The grading pattern

Rules have to be followed.

Take the rays in the example below. Each ray simply follows the laws of reflection. The underlying symmetry creates a pattern. Similarly, single events are a part of a larger pattern; a pattern that is understood only when we see a whole.

A cardioid in the making, captured by the author

My first time

I consciously thought about patterns when I was handed a B+ in Electrodynamics in my master’s course. I intuitively understood that the 70% marks was a reflection of the level of my understanding in my assigned material. That was the easy part. But when my instructor said that he graded on a bell curve and gave me a B+ since I was in the 85th percentile, I was lost.

I was more lost when I got an A with 65% marks in a math course which had the most ridiculous 10 hour open book exam. Clearly, grades were relative. Somehow the information about the highest marks and the number of people above me are rolled into grade and percentile. I just did not understand how.

Though I kept seeing this bell curve, commonly known as the Gaussian or normal distribution, it was only during the first year of my PhD that I understood what it meant. I had to assign the grades based on the same curve, you see. Nothing teaches a concept better than learning to use it correctly in a real world scenario!

As you can expect, once we understand something, we see it everywhere. Now, I cannot un-see it. Please join me.

Anatomy of a 1D graph

Before we delve deep, it is always good to state the basics, right?

Let’s say we are examining a phenomenon by measuring a variable, say Y1. This Y1 in-turn depends on independent variables X1,X2, X3,… and so on. Y is measured by varying X1, while keeping X2, X3,… and the rest of the variables constant.

Now for the plot, independent variable X1 is the X axis and dependent variable Y1 is the Y axis. Therefore, Y axis is the dependent variable that you are measuring; during this investigation, a bunch of other parameters are held constant. Find detailed notes on plots here.

Underlying symmetry and a graph

When does a measurement get a bell-shape and not a straight line? What is the underlying cause? Typically, any set random processes operating in the background affecting the quantity being measured yields a Gaussian curve. For instance, collect all the leaves from a tree, tabulate the number of them in different ranges of their height, and voila, you have a bell curve, again. Similarly, the path followed by the dust particles in muddy water, called Brownian motion, also shows a Gaussian distribution. Same with individual atoms being kicked around by laser beams!

Why the name?

It is a symmetric curve shaped like a bell (hence the name, duh!) where 50% of the values lie above and 50% lie below the mean x-value (denoted by Greek letter mu). The spread of the distribution is measured by the standard deviation (denoted by Greek letter sigma).

Distribution of marks in the Electrodynamics course.

See, for example, the marks distribution above (each blue data point denotes the number of students within a 5-mark range of the corresponding value), where 68% of the values fall under the width within 1-sigma on either side of the mean. 2-sigma width on either side of the mean already contains 95% of the values and 3-sigma is almost hundred percent (99.7%), but not quite. The well-developed semiconductor/electronic chip industry has “6-sigma level” processes where only 3.4 defects per million products on average occur! The famous Mumbai dabbawalas delivering lunch boxes on bicycles have also been found to operate at this same level of precision! Crazy, right?! For us mortals, knowing till 3-sigma is good enough.

Please add 68–95–99.7 to your list of favorite numbers.

How are grades assigned?

In the Gaussian plot in my grade problem, X axis is the random variable, marks. Y axis is the number of students getting that particular mark. We were a class of 50 students.

Percentile calculation in the Electrodynamics course

Now we take the integral of this curve to compute the “percentile”. Here we add up the number of students getting less than or equal to a particular mark, and convert it as a percentage of the total number of students in the class; thus the highest score is in the 100th percentile.

Ready to play true or false?

1. Only 1 person got an A+ (>95th percentile) in the Electromagnetism course.
2. The person who got B (50th percentile i.e. average) scored in the vicinity of 60 marks.
3. There were no exceptional students outside of the 3-sigma range in the class.

Fun, right? Happy Gaussian spotting!

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