# How to understand if the points are constant around a line?

I would like to understand how can I check if some points around a line have constant distance between the point and the line.

So image a normal chart(cartesian plane) with a line that is formed with points where its coordinates are: x > 0 and y > 0 so always numbers bigger then zero.

Then, when I have this line i need to check if the distances of a list of points(x,y) are constant. With "constant" I mean that the distance between the points and the line is similar. Example:

LINE: (x - y coordinates)

1 - 1
2 - 2
3 - 3
4 - 4
5 - 5
6 - 6


POINTS: (x - y coordinates)

1 - 1.1
2 - 1.9
3 - 2.9
4 - 4.2
5 - 5.1
6 - 5.8


How can I check if the difference between the point on the line and the point (of the list of points) are constant?

Thanks

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Project the data orthogonally onto the line and inspect the residuals.

In this case, where the reference line is given (independently of the data) as $y=x$, the projection is tantamount to plotting the points $\{((x_i+y_i)/2, y_i-x_i)\}$. To illustrate, here are your points and the reference line:

Here is the derived plot:

You can now apply any test of homogeneity and independence of residuals you like, such as the Fligner-Killeen test or Breusch-Pagan test. (Many such tests assume the first coordinate is fixed, not random, so they will be only approximately correct, but in general--at least when the data aren't terribly noisy--they should work fine.)

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thank you for your answer, I have one doubt. I'm reading how FLigner-Killeen works and I found: stat.ethz.ch/R-manual/R-patched/library/stats/html/… I have to pass a "group" at the second parameter, what does it mean? thank you! –  Dail Sep 27 '11 at 16:40
That's an interesting solution, but what are the alternatives that the BP test would have power against? I guess that you could test against a polynomial function of x, depending upon how many data points the OP has. I'm not familiar with the FK test that you mention. –  Charlie Sep 27 '11 at 16:42
@Charlie what do you mean with "you could test against a polynomial function of x" ? could you give me a simple example with R code? –  Dail Sep 27 '11 at 16:54
@Dail, you need to specify the form of the BP test. The test requires that you regress the residuals on something. It could be just a constant term (the coefficient on this term should be exactly 0 if the line in question is from OLS on the points) or include other variables as well. No variables besides x were mentioned in the question (taking y to be the outcome, so it doesn't count), so the only other things that you could include in the model are functions of x---I suggested polynomial functions. The BP test judges whether the coefficients on these terms are 0. –  Charlie Sep 27 '11 at 17:10
@Charlie, do you know how to calculate OLS regression with R? I didn't find it, at the moment i use lm() (linear regression), is not good? –  Dail Sep 27 '11 at 17:17