correlation and regresssion question- What does each of the followings tell you? a) Y-Y' b) X-X' c) Σ(Y-Y')

Question

correlation and regresssion question-
What does each of the followings tell you?
a) Y-Y'
b) X-X'
c) Σ(Y-Y')
d) Σ(Y-Y')^2
e) Σ(Y'-Ybar)^2
f) Σ(Y-Ybar)^2

Most importantly, is there any formula to find Σ(y-Y)^2?

Answer 1

Well I will try my best with this, although I am baffled by the notation. I'm assuming that Y' is the fitted value of Y; it is what you get when you plug in the X value.

a) This is what is called a residual. It is the vertical distance a point is from the regression line.
b) As far as I know, there is no such thing at X'. The only guess that I have is that X-X' is the horizontal distance a point is from the regression line. But I don't know of any use for this computation.
c) This is always equal zero. A property of the regression line is that the average residual is zero.
d) The is the sum of squared residual, or sum of squared error. The regression line is such that it makes Σ(Y-Y')^2 as small as possible. Using any other line to compute Y' will make this bigger. This computation also tells you how much variation in Y is not explained by its relationship with X.
e) This is the sum of squared regression. This is the amount of variation in Y that is explained by its relationship with X, meaning that Y varies because X varies.
f) This is the sum of squared total. This is how much variation there is in Y.

I don't know what y means in Σ(y-Y)^2. If you are talking about Σ(Y-Ybar)^2 though, you can find it this way.

Σ(Y-Ybar)^2 = Σ(Y'-Ybar)^2 + Σ(Y-Y')^2

In other words.

Sum of squared total = sum of squared regression + sum of squared error.

Answer 2

<< Most importantly, is there any formula to find Σ(y-Y)^2? >>
If Yj is experimental datum for xj, then you may specify y(x) with unknown parameters a, b, c, ,,,, e.g. y=ax^2 +bx +c is parabola;
In this case S= Σ(y(xj)-Yj)^2 and
dS/da = 2Σ(y(xj)-Yj)*(xj)^2 = 0;
dS/db = 2Σ(y(xj)-Yj)*xj = 0;
dS/dc = 2Σ(y(xj)-Yj) = 0;
solving this system with respect to a, b and c, you find the best function y(x) approximating your experimental data.