Normal, Not-normal, and Significantly-different

The word normal is loaded with meaning. The Merrian-Webster dictionary explains that it derives from the Latin word normalis meaning “made according to a carpenter's square, forming a right angle.” That can be, but with time the word also comes to encompass the idea of rules of behavior and norms of social interaction. Normal today entails a broad definition of “what is to be expected” and for many what is normal defines what is good. This idea has the name normativity.

Normativity is one of these ideas that can be right in some contexts and wrong in others. For a supermarket to estimate the normal number of toilet paper rolls sold per week can be very useful to avoid having too much/little of it. But if a pandemic hits, as we learned, as a society we buy more toilet paper than usual. Many of us were grateful to the few supermarkets that had an above-normal of toilet paper in stock.

Just as in this post, in this blog, I use the word normal a lot. Specifically, within the context of statistics and data analysis. I am utterly unqualified to discuss normalcy within the context of our social life but with a bit of website scrubbing, I can calculate statistic estimates with relative ease. Things like what is the average human skin color which is something we can somehow provide a ball-park approximation.

Statistics is a double-edged sword. In the sense that it allows us to account for whatever we want. We can, for example, find that on “average, humans have one testicle”. or that “the average human being has an above-average number of legs”. It is idiotic to advocate for an optimal number of testicles, but trying to keep the average number of legs closer to two per person could save many people hurt from wars or theIr aftermath.

What is important though is that in calculating what is normal I mean only what is average, i.e., what is close to the mean. Within the use of the word, I will make in this blog, normal entails every observation that is within one standard deviation from the mean. If the data follows a normal distribution or has an aggregation mechanism that follows the central limit theorem, then my definition of normal includes around 68% of all observations.

What lies outside the range between the mean minus one standard deviation and the mean plus one standard deviation is what I call not-normal. I use this word specifically to avoid bias. A value that is not-normal is not necessarily undesirable. Shaquille O’Neal has a not-normal height and this height opened doors for him to achieve not-normal wealth and education level – he has a doctorate.

A subset of not-normal values is what was known as significantly-different these represent values that are either two standard deviations above or below the mean. Just as Shaquille O’Neal any significantly-different value is also a not-normal value. The opposite is not always true. Scientists routinely make value claims about significantly-different metrics, I will try to avoid discussing these in this blog as I find them intrinsically more boring than approximating what is normal and what is not.