Can someone in plain english explain what R^2 is?

Question

Can someone in plain English without any complicated words explain the meaning of R^2???
I am trying to explain the relationship of r^2 and an equation I had for a parabola. But I dont understand it myself... can someone explain it to me clearly?

I have an equation for a parabola and the value of r^2= 0.998

Does it mean the equation is 99% accurate?? How does the value of r^2 calculate that?

When I am talking of r^2, I am talking of the statistical measure and the correlation coefficients

Answer 1

Are you talking about statistics? Where R-squared is the square of the coefficient of linear correlation? If so, I am not sure where a parabola fits in, as that is a nonlinear relationship.

Are you fitting data to a parabolic model to find estimates of the coefficients? those appear linearly.

At any rate, here is the general answer to your question. I will state it in terms of single-variable regression.

You have gathered ordered pairs of data (x, y) . The hope is to find a linear relationship by which the variable x can be used to predict the value of y. Of course, we don't expect this relationship to be exact, so the predictions will not be perfect. R^2 is a measure of how fuzzy we expect those predictions to be. When R^2 is close to 1, the relationship is almost exactly linear and the predictions will be almost perfect. When R^2 is close to 0, the linear equation we get does a lousy job of predicting y when x is given.

More precisely, it is possible to quantify the amount of variability in the measurements of y (when x is ignored). R^2 gives the percent of that variability that is "explained" by x.

So if R^2 = .98, then the introduction of x explains 98% of the variablity in y, leaving 2% of the variablity as yet "unexplained" and presumably accounted for by other random phenomena not taken into account.

Answer 2

Okay... how weird.

y = ar^2 + br + c

Doesn't really matter what a, b, and c are. They could be any real number on the number line, except "a" MUST NOT be 0.

There are tilted parabolas in a form like:
y = ax^2 + bxy + cy^2 + dx + ey + f, such that the angle of rotation is [(1/2)arctan b/(a-c)], but I don't think we should get to that.

Okay, let's explain why y = ar^2 + br + c, when a is NOT 0, is always a parabola.

Let's first keep in mind that a parabola is also:
a(y-k) = (x-h)^2

So every parabola is only a translation of the other. So if I moved all of them so that their vertex is at the origin...

They would clearly be a parabola like:
y = ar^2

This function is even, meaning that it is symmetrical from the y-axis. If the x values are opposite each other, their y-partner is always the same. And you can say that this is the definition of a parabola (not the mathematical one, they might include words like locus).

And this is why ALL y = ar^2 + br + c, such that a is NOT 0, is always a parabola.

Answer 3

R^2 means two-dimensional. On your cartesian place it would mean you have an x and y component.