The Bigger, the Better

What a scary equation.

However, the law itself is quite self-explanatory and well-known; its nickname is “the strong law of large numbers.” It dictates that an observed sample average from a large sample will get closer to the true population average as the sample grows larger. For instance, if we index more people, the ratio between men and women will get closer to about fifty percent.

Then what about the law of small numbers?

Like Loki and Thor (before Loki as a TV series came out), the law of small numbers is quite a frivolous theorem compared to the law of large numbers—and psychologists, instead of mathematicians, are interested in the “theorem” and the “cognitive bias.”

Amos Tversky and Daniel Kahneman first introduced the concept, as they found that people tend to consider a small sample size to be a highly representative sample, which is not true. For instance, when visiting colleges and universities, it is rather preposterous to evaluate a school based on the tour guide’s comments, as they are not highly representative of the real visage of the school. Moreover, it is rather unintelligent to bet on tails even if there have only been heads for the past decades; this, of course, is the gambler’s fallacy.

Then, one may ask, what are the results of the law of small numbers?

According to Tversky and Kahneman’s essay “Belief in the Law of Small Numbers”, part of the collection Judgment Under Uncertainty, it is heuristics, or shortcuts of the brain, that result in cognitive biases such as the law of small numbers. Specifically, it is the representative heuristics that lead to the belief in the law of small numbers. Representative heuristics works by comparing an event to a prototype or stereotype that we already have in mind. For example, when someone sees a student wearing a blazer and maybe carrying a briefcase, their brain matches the images with the stereotypical Deerfield student image, while when someone sees a student wearing pajamas who hasn’t combed their hair for the past three years, the observer’s brain will match the images with the stereotypical Choate student image.

Believers in the law of small numbers will always consider observed facts (even with a small sample size) highly representative of a certain parent population. For example, when someone sees a student wearing pajamas wandering around Albany Road, it is not wise to conclude that the student is a Choate student who forgot to get on the bus because of their outfit—maybe they just lost all their clothes in the E&R laundry service. But for the Loki believers, they must be a Choate student because the sample (the student) should always be highly representative of the parent population (Choate students).

In conclusion, when you have a good grade on some test, don’t worry about it because one sample size means nothing; likewise, when you have a bad grade on some test, do not fully trust it because one sample size means nothing. When you are doing a physical experiment, expect errors—because one instance is not always representative of the general model; when you are writing for Focal Point, don’t believe an author is omniscient just because they did well on their last essay, as one sample size means nothing.