Why variance in model and residual add up the way they do?

a. Standard deviation of a centered score s_x=\sqrt{\frac{\sum{x_i^2}}{N-1}}.

Hence if one thinks of the variable as a N-dimensional vector then the standard deviation corresponds roughly to the length of the vector.

b. When you regress y onto x, there is a component of y \widehat{y} lying along x, and then there is the remaining component e which is orthogonal to x.

c. \widehat{y}, e and y form a right-angled triangle with y being the hypotenuse. Hence,

|y|^2 = |\widehat{y}|^2 + |e|^2.

In plain english, sum of squares of dependent variable can be partitioned into squares that sum to the total. End of story.

Long live Pythagoras!

p.s. and the author of the book “The Geometry of Multivariate Statistics”, Thomas D. Wickens.

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