Benford's Law is explained in several helpful sources on the Internet. For the mathematically inclined, see Eric W. Weisstein's article, "Benford's Law" at MathWorld--A Wolfram Web Resource, where we read that this law refers to:
A phenomenological law also called the first digit law, first digit phenomenon, or leading digit phenomenon. Benford's law states that in listings, tables of statistics, etc., the digit 1 tends to occur with probability ∼30%, much greater than the expected 11.1% (i.e., one digit out of 9). Benford's law can be observed, for instance, by examining tables of logarithms and noting that the first pages are much more worn and smudged than later pages (Newcomb 1881). While Benford's law unquestionably applies to many situations in the real world, a satisfactory explanation has been given only recently through the work of Hill (1996).Another good source is an article by Alexander Bogomolny, who writes:
With the view to the eerie but uniform distribution of digits of randomly selected numbers, it comes as a great surprise that, if the numbers under investigation are not entirely random but somehow socially or naturally related, the distribution of the first digit is not uniform. More accurately, digit D appears as the first digit with the frequency proportional to log10(1 + 1/D). In other words, one may expect 1 to be the first digit of a random number in about 30% of cases, 2 will come up in about 18% of cases, 3 in 12%, 4 in 9%, 5 in 8%, etc. This is known as Benford's Law. . . .Benford's law has been used in several cases to detect fraud. See, for example, discussions of Benford's law and cheating by Malcolm W. Browne and another by Jason Kottke. For example, falsified numbers generated by cheaters and frauds will rarely follow Benford's law in situations when real physical data will tend to do so. Careful cheats, unaware of Benford's law, may craft numbers that are relatively uniform in the distribution of leading digits. But when it comes to recorded numbers for populations and other measurements, especially measurements that have dimensions (things like feet, pounds, years, etc.), Benford's Law will often apply, and large unexplained disparities between the data and Benford's law can be a warning sign of possible fraud.
The law was discovered by the American astronomer Simon Newcomb in 1881 who noticed that the first pages of books of logarithms were soiled much more than the remaining pages. In 1938, Frank Benford arrived at the same formula after a comprehensive investigation of listings of data covering a variety of natural phenomena. (Benford's original data table can be found on Eric Weisstein's Treasure Troves of Mathematics - Benford's Law page.) The law applies to budget, income tax or population figures as well as street addresses of people listed in the book American Men of Science.
I've known of Benford's law for some time, but only today did I take them time to jot down the numbers given in the Book of Mormon to see how they compare to Benford's Law. I've only spent about a little over an hour on this project, so my findings are preliminary, but here's what I did. From the onset, I recognized that a fair comparison using small numbers like one and two would be problematic, since there a hundreds of references to "a man" or "a person" = do all those count as the number 1? I felt that a fair analysis would require consideration of numbers beginning with 10. Thus, all groups of people of 9 or less are discarded, as well as units of time of 9 or less (a day, a month, etc., are thrown out).
Here are the results, showing the distribution of leading digits for periods of time:
Occurrences of Leading Digits in Measures of Time
I would say that the time-related numbers show reasonable agreement with Benford's Law.
For numbers of people, there were fewer numbers to work with and thus a choppier distribution:
Occurrences of Leading Digits in Counts of People
In these people-related numbers, I have deliberately ignored all references to the 12 Tribes, the 12 Disciples, or the 12 Apostles. There were 19 such references that I have left out, feeling that they were too "non-random" and would inflate the number of 1s as leading digits. On the other hand, the leading 1s include 10 references to groups of 10,000 in Moroni 6, which arguably could be viewed as "inflationary." Perhaps 9 of those could be discarded, in which case the number of leading 1s would be 11. As another downward adjustment, the statistics for the leading digit 2 use only 1 of 12 references to the 2000 stripling warriors and 1 of 3 references to the 2060 of Helaman's expanded army. Several of the other references appear related to the preference of 2000 as a base unit of soldiers in an army (interestingly, I think that all large military groups whose numbers are stated are always multiples of 2000). Note that the large number of leading 5s is due to 8 occurrences of the number 50, 5 of which refer to a military unit, as in "Laban and his fifty." Units of 50 men also played an important role in ancient Jewish systems. Given these considerations and the smaller sample size, the numbers for people still seem reasonably compatible with Benford's Law.
Update (April 2): Please note that this is not necessarily a confirmation of Book of Mormon authenticity, for it is entirely possible for ordinary fiction to have a numerical distribution similar to Benford's law. Numbers in the teens, hundreds, thousands, etc. tend to be more important to us than numbers in the nineties, nine hundreds, and nine thousands, for one thing. It would be interesting to look at distributions of leading digits in several works of fiction and see if any broad generalities might be drawn. At the moment, though, this post is a confirmation of Benford's Law rather than a confirmation of the Book of Mormon, though it may at least suggest that the numbers in the Book of Mormon were not consciously fabricated by someone trying to make them look randomly distributed. But I am sure that you can find Benfordesque distributions in ordinary fiction as well.
As always, do your own due diligence, and don't give too much weight to anything I say, especially since this is preliminary and subject to errors of several kinds. But so far, I think it's interesting. I have not tweaked the numbers to get a desired result, but have tried to be fair. In particular, the choice to start with 10 and higher was made before doing any counting to prevent artificial inflating of the number 1.