was seeking a convenient method for predicting the final grade. Also, it was inconvenient to determine the current average for this study. The variable exam average (EA) is obtained by adding the first three exams and dividing by three. Any student with a missing exam was omitted. Trigonometry 90 80 70 60 Course Average The following scatter plots and regression equations are for (3-exam, course average). The significance levels for the courses were all 0.000. Note that R2 decreases as the course level increases. I think this is because upper-level students are more flexible in improving their study habits if they do poorly on the first exam. 50 40 30 30 40 50 60 70 80 90 Exam Average C = 0.202 + 1.001EA, R2 = 0.83, n=69 College Algebra Precalculus 80 90 70 80 60 70 50 60 Course Average Course Average 40 30 20 10 10 20 30 40 50 60 70 50 40 30 20 80 2 C = -0.398 + 1.002EA, R = 0.90, n=103 50 60 70 80 C = 6.87 + 0.898EA, R2 = 0.79, n=104 Probability/Statistics Statistical Methods 90 80 80 70 70 60 Course Average Course Average 40 Exam Average Exam Average 60 50 40 30 50 60 70 80 50 40 40 Exam Average 50 60 70 Exam Average 2 C = 17.0 + 0.759EA, R2 = 0.53, n=43 C = 11.0 + 0.895EA, R = 0.64, n=37 65 80 Academic Forum 22 The following tables give the probability of passing or failing the course conditioned on the 3-exam average. Except for Statistical Methods, about 8% of everyone who has a passing 3-hour exam average will fail the course. College Algebra Pass Course Fail Course Passing Avg. 0.93 0.07 Failing Avg. 0.26 0.74 Statistical Methods Pass Course Fail Course Passing Avg. 1.00 0.00 Failing Avg. 0.33 0.67 The following table gives the probability that the 3-exam average underestimates, predicts exactly, or overestimates the course grade. The adjacent table indicates the rare event when the 3-exam average was off by two or more letter grades. 2004-05 Plane TrigonometryPass Course Fail Course Passing Avg. 0.92 0.08 Failing Avg. 0.25 0.75 Precalculus Pass CourseFail Course Passing Avg. 0.92 0.08 Failing Avg. 0.35 0.65 Probability/ StatisticsPass Course Fail Course Passing Avg. 0.91 0.09 Failing Avg. 0.73 0.27 UnderestimatesSame Overestimates College Algebra 0.17 0.66 0.17 Plane Trigonometry 0.12 0.78 0.10 Precalculus 0.19 0.66 0.15 Statistical Methods 0.43 0.49 0.08 Probability/Statistics 0.42 0.44 0.14 College Algebra Plane Trigonometry Precalculus Statistical Methods Probability/Statistics A more sophisticated method for using the first three exams would be to use multiple linear regression. The following are scatter plots of (multilinear prediction, course) average. never underestimated once overestimated once underestimated 4 times underestimated once probability 0.00 0.01 0.01 0.11 0.02 Trigonometry 90 80 70 Note that for the semesters used in the study, College Algebra and Precalculus used essentially the same book. Also, the middle exam (E2) covered logarithms and exponential functions had the most influence in the regression model. Course Average 60 50 40 30 Rsq = 0.8371 40 50 60 70 80 90 Predicted Avg C = -0.6+0.311E1+0.300E2+0.403E3,R2=0.84 66 Academic Forum 22 2004-05 College Algebra Precalculus 80 90 70 80 60 70 50 60 Course Average Course Average 40 30 20 10 Rsq = 0.9059 10 20 30 40 50 60 70 50 40 30 80 Predicted Avg 60 70 80 Probability/Statistics Statistical Methods 90 80 80 70 70 60 Course Average Course Average 50 C = 7.5+0.251E1+0.373E2+0.266E3,R2=0.80 C = -0.4+0.280E1+0.373E2+0.350E3,R =0.91 60 50 Rsq = 0.7278 60 40 Predicted Avg 2 50 Rsq = 0.7999 30 70 80 90 50 40 Rsq = 0.5388 50 Predicted Avg 60 70 80 Predicted Avg 2 C = -8.9+0.661E1+0.334e2+0.189E3,R =0.73 C= 17.5+0.225E1+0.313E2+0.212E3,R2=0.54 For trigonometry, the last exam (E3) included the law of sines, law of cosines and vectors was most influential variable. For Statistical Methods, the first exam (E1) was the most important. This exam covers descriptive statistics and interpretation of basic statistical graphs and measures. For Probability/Statistics, the second exam (E2) was the most important. That exam primarily covers probability word problems. There was only a slight improvement in R2 over the using the 3-exam average, so the use of this more sophisticated approach is not justifiable. The following tables give the probability of passing or failing the course conditioned on the grade predicted by the multilinear regression model’s prediction. As expected, this does not improve much on the 3-exam average approach. 67 Plane Trigonometry Pass Course Fail Course Predict Passing 0.92 0.08 Predict Failing 0.21 0.79 Academic Forum 22 College Algebra Pass Course Fail Course Predict Passing 0.92 0.08 Predict Failing 0.32 0.68 Precalculus Pass Course Fail Course Predict Passing 0.91 0.09 Predict Failing 0.14 0.86 Statistical Methods Pass Course Fail Course Predict Passing 0.97 0.03 Predict Failing 0.00 1.00 The adjacent table gives the probability that the grade predicted by multiple linear regression underestimates, predicts exactly, or overestimates the course grade. Note that the table is more balanced and improved for upper level courses. 2004-05 Probability/ Statistics Predict Passing Predict Failing Pass Course 0.89 0.67 Fail Course 0.11 0.33 UnderestimatesSame Overestimates College Algebra 0.17 0.64 0.19 Plane Trigonometry 0.13 0.77 0.12 Precalculus 0.13 0.70 0.17 Statistical Methods 0.24 0.52 0.24 Probability/Statistics 0.22 0.54 0.24 The multiple regression model was off more than 2 letter grades only once each, and those were in Plane Trigonometry and Precalculus. Conclusion The following were determined in this study: • Precalculus students have on average higher grades in the spring than the fall. • In every course except Probability/Statistics, an A on the first exam implies passing the course. • For Probability/Statistics, passing the first exam is independent of passing the course. • The 3-exam average is more correlated to the ultimate course grade for lower-level courses. • A student with a passing 3-exam average is at least 91% likely to pass the course. Some of the patterns discovered in this paper may be included on my syllabi or on the class website. This would be done with the intention of helping the students make decisions like whether it would be in their best interests to drop a course. However, revealing this information may adversely affect their attitudes. 68 Academic Forum 22 2004-05 Ideas for Further Study The summer classes could be removed and the current homework average used for the prediction. (I do not assign homework in the summer.) Also, it might be interesting to compare the students’ actual course grades with what they think they will receive. Finally, it might be determined if gender has any effect on a student’s course grade. Biography Michael Lloyd received his B.S in Chemical Engineering in 1984 and accepted a position at Henderson State University in 1993 shortly after earning his Ph.D. in Mathematics from Kansas State University. He has presented papers at meetings of the Academy of Economics and Finance, the American Mathematical Society, the Arkansas Conference on Teaching, the Mathematical Association of America, and the Southwest Arkansas Council of Teachers of Mathematics. He has also been an AP statistics consultant since 2002. 69

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