Question

when a cross tile is placed on a number chart, the number covered by the center sqare is the average of all the numbers covered, no matter how large the cross is. Why is this? How is this ?

Answer 1

The number chart has to fall within certain parameters:
It will be a square with odd number of tiles on the side (3*3, 5*5 etc)
The numbers must be evenly spaced ( 1 , 2 , 3 ) ( 2 , 4 , 6 ) etc
The centre square value will always be the average under these circumstances.
Its also the median value.

Answer 2

(or)
The numbers around the centre (k) are k-1, k+1, k-10 and k+10 (though it need not be 10s, but that is surely common - on a calendar it would be 7 as a counter example), which sums to 5k and averages k.
The longer the cross is adds more pairs of values equidistant from k ... 9k/9, 13k/13, etc.