Mean and Variance

During the Montage dynasty, the land of Improvia was ruled by a monarch who would assign a number to every scene. Given any two scenes, the scene with a higher number was the one the monarch preferred, and every improviser in Improvia sought the favor of the monarch.

One might think that given the monarch’s numbers, Improvians had an easy way to rank improvisers. However, this couldn’t have been further from the truth. The problem was that improvisers were in multiple scenes, and there was variation across the numbers.

Some Improvians considered the average of one’s scenes to be the right way to rank improvisers. Others noted that improvisers with lower averages sometimes ended up in scenes with significantly higher numbers than those with high averages that were consistent. Still others wondered whether scenes from more than a few months ago should count to the ranking of an improviser, and perhaps looking at the trajectory of an improviser’s scene numbers over time was the right way to go.

Improvisers formed schools around their preferred philosophies of improv, some stressing techniques that could lead to scenes that were at or close to a particular number deemed respectable by the monarch (the consistent schools) and others emphasizing techniques that led to the same average but with higher variance, sometimes resulting in scenes with significantly higher scene numbers as well as significantly lower ones than those of the consistent schools. Some members of the consistent schools derided these “loose cannons” as pitching false promises in improv.

Of course, most improvisers wanted to be in scenes with higher numbers, but they didn’t want to risk losing the respect of the monarch by having scenes with low numbers too often.

The question put forth to improvisers was how to trade off between techniques that led to a consistent performance against those that could sometimes yield better results when they worked but could also cause spectacular failures when they didn’t. Not surprisingly, everyone had a slightly different perspective.

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