# Quantiles applied to two separate evaluation components vs applying to sum

Our school has a mandate that in any evaluation component (say quiz1, quiz2, etc.), the grades to be given are A, B, C and D to students. There is a further mandate that in any evaluation component (quiz1 for instance), the % of students receiving A should not be greater than 20% of the class and the % of students receiving A or B should not be greater than 60% of the class. [This implies that suppose all 100 students in my class had scored full points in both quizzes, all of them, unfortunately, necessarily have to receive a grade of C or D. (This has not happened thus far, thankfully!)]

We had two quizzes, quiz1 (15% weightage in GPA) and quiz2 (15% weightage in GPA). Technically, the mandate above applies to quiz1 separately and to quiz2 separately.

Can it be shown that if we do combine quiz1 and quiz2 into a single component QUIZ (say quiz1 was out of 50 points, and quiz2 out of 50 points, QUIZ = quiz1 + quiz2 out of 100 points) with QUIZ having weightage of 30% of the GPA, and apply quantiles to QUIZ to figure out the atmost 20% As and the atmost 60% As or Bs, that student grades would be better:

(a)on average, or

(b)even stronger, individually that no student is worse off in his final GPA for the course?

I'm not defending the edict. However, if you give six quizzes of equal weight on which students get grades 1 through 5 (worst to best) with probabilities .1, .2, .4, and .2, respectively, and sum the six scores, then you'll likely have a reasonable spread of total scores to which final grades might be somewhat equitably assigned.

One simulated experiment with six such quizzes in a group of 100 students is summarized below:

set.seed(925)
t.5 = replicate(100, sum(sample(1:4, 6, rep=T, p=c(1,3,4,2))))
stripchart(t.5, meth="stack", ylim=c(0,5), pch=20)
table(t.5)
t.5
10 12 13 14 15 16 17 18 19 20 21 22  # total scores
1  2 10  7 16 14 18 19  8  2  1  2  # numbers of students


The edict may be a way of suggesting not to construct quizzes on which all students get the same grade.

• Thank you. It is clear to me that combining will give a better spread of scores, can it be argued that students will be no worse off if they are assigned a grade (and hence a numerical gpa) based on the total sum instead of assigning individual grades to each of 6 components which are then combined in the usual way to obtain the final numerical gpa for the course? – Tryer Sep 25 '20 at 16:31
• It would depend on what the 'usual' way of getting a numerical gpa is. – BruceET Sep 25 '20 at 16:34
• What I meant by the 'usual' way was, suppose A=4, B=3, C=2, D=1. Suppose quiz1 (15%) student gets A, quiz2 (15%) student gets B, final exam (70% weightage) student gets A, then student's gpa = 0.15 x 4 + 0.15 x 3 + 0.7 x 4 = 3.85 – Tryer Sep 25 '20 at 17:17
• If quiz scores and grade values are the same, then of course it can make no difference. // If you have a grading system in which no one can 'fail', then I wonder about its purpose? Merely to demoralize those who get D's, before 'passing' them on to get more D's? – BruceET Sep 25 '20 at 18:55