Statistical Analysis of Month-dates in the Book of Mormon
by Duwayne Anderson (email)
Abstract
The Book of Mormon claims to be a literal history of the ancient Americans. However, a statistical analysis of month-dates in the Book of Mormon strongly suggests that the dates are none random and thus probably fabricated. This document explains in detail the process of arriving at the relevant probabilities, and the rational behind the resulting conclusion that the Book of Mormon is a fake.
Discussion
In his book Conned Again, Watson, Colin Bruce has an interesting story in which Sherlock Holmes is asked to investigate a possible plot to steal an inheritance. To see through the scheme and uncover the perpetrator, Holmes must examine two lists of 60 birthdays and determine which is legitimate and which is a phony. As his client, Madam Zelday, puts it:
For he claims that, far from his being the only one, there are another fifty-nine of his branch of the family living in various parts of the Americas. He sent me a list of birth dates. But I have no way of determining whether these people actually exist. Most impertinently, my cousin will not provide a list of the associated names and addresses.True, the story looks strained. But the point of Bruce’s book is to explain – in an entertaining way – basic principles in mathematics and statistics, in particular. In this story, Holmes saves the day by performing a statistical analysis on the list of birth dates. As Holmes puts it:
It is a problem in pure logic, Watson. Of these two lists of birth dates, one describes real people and the dates will be genuinely random. The other has probably been invented in the mind of an unscrupulous man. Fortunately, the human brain is a very bad generator of random numbers.With characteristic brilliance, Holmes then explains the statistical fact that, in a list of 60 truly random birth dates, there is roughly 1 chance out of 170 that no two birth dates will fall on the same day. Armed with this unlikely bit of information, Holmes examines the two lists and finds, sure enough, that one of them has duplicate birth dates and the other does not. But contrary to popular intuition and common sense, the list with duplicate entries is the legitimate one. Logic wins, Holmes discovers the con man, and all retire to the library for refreshments.
When I first read Bruce’s book the thought occurred to me that a similar type of statistical analysis of dates in the Book of Mormon might provide some insight into its origins. The Book of Mormon purports to be a real history of the ancient Americans. An angel, so the story goes, told Joseph Smith, in the early 1800s, where to find buried gold plates with ancient inscriptions derived from Egyptian. With the help of the angel, and by the "gift and power of God," Joseph Smith claims to have translated the Book of Mormon from an indecipherable language into English. The book has since been translated from Smith’s original English version into over 100 languages, with millions of copies sold throughout the world.
But is the book what it claims to be, and how might we test its claims? The obvious test is to have the gold plates examined by scientific experts who could vouch for their authenticity. Conveniently, however, Joseph Smith explained that the angel took them back. So the plates are unavailable for independent verification. Since the book claims to be literal ancient history, a second obvious test is to compare the archeological evidence with the claims the Book of Mormon makes about the ancient Americans. Indeed, this has been done, with the Book of Mormon failing badly on virtually all counts (see, for example, Quest for the Gold Plates, by Stan Larson.). Yet another method of examination is to look at the statistical distribution of dates in the Book of Mormon, as Sherlock Holmes did with the birthday list, to see if it really looks random.
To see how we might do this, let’s assume there are 30 days in the month, and we pick a day at random. [According to the Book of Mormon, the early inhabitants of the ancient Americas were Hebrews, and they followed the Law of Moses. The Hebrew calendar has 13 months, each with either 29 or 30 days. You can see a discussion about the Hebrew calendar at http://www.jewfaq.org/calendar.htm.] The probability of picking any particular day is 1/30. The mean of the probability distribution is 15.5 and the standard deviation is about 8.655 (see Appendix A for some basic background information regarding statistics):
If we pick two days at random and add them, the expected (mean) value is 31 (two times the expected value for one day) and the standard deviation is root-two times 8.655, or 12.241. Similarly, if we pick N days of the month at random, the expected value and standard deviation will be N times 15.5 and root N times 8.655, respectively. Furthermore, we know from the Central Limit theorem that if N is relatively large (more than about 5) the probability distribution is approximately Gaussian. For example, if N = 8, the expected value is 124 and the standard deviation is 24.481. Figure 1 validates this reasoning with results from a simple Monte Carlo simulation (the Mathcad document used in the Monte Carlo simulation is in Appendix B).
The distribution in Figure 1 can be used to draw statistical inferences about the likelihood of finding a given sum when adding 8 days of the month, each chosen at random. We do this by integrating (summing over the possibilities). For example, suppose we pick 8 days at random, and the days add up to 150. We see that the difference between the value of 150 and the expected value of 124 is 26, or slightly more than one standard deviation. So, this is not an unlikely value. If, on the other hand, we pick 8 days of the month at random, and the sum comes to 10, we find that the difference between this value and the expected value is 114 or nearly 4.7 standard deviations. The probability of picking a string of 8 days, whose sum equals 10, is very small. In fact, we’d expect to find this sum in only about one out of every millions tries.
We can put this on a firmer basis by looking at the area under the probability distribution curve that lies outside the measured value. Figure 2 illustrates the results. From Figure 2 we see that the probability of a sum lying outside of the mean is 1 since this region encompasses the entire curve. As we move further and further from the mean, however, the probability of finding a sum in the remaining territory diminishes. For a Gaussian distribution, at one standard deviation the probability drops to about 31.7%, indicating that about 68.3% of all the sums lie within one standard deviation of the mean. At two standard deviations the probability of being outside the region drops to 4.6%, at three standard deviations it drops to 0.27%, and so on. [Note: The probability distribution for the sum of 8 days chosen at random is only approximately Gaussian, and the deviation is most pronounced in the distribution’s tails. The Gaussian approximation works reasonably well out to about 3 standard deviations. For the analysis presented here (Figure 2), I’ve relied upon the results of the Monte Carlo analysis, which gives lower probability for the extreme tail region than the Gaussian distribution.]Figure 1. Monte Carlo simulation showing the mean and standard deviation resulting from choosing 8 days of the month at random (1,000,000 samples used in this simulation). These numbers agree reasonably well with the theoretical approximations determined by the Central Limit theorem, which predict a mean and standard deviation of 124 and 24.481 respectively.
This is a fun exercise in statistics, but what’s the connection with the Book of Mormon? To address this question we digress for a moment. Suppose you are looking at the dates when various states entered the union. These dates are essentially un-correlated with each other, and with the day of the month. This means there is no reason to expect, for example, that the day of the month when Arizona entered the union was affected by the date that Utah entered. Thus, we’d expect a sampling of the dates when various states entered the union to be randomly distributed among the days of the month. This turns out to be the case, as you can see in Table 1.
Figure 2. Probability that the sum of 8 days of the month, drawn at random from months having 30 days each, lies a outside a given range from the mean. The horizontal axis gives the range in units of the standard deviation. Note that the vertical axis is logarithmic. For example, taking the sum of 8 days chosen at random from a month with 30 days, the probability that the sum exceeds 2 standard deviations is about 0.054 [Note: This curve generated from the Monte Carlo analysis.].
Day |
Month |
Year |
State |
14 |
2 |
1912 |
Arizona |
15 |
6 |
1836 |
Arkansas |
9 |
9 |
1850 |
California |
1 |
8 |
1876 |
Colorado |
3 |
7 |
1890 |
Idaho |
28 |
12 |
1846 |
Iowa |
29 |
1 |
1861 |
Kansas |
30 |
4 |
1812 |
Louisiana |
Table 1. Trans-Mississippi West: statehood. This table shows 8 states and the associated days of the month when each state was admitted into the union. The sum of days of the months is 129, which is close to the expected value of 124, lying less than one standard deviation from the mean (dates in Table 1 and Table 2 were taken from The Illustrated Encyclopedia of the Old West, by Peter Newark).We could try another example. Table 2 lists some of the dates for territorial status for states west of the Mississippi.
Day |
Month |
Year |
Territory |
24 |
2 |
1863 |
Arizona |
2 |
3 |
1819 |
Arkansas |
28 |
2 |
1861 |
Colorado |
4 |
3 |
1863 |
Idaho |
12 |
6 |
1838 |
Iowa |
30 |
5 |
1854 |
Kansas |
26 |
3 |
1804 |
Louisiana |
3 |
3 |
1849 |
Minnesota |
Table 2. Trans-Mississippi West: Territorial status. Again, we have 8 regions and the dates each achieved territorial status. The sum of the days of the month is 129, which is within one standard deviation of the expected value, which is 124 (and, interestingly enough, the same sum as in Table 1).Let’s try one more example. Table 3 lists the dates when 8 of the LDS Church presidents died. As with the other tabulated data, we see that the sum of the days of the month is close to the mean. In this case the sum is 130, which is within one standard deviation from the expected value.
Day |
Month |
Year |
Name of LDS prophet |
27 |
6 |
1844 |
Joseph Smith |
29 |
8 |
1877 |
Brigham Young |
25 |
7 |
1887 |
John Taylor |
2 |
9 |
1898 |
Wilford Woodruff |
10 |
10 |
1901 |
Lorenzo Snow |
19 |
11 |
1918 |
Joseph Fielding Smith |
14 |
5 |
1945 |
Heber Jeddy Grant |
4 |
4 |
1951 |
George Albert Smith |
Table 3. Death dates of 8 LDS Church Presidents. As with Table 1 and Table 2, the statistics here follow the predicted values, with the sum of the month-dates (130) being within one standard deviation of the mean (dates taken from Essentials in Church History, Joseph Fielding Smith, 1950, page 574).This methodology provides a possible tool for evaluating the claims of books that purport to contain a real history of events. As we’ve seen in these examples (and it should be clear from examining basic assumptions) the dates of non-correlated (and non-periodic) events in history are randomly distributed among the days of the month. If, therefore, we were to find a book with dates, purporting to be a real history, and if the sum of those days-of-the-month is far from the mean, we have a statistical tool for questioning whether the dates are really authentic.
This brings us to the Book of Mormon. Although month-dates in the Book of Mormon are relatively rare, there are enough (8) to provide a reasonable basis for statistical analysis. Table 4 summarizes the results:
Day |
Month |
Year |
Verse |
Circumstances |
1 |
1 |
66 BC |
Alma 52:1 |
Assassination of Lamanite leader, Amalickiah. |
2 |
1 |
62 BC |
Alma 56:1 |
Heleman sends an epistle to Moroni. |
4 |
1 |
34 AD |
3 Nephi 8:5 |
Destruction before Jesus visits the Nephites |
5 |
2 |
81 BC |
Alma 16:1 |
Lamanites begin a war against the Nephites |
3 |
7 |
About 64 BC |
Alma 56:42 |
Account of a war between the Nephites and Lamanites. |
4 |
7 |
81 BC |
Alma 10:6 |
Alma the younger is converted |
12 |
10 |
81 BC |
Alma 14:23 |
Alma the younger freed from prison |
10 |
11 |
72 BC |
Alma 49:1 |
Lamanites march on the city Ammonihah |
Table 4. Month-dates from the Book of Mormon.In this case, unlike the real historical dates from the previous examples, the sum of month-dates is only 41, or nearly 3.4 standard deviations from the mean. Even a casual observation shows that the author of the Book of Mormon had a tendency to pick days in the first week of the month. The probability of this happening by random chance (see Figure 2, which was calculated using the Monte Carlo analysis) is about 1 out of 2,000. This is sufficiently unlikely to call into question the validity of the claim that the month-dates are based on a real history and not fabricated. However, since the Book of Mormon presents the month-dates matter-of-factly, as being part of a literal history, we conclude that the Book of Mormon itself is a fabrication. That is, we conclude that the Book of Mormon is a unique product (most likely) of the 19th century, and (as Holmes might have put it) "probably … invented in the mind of an unscrupulous man."
Summary
Let’s summarize what we’ve developed, and carefully list our assumptions:
Throughout all of this analysis we should be careful to remember that conclusions based on statistical inference do not constitute proof. Indeed, given a sufficiently large sampling of statistical data from real historical events, there is a finite probability that some will have the same distribution found in the Book of Mormon. These, however, will be very rare; it is far more likely that the dates in the Book of Mormon were concocted as part of a fabricated story. This conclusion is buttressed by other important information, such as the Book of Mormon’s almost universal failure to properly describe the realities of ancient-American life.
Basic resources in statistics
The statistical analysis in this paper is pretty basic, and can be verified with most introductory texts. I’ve used Bulmer’s Principles of Statistics, which is published by Dover. Bulmer gives equations for the mean and standard deviation of statistical distributions, and he has a very nice discussion about the Central Limit theorem. There is also a great deal of useful statistical information on the Internet. There is a nice applet that illustrates the Central Limit theorem through simulation at http://www.stat.sc.edu/~west/javahtml/CLT.html, and another one at http://www.math.csusb.edu/faculty/stanton/m262/central_limit_theorem/clt.html. There is a useful dictionary of statistical terms at http://www.animatedsoftware.com/statglos/statglos.htm, and some equations for things like mean and standard deviation at other sites. Finally, you can find some good background information on the Gaussian distribution at http://mathworld.wolfram.com/GaussianDistribution.html, and also at http://hyperphysics.phy-astr.gsu.edu/hbase/math/gaufcn.html.
Mathcad function used for Monte Carlo analysis