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

Question

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

Answer 1

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.)