Why are there squares everywhere in statistics (e.g., normal density, variance, least squares, etc.)?

I remember asking my colleagues at Carnegie Mellon this very same question as I was first learning basic statistics in the early 1990s and they gave the same kind of answers as I found more recently in the AskStatistics subreddit. It’s an evergreen question, coming up regularly enough I think it’s fair to say it’s a zombie. This is probably going to be familiar material to most blog readers, but if you’re like me when I first started reading this blog decades ago, then read on. AskStatistics subreddit: Why is everything always being squared? AskStatistics subreddit:Why square in variance not absolute value? The answers are all over the place ranging from “the central limit theorem” to “it’s distance” to “it makes everything positive” to “it is smooth unlike absolute value” (another path to positivity) to “mathematical convenience” or “the math says so” (sure, but how?) to the “central limit theorem” (a more specific form of “the math says so”) to the link to entropy (it is the…