My "wiriting through the curriculum" answer is that the average distance between prime numbers is ~3.75.
All prime numbers >5 are distributed evenly (quantifiably speaking) along 8 diagonals in a 30-sectioned spiral (or as arrayed in 8-columns in a rectangular matrix) populated by all natural numbers not divisible by 2, 3 and 5: Modulo 30 for all numbers in this array, and therefore all primes >5 must be 1, 7, 11, 13, 17, 19, 23 or 29.
It follows, therefore, that the distances between distributed primes are 6, 4, 2, 4, 2, 4, 6 then 2 to the next rotation of the spiral (or row of a matrix). The sum of these intervals = 30; the average distance between them = 30/8 = 3.75 (excluding a micro-adjustment for the 1st 3 primes, 2, 3 and 5)
The full story, including why I can state with authority that all primes >5 are distributed evenly along the 8 diagonals described above, is here: http://www.primesdemystified.com
