What is the intuition behind the formula for the average? Why is the average for $n$ numbers given by $(a+b+c+\cdots)/n$? I deduced the formula for the average of 2 numbers which was easy because its also the mid point, but I couldn't do it for more than 2 numbers.

Suppose all of us gathered here in this room take all the money out of our pockets and put it on the table, and then we divide it among us in such a way that we all have the same amount. The total amount is still the same.  Then the amount we each have is the average. That's what averages are.
The total amount is $a+b+c+\cdots$. The number of us gathered here is $n$. So the amount that each of us gets is $(\text{total}/n).$
