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This tutorial describes a technique for curving class grades using a normal curve.
The Normal Distribution
A distribution is the manner in which a set of values are distributedacross a possible range of values.
Many human and environmental phenomena follow a normal distribution,The smoothed histogram associated with the normal distribution is popularlyknown as the bell curve. Examples include:
- Standardized test scores
- The heights and weights of frogs
- The heights and weights of people in the United States
- The deviations of actual weights of potato chips from the weights marked on the package
While the formula for the curve is ugly, the concept is fairly simple.
The bell curve is a density curve, where the x axis representsvalues from the distribution. The area under the bell curvebetween a set of values represents the percent of numbers in the distributionbetween those values.
The middle value of a normal distribution is the mean, commonly calledthe average value. The mean value is represented with the lower-case Greek lettermu (μ), which looks like an italicized lower-case English letter "u".
The width of the bell curve is specified by the standard deviation.Standard deviation is represented by the lower-case Greek letter sigma (σ), which looks like a lower-case English "b" that's had too much to drink.
The number of standard deviations that a value departs from the meanis called the z-score. The z-score of the mean is zero. Convertingvalues to z-scores is called standardizing.
The area under the bell curve between a pair of z-scores gives the percentage of things associated with that range range of values.For example, the area between one standard deviation below the mean and one standard deviation abovethe mean represents around 68.2 percent of the values.
Male Height Example
For example, in the USA the distribution of heights for men follows a normal distribution.
The average height for men in the US is around five feet, ten inchesand the standard deviation is around four inches.
The number of very short men is the small area under the left tail of the bell curve,and the number of very tall men is the small area under the right tail of the bell curve.But most men are right in the middle around the mean.
68% of the distribution is within one standard deviation of the mean.So, 68% of American men are between five feet, six inches and six feet, 2 inches tall.
The name normal curve is related to the everyday concept of normal asconforming to a type, standard, or regular pattern.Mathematically, if you are right around the mean, you can be called normal.
Grade Distributions
In many school courses, the distribution of grades also roughly follows a normal curve. Forexample, this is a histogram of final point totals for a 28-person class.
With small numbers of values, the normality of the distribution is notoften obvious with a histogram.
However, if you take the integral of the normal curve, you get thecumulative area, which forms a sigmoid curve. This curve is often called ans-curve because it resembles an S. The X-axis is the grade, and the Y-axis isthe percentage of the class.
If you turn the curve on its side, you reverse those axes.The X-axis is the percentage of the class and the Y-axis is the grade.Each student represents a percent of the class.If there are 25 students, each student represents 4% of the class.
This means you can place bars under the S-curve by rank order withthe score of each student. If the distribution of scores is normal, the barswill line up with the curve.
Given this mean and standard deviation, you can convert point totalsto grades with a simple formula. For example, at an institution with a traditional percentage definition of letter grades, a typicalformula that places the mean of the class at 85% representinga B letter grade and a standard deviation of 10% givingstudents one standard deviation or greater with an A letter grade:
μ = mean of all student point totals
σ = standard deviation of all student point totals
z = (student_point_total - μ) / σ
percentage = 85 + (10 * z)
The normal curve is an abstract mathematical ideal, and reality rarely exactly matches thisideal. So, the bars will rarely exactly line up with the curve.However, if they are close, you know you have a normal distribution, and thisrepresents a useful technique for modeling and quantifying characteristics andperformance.
Choosing Coefficients
While the normal curve is a fairly rigorous model, the choice ofcoefficients used to convert the z-score to the grading system at anyparticular institution is dependent on the norms of the institution and theextent to which a course is expected to be "hard" or "easy"relative to other classes.
For example, at an institution where 93% = 4.0, 83% = 3.0, etc., the 85% /10% coefficients used above give a rough approximation of the GPAs within thestudent population, with slight stretching to represent an above-average level ofcourse difficulty for this particular population.
Change Over Time
The performance of individuals over a semester can fluctuate widely,especially in cases of students who have to deal with particularlydifficult personal challenges over the semester. However many studentsalso stay within a surprising narrow range over the semester.
Critiques
The learning process is complex and no evaluation system can ever beconsidered perfect. The following are some critiques of curving and responsesto those critiques.
Fairness
Some students may question whether it is fair to evaluate relativeto other students in the class.
Because the grading formula is applied equally to all students, it is fair,although this does raise the question of whether a curve is appropriatefor a given population and/or course topic.
In situations where there is a clear standard for what needs to be masteredto pass a course, curving may not be appropriate. Courses in the hard sciencesor courses that involve specific professional skills are possible examples.
However, in non-critical situations like general education courses where the studentsoften have extremely wide ranges of skills and engagement, gradingnormatively may be appropriate to avoid either failing large numbersof students, or giving high marks to students who are nothigh achievers within the norms of the institution.
Indeed, instructors commonly adjust assessments over the course of a semesterto normalize the distribution of scores. Use of a curve simply moves this adjustmentfrom assessment design to assessment calculation, shifting the focus fromthe metric to the material.
Effort
Students may express concern that their curved scores do not reflectthe amount of effort put into a class. This is especially common with low-performingstudents who have not been held to high standards in the past, or whoare seeking to leverage philosophical ambiguity into a scoring advantage.
It is, of course, difficult to quantify effort, especially when workingwith academically-diverse populations where some students will find masteryof the material requires little effort, and other students consistently struggle.
Use of a curve places the focus on performance. If a student has scores that arecoming in below the norms of the class, the focus can then be shifted to what specificallythat student needs to modify in their preparation to come up to the standardsof the rest of the class.
Competition
The use of a curve turns class performance into a competitive zero-sum game.This has been observed as a disincentive to study, and can be a disincentiveto the development of collaborative skills needed for most careers(Dubey and Geanakoplos 2016).
A similar dynamic has been observed inthe professional world in the process of evaluating employees on a curvewith stack ranking, where certain percentages of employees aregiven normally distributed rankings of excellent, acceptable, or poorregardless of absolute performance. This leads to a destructive focus on competition within the internal bureaucracy rather than an external focus on the core mission of the business.(Eichenwald 2012).
This critique may be especially valid in cases where collaboration isintegral to the class procedures, or in honors classes where the standarddeviation will often be small and curving would give low scores to students whose performance is only marginally different from their peers.
However, in situations like freshmen-level survey or general-educationclasses, where students largely function as independent agents, suchcompetition may arguably be a realistic mirror of the real world, encouragingcompetitive, high-performing students to excel.
Discouragement
When working with low-performing students who often have low academicself-esteem, and/or multiple layers of extrinsic factors that havecontributed to low performance, curving creates a positive feedbackloop that can reinforce poor self-image and lead to discouragement.
This critique will have different levels of seriousnessdepending on the population of an institution and the broader social justicemission of the institution.
The draconian argument is that education is part of a social filteringprocess where students are sorted into life paths appropriate to theircapabilities.
However, this could again be translated into a renewed focus on the materialand performance rather than the metric. Because curving turns scoring into amechanistic process, the focus can be turned to what caused the scores to below relative to peers, and on how those issues can be addressed to improveperformance. This requires individual initiative and attention on the part ofboth the student and the instructor, and may not be practical in situationswith high workloads and student-teacher ratios.
