# There oughta be a law without words

There really oughta be a law.

There really is!
$\require{begingroup}\begingroup \small\sf \def \& { & \kern-.5em = \kern-.5em & } \def \l #1 / #2 { \log \frac {\normalsize #1} {\normalsize #2} } \def \_ #1{ \underline{ \raise.5ex{ ~~~ #1 ~~~ } } } \def \- #1{ \overline{#1} } \begin{matrix} & & \_{ 1 - 9.\-9 } &~~~ \_{ 10 - 99.\-9 } &\cdots&~~~ \_{ 100,000 - 999,999.\-9 } \\[1.5ex] \matrix{ \sf\scriptsize I, X, C, \\[-.5ex] \sf\scriptsize M, \-X, \-C } & 74.5\% ~ = & \l 4.\-9 / 1 + \l 9.\-9 / 9 ~~~& \l 49.\-9 / 10 + \l 99.\-9 / 90 & & \l 499,999.\-9 / 100,000 + \l 999,999.\-9 / 900,000 \\[1ex] \matrix{ \sf\scriptsize V, L, D, \\[-.5ex] \sf\scriptsize \-V, \-L, \-D } & 25.5\% ~ = & \l 8.\-9 / 5 ~~~& \l 89.\-9 / 50 & & \l 899,999.\-9 / 500,000 \\ \end{matrix} \\ \kern7.5em \raise2ex\strut \scriptsize \-V = 5,000, ~~ \-X = 10,000, ~~ \-L = 50,000, ~~ \-C = 100,000, ~~ \-D = 500,000 \endgroup$

Just what is that there law?

When could it come into play?

The table above is a justification-without-words of a statistical law that might be applied by authorities when foul Roman numbers are afoot.

This is

Benford's Law, which states that the digit 1 is more common than any other digit as the first digit. Here, Roman numerals are compared, and the ones that start with a power of 10 appear just under 3/4 of the time (using a logarithmic distribution).

It can sometimes be applied

to check the validity of a large set of numbers from one source. If the data don't approximately fit, then it's likely that the data set was faked. Not sure what that has to do with Romans, though - possibly something to do with tax collection?

Notes from this puzzle’s poser

The column under $\small\underline{ ~ 1 - 9.\overline9 ~ }$ calculates...

...the expected frequencies of leading Roman digits I and V, from the integer parts of all measurements 1 through 9.999... .
$\small \log {\normalsize\frac{ 4.\overline9}{1}} {+} \log {\normalsize\frac{9.\overline9}{9}}$ accounts for numbers that begin with I:   I, II, III, IV and IX.
$\small \log \normalsize{\frac{8.\overline9}{5}}$ accounts for numbers that begin with V:   V, VI, VII and VIII.
These calculations differ interestingly from those for decimal numbers due to grouping and the out-of-order inclusion of IX in the group with I − IV. (Use of VIIII for 9 would change this but use of IIII for 4 would not.)

Calculations are logarithmic and have equal results across columns because...

...no assumptions are made about the units of measurement that might produce a set of numbers. For example, the same distribution of leading digits is expected for a random set of Roman distances whether those distances are measured in paces, stadia, miles or leagues.

Such reasoning for logarithmicity is simple but adequate and the resultant skew in favor of I,X,C,M,... over V,L,D,... is surprisingly strong.