The sum of the squares of the first $n$ natural numbers is given by the formula:
Sum = $\frac{n(n+1)(2n+1)}{6}$.
The average of these squares is the sum divided by the number of terms, $n$.
Average = $\frac{\text{Sum}}{n} = \frac{n(n+1)(2n+1)}{6n} = \frac{(n+1)(2n+1)}{6}$.
In this problem, we need to find the average for the first 50 natural numbers, so $n=50$.
Substitute $n=50$ into the average formula:
Average = $\frac{(50+1)(2 \times 50+1)}{6}$.
Average = $\frac{(51)(100+1)}{6}$.
Average = $\frac{51 \times 101}{6}$.
We can simplify the fraction by dividing 51 and 6 by their common factor, 3.
Average = $\frac{17 \times 101}{2}$.
Now, calculate the product in the numerator: $17 \times 101 = 1717$.
So, the average is $\frac{1717}{2}$.