Statistics shows that if you put lots of random points on a bounded flat surface you can find **alignments of random points**. Some people think that this shows that such things as ley lines naturally exist, and are therefore not interesting phenomena. Other people see this as a failure to understand scientific method which is about setting particular criteria for particular comparisons which can then reveal levels of probability.

One precise definition which expresses the generally accepted meaning of "alignment" as:

*a set of points, chosen from a given set of landmark points, all of which lie within at least one straight path of a given width w*

The width *w* is important: it allows the fact that real-world features are not mathematical points, and that their positions need not line up exactly for them to be considered in alignment.

For example, using a 1mm pencil line to draw alignments on an 50000:1 Ordnance Survey map, a suitable value of *w* would be 50m.

Statistically, finding alignments on a landscape gets progressively easier as the area to be considered increases. One way of understanding this phenomenon is to see that the increase in the number of possible combinations of points in that area overwhelms the decrease in the probability that any given set of points in that area line up.

The number of alignments found is very sensitive to the allowed width *w*, increasing approximately proportionately to *w*^{k-2}, where *k* is the number of points in an alignment.

For those interested in the mathematics, the following is a very approximate estimate of the likelihood of alignments, assuming a plane covered with uniformly distributed "significant" points.

Consider a set of *n* points in an area with approximate diameter *d*. Consider a valid line to be one where every point is within distance *w*/2 of the line (that is, lies on a track of width *w*.

Consider all the unordered sets of *k* points from the *n* points, of which there are

So, the expected number of k-point ley lines is very roughly

Then we have the expected number of lines equal to: