# Least-Squares fit for a line Concept

1. Jul 21, 2007

### cycam

1. The problem statement, all variables and given/known data

Prove the following useful fact: the least-squares fit for a line through any set of points (x_1,y_1) .....,(x_N, y_N) always passes through the "center of gravity" (x-bar, y-bar) of the points, where the bar denotes the average of the N values concerned. [Hint: you know that A and B satisfy the equation:
AN + B(sigma)x_i = (sigma)y_i ]

2. Relevant equations

A(sigma)x_i + B(sigma) x_i^2 = (sigma) x_i*y_i

A = [(sigma)x^2(sigma)y - (sigma)x(sigma)xy]/ [N(sigma)x^2 - ((sigma)x)^2]

B = [N(sigma)xy - (sigma)x(sigma)y]/ [N(sigma)x^2 - ((sigma)x)^2]

3. The attempt at a solution

I think the solution has to do with the solved equations for A and B and their relation to dividing by N, the number of values concerned, which eventually yields the "center of gravity," that is (x-bar, y-bar). I'd appreciate any help. Thanks in advance!

2. Jul 22, 2007

### cycam

...........bump!