J = (true_positives x true_negatives - false_positives x false_negatives) / (positives x negatives)
where positives and negatives are the number of real positive and real negative samples.
Its value ranges from 0 through 1 (inclusive)
This is simply a result of assuming that the test/classifier performs better than with its answers swapped. As it is said in the original paper (Youden, 1950):
1. The possible range of values for the index is from zero to one inclusive. (It is expected that the test will show a greater proportion of positive results for the diseased group than for the control.)
But theoretically Youden's J statistic can be negative and whether the test/classifier needs class-swapping that's another issue I would say.
Why is this assumption assumed by default? For me it is not so obvious.