Roman digit count

Consider the Roman numerals for 0 through 9: $\begin{array}{r|cccccccccc} n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \text{Roman} & & \mathrm{I} & \mathrm{II} & \mathrm{III} & \mathrm{IV} & \mathrm{V} & \mathrm{VI} & \mathrm{VII} & \mathrm{VIII} & \mathrm{IX} \\ \text{digits} & 0 & 1 & 2 & 3 & 2 & 1 & 2 & 3 & 4 & 2 \end{array}$

The average is 2 "digits" to represent each unit (number 0-9). In the same way, 2 "digits" are needed to represent tens (0-90), and 2 "digits" to represent hundreds (0 - 900).

Adding these together, we see that the average number of digits for a Roman numeral between 0 and 999 is precisely $2 + 2 + 2 = 6$ .

Excluding 0, the average becomes slightly higher, namely $6 \times \frac{1000}{999} \approx 6.006 \approx \boxed{6.0}.$

Mathematica

N@Mean@Array[StringLength@RomanNumeral@#&, 999]

6.00601