A PARADOX?

by Antisthenes

“You can prove anything by statistics” – Anon. In the Journal of Geomancy (1/2 p. 39) Keith Raison criticizes Robert Forrest’s statistical model of Ley Hunting because it totally ignores fieldwork. Robert Forrest has replied (1/3 p. 57): “Fieldwork uncovers sites ‘not on the map’. A new statistical model then becomes necessary to compute the alignments not just among ‘shown’ (on the map) sites, but amongst the ‘not shown’ ones as well”. {5} If we make the computations indicated by Forrest we are likely to be led into a paradox, as I shall show by an imaginary example.

Suppose an O.S. 1:50 000 map, covering an area 40 km square, has 100 ancient stones marked on it. If a stone is allowed to be up to 20 metres off-line then the expected number of 3-point alignments on this map will be 506. Suppose also that the region contains 100 more stones, which are not on the map. The expected number of 6-point alignments among the whole set of 200 stones will be 1·635. On average, each 6-pointer will contain 2·5 triads of marked stones; so we expect that 4·09 of the 3-pointers shown on the map will be extensible to 6-pointers when the 100 unmarked stones are taken into account. A ley hunter investigates the map and finds a 3-point alignment. He knows this is not significant in itself, but for some reason (eg. metric properties or folklore) he decides it is worth investigating further. Going out into the field, he finds 3 unmarked stones in the same line.

The ley-hunter (eg Raison) argues: “Although chance alone would yield about 500 3-pointers on the map, only about 4 of these would be extensible to 6-pointers. Therefore, my hunch that this is a special alignment, or ley, has been confirmed at the 1% level.”

The skeptic (eg. Forrest) argues: “You have merely found a 6-pointer, of which 1·6 would be expected by chance. Therefore, the alignment is not significant.” Who is right?