All Curves Are Not Created Equal

Grading yesterday’s real estate midterm I decided to curve the grades. I could have increased everyone’s score by some number, say 2.56 points. It’s simple. But that’s not what I did.

The exam consisted of 20 multiple choice questions worth 3 points each, 60 points total, and 8 matching questions worth 8 points each, 20 points total. The impetus to curve came from reviewing the matching questions, where I saw that some students had unexpected difficulty distinguishing one of the answers from another. I started by tinkering with the weight of correct matching answers. I started with 8 matches worth a total of 20 points, or 2.5 points each. First I increased the weight so 6 correct matches equaled 20 points–3.33 points each–but abandoned it when I saw that it added 6.6 points to the scores of students with 8 correct answers. That was nuts.

I played with other weights, from 2.8 to 3.03 points, but didn’t like how much they tilted the overall exam score towards the matching section. The mean score for the matching section is 82.5% correct, 1.75% higher than the multiple choice mean of 80.75%, but far more students–49%–correctly answer all the matching questions than correctly answer all the multiple choice questions–8%. It is fairer to employ a curve that respects the original 75%/25% weights of the two sections and does not discriminate in favor of students who perform better on the matching. I increased the weight of every correct answer 2.56% by changing the percentage calculation denominator from 80 to 78. In other words, ¬†a raw score of 70 points went from a percentage score of 87.5% to 89.7%–which my spreadsheet rounds to the nearest multiple of 1, for a grade of 90/100.

Increasing all raw scores by 2.56% does not, of course, increase all scores by the same number of points. A perfect score of 80 points goes from 100 (80/80) to 103 (80/78, ¬†rounded), while a score of 40 points goes from 50 (40/80) to only 51 (40/78, rounded). This made me pause. Is it fair? Increasing each grade by 3 points would increase the perfect grade by 3% and the half-perfect grade by 6%. Increasing each grade by 3% would increase the perfect grade by 3% and the half-perfect grade by 1.5%–less, if rounded to the nearest multiple of 1.

It becomes a policy decision: should the reward favor lower performance or higher performance? (Phrasing the question thus reveals its bias, compared to reward those who need it more or those who need it less?) After thinking and writing about this for inordinate time I employed a curve that rewards higher performance–but I also removed the rounding hit to the lower scores.

No doubt there are other approaches, which I am soon to be educated.

2 Replies to “All Curves Are Not Created Equal”

  1. victorpan

    Your first approach is actually the most statistically valid method without skewing the results of the best and worst performers. All other approaches mentioned seem to change the spread of the overall distribution of grades (imagine the bell curve going from flat to pointy)….

    Well, there’s also the ranking methodology (which isn’t a curve) where you disregard the absolute scores of all the tests and rank them relatively – you know, the original bell curve.

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