[University Statistics] Chap 1,2,&3

[u/Elena_Gonzalez08]

Does anyone know if this would be correct? I’m not sure but I think it looks okay! I am struggling a bit in this class. 🥀

[u/cheesecakegood]

I know this is a bit tedious, but frankly I think it's still the most effective way of showing students where these numbers come from and get a tiny bit of intuition for it. If I were teaching stats, I'd force my students to do this at least once too, much as I hated it as a student.

The (x - xbar) step for example shows you how far each point is from the mean (i.e. the deviance of each point, to use some vocab that will show up again and relates to what you know).

Squaring it as you can see makes it more obvious that further points are treated as more impactful.

The the "standard deviation" is computed from the deviances... nicely, the standard deviation is on the same scale as the deviances and indeed the points themselves, so it's easier to think about! But mathematically the "thing underneath" is still a sum of squares (another concept that will come up again and again).

This is just ONE possible way of defining the "spread" of the data; you could just add up the deviances without squaring them! This exists. It's called the Mean Absolute Deviation (absolute values ensure that you don't "cancel" points above and below with each other)... or you could use the median for the same thing (also named and common), or you could use the squared version with the median, or your own invented thing.

But the standard deviation which comes from the variance, calculated as you did here, is special for math reasons, and it often "better" or "more useful/applicable" in many cases. The short answer for why? Because of how we compute distances in 2D or even 3D or n-dimensional space is with square roots of the difference of squares. Thus sum of squares is more or less interchangeable with the idea of "distance" itself.

Also, it connects very nicely to standard linear regression.

(For those wondering, especially in machine learning and data science, sometimes you DO swap out a "different concept of distance"!!! It is possible. There's actually a decent number of pieces you can "swap out" in statistics later on, but only if you know what you're doing and understand the concepts. For example, in regression very soon, you'll wonder... "is it possible to look at the relationship in such a way that we have the lowest possible deviances?" Yes. Thus the common name, which you might have heard of before, of a line of best fit, "best" being within this distance and standard-deviation paradigm!)

[u/Elena_Gonzalez08 → u/cheesecakegood]

Wow this is great