The final bit is complete bunkum for an unknown distribution. Whether the distribution is monotonic and if not, how symmetric, is the major determinant of occurrence of such result. Giving a flat number is completely bogus.

I don't understand what you mean. It doesn't actually matter what shape the distribution takes. That figure is derived entirely from what the "median" means, i.e. a random sample has 50% chance of being greater than the median, and 50% chance of being lower than it.

A bit of maths would show that the probability that all 5 samples lie entirely above or entirely below the median (i.e. the median is NOT between the greatest and the smallest value) is 1/16. That gives you a 15/16 (= 93.75%) chance that the median is contained within the bounds.