During a recent visit to the vault* of unshared musings, I discovered the pen-sketched framework of what I expound on here. It is common, I’m sure, when examining for the first time in years an artifact of one’s own creation, to be cognizant of a mild and sustained excitement that results from experiencing the fragments of one’s former self that the artifact evokes. Insofar as it possessed this characteristic, the finding of which I write was unremarkable. However, the finding’s timing, in combination with the content of what had been found, did achieve a level of peculiarity that, if inadequate to threaten the reader’s conceptions of reality, at least prompted me to wax curious. Here, then, is the formerly unshared musing.
Consider scenario A, illustrated:
At time T = 0, Prof. 1 begins a sixty minute lecture on topic ‘alpha’, which is received, with varying degrees of enthusiasm, by students ‘a’ through ‘f’. At the conclusion of the lecture, significant overlap will exist between each student’s understanding of topic ‘alpha’, yet, for myriad reasons, none will be the same. For example, small deviations in understanding will exist between the students as a consequence of their having ascribed to any particular portion of the lecture different types of meaning. Define Single Scenario Discrepancy (SSD) as the difference in the understanding of topic ‘alpha’ between any two students ‘a’ through ‘f’ at the conclusion of the lecture.
Next, consider scenario B:
Scenario A never happened. At time T = 0, Prof. 2 begins a sixty minute lecture on topic ‘alpha’, received, as before, by students ‘a’ through ‘f’, where ‘alpha’ corresponds to the same topic as in scenario A and students ‘a’ through ‘f’ likewise correspond to the same students. The only difference between scenarios, then, is that a different Prof. is lecturing, with the implication that the coverage of the topic will be slightly different. Define Double Scenario Discrepancy (DSD) as the difference in the understanding of topic ‘alpha’ between a particular student at the conclusion of the lecture in scenario A and that same student at the conclusion of the lecture in scenario B.
The question is whether it can ever generally be said that SSD is larger than DSD or vice versa. Certainly, if the lectures of Prof. 1 and 2 differ significantly in comparison to the differences between who the students ‘a’ through ‘f’ are as individuals, one would expect DSD to exceed SSD. But in lecture halls across the world, what is typically true?
What makes finding the aforementioned framework, and the framework itself, so curious is that I had no conception of random variables as a mathematical class of entities at the time that I sketched it in pen, yet not only do random variables most assuredly pertain to the scenarios (what I define above as discrepancies are referred to in the language of random variables as variances), they have underpinned the majority of my graduate coursework.
*Through its implication of grandiose impenetrability, the term ‘vault’, I concede, is a misnomer for the top drawer of my bedside nightstand, and was chosen owing to the allure of alliteration.