Re:Math Intuition Cheatsheet
by chestervonwinch on Friday July 25, 2014 @01:56PM
(#8086486)
That's pretty amazing you came up with that in 9th grade. You may know this already but for anyone who doesn't, this is a result of covariance being an inner product [0]. Remember that the inner product of a vector with itself is the length of that vector squared. So if you had another data set, y1, ..., yN, the correlation ( = covariance / (std.dev(X)*std.dev(Y)) ) is the inner product divided by each length which, if you remember, is the angle between your vectors (datasets after removing the means.)
http://en.wikipedia.org/wiki/Covariance#Relationship_to_inne...
Great example, I like that. I started realizing that many instructors don't necessarily have an intuition for the topic they're teaching, and it's often up to the individual student to seek it out. I'm hoping to do more stats concepts down the line, this is a nice analogy for why standard deviation is defined as it is.
Personal anecdote related to Pythagorean theorem/euclidean distance.
When in ninth grade, I asked my math teacher to explain why why the the standard deviation is like it is and why the square root and not something else. He could not explain that to me satisfactorily.
I spend few hours looking at the formula myself and plotting and drawing different datasets until I made the following association: if dataset is like N dimensional vector and its values x1, x2, .. xN are coordinates, then I'm looking at euclidean distance from mean plus adjusting for the number of dimensions. I think this single moment of cryptic formula making sense changed my attitude towards statistics and math. Dataset ~ vector and data-point ~ vector-coordinate association helps to understand statistics using geometrical intuition.