The Use of Statistics to Analyze the Qur’ân

Analyzing the mathematical structure of the Qur’ânic text involves generating data sets using concise, meticulously crafted definitions and then computing probabilities associated with specific phenomena. Although there are several anomalies that can be looked at in the Qur’ân, our attention will mainly be focused on looking for multiples of 19 specifically. To the extent that other structural features are examined, the appropriate statistical tools will be employed and explained at that time, God willing. Meanwhile, let us look at how we would generate and analyze a data set where 19-divisibility is central to our research problem.

If you have had a chance to examine the table of muqaţţa‘ah counts, you will have found that some totals are multiples of 19, but most of them are not. The question that the statistical approach intends to answer is whether the frequency of multiples of 19 in a given configuration of these numbers is what we should expect from random chance or not. If so, then we cannot say there is anything meaningful in them, for they would then be no different from what we would find in any other book. But if they do indeed deviate from the norm in any striking fashion, then we have to ask ourselves: Why are they there? Is it a freak accident or is there other evidence in the Qur’ân that this was intentional? Lastly, if it was intentional, how was it accomplished? The answer to these questions could conceivably say a lot about the origin, nature, and ultimate purpose of the Holy Scripture of Islâm.

Defining Data Sets. In order to test any hypothesis, a concise definition must generate a data set—the set of numbers that is be analyzed. If you peruse the table of muqaţţa‘ât, you will notice that several chapters (sûrât) have identical muqaţţa‘ah sets, while others have unique sets, sometimes comprising a single letter. We might generate our data set using the definition, "total quantity of muqaţţa‘ât in a sûrah." If so, then we will generate a total of thirty data points, one for each initialed chapter, plus one for the additional muqaţţa‘ah set in Sûrah 42. On the other and, we might generate our data set using the definition, "total muqaţţa‘ât in a group of sűrât introduced by the same muqaţţa‘ah set." If so, then we will generate a total of fourteen data points, one for each group of sűrât introduced by the same set of initials. It is possible to generate only a few logical configurations from the table of muqaţţa‘ât without diverging from cohesive definitions. However, more than one data set can conceivably be generated from the raw numbers. Consequently, we must be concerned with how we define our data sets.

Implications of Our Findings. Even if there are errors in our data, statistical tests can provide enough information for us to draw informed conclusions about the basic nature of what we are examining. Are the numbers of letters an intentional part of the text? In other words, did the Author plan it that way? If so, what do they seek to tell us? If we notice a statistically significant pattern, are we actually looking at it directly or are we just close enough to see an indirect reflection of it? If it is an indirect reflection, is this because we have not identified the proper configuration? Is it because there are errors in our counts? Is it because the text has been altered? Statistics is capable of informing our judgment about such matters and giving us clues about how to proceed with each subsequent step based on objective conclusions.

A Final Note. I invite any and all attempts to count the letter alif anew and send me your findings. Please perform your counts on a verse-by-verse basis so I can resolve discrepancies. God Willing, I will provide more information on my methodology in due course, along with my statistical tests. Lastly, I would greatly appreciate your informing me wherever I happen to have made a mathematical error in my numbers.

In the above formula, n is the number of observations (numbers in the data set, or "trials"), y is the number of multiples of 19 discovered in our data set ("hits"), and p is the probability of finding one multiple of 19 in a random sample, i.e., 1/19.

In order to assess whether the frequency of multiples of 19 in a given data set is unusual or not, we must first count up all of the multiples of 19 we encounter and then compute the probability of finding that many in a random sample. It goes without saying that about 1/19 of a completely random sample of numbers consists of multiples of 19. Consequently, if, say, we find a multiple of 19 in a data set of fifteen to twenty-five, we should not at all be surprised.

Analyzing the Data Set. The formula given above is the standard binomial formula found in basic statistics textbooks. However, we cannot simply apply this formula alone in order to test our data. There is one more step involved. This formula will only tell us the probability of getting exactly so many multiples of 19. For example, if we observe 3 multiples of 19 in a given data set, the binomial formula will tell us the probability of getting exactly 3 multiples, not 1, or 2, or 4, or 5. This is known as the "point" probability. But this makes no sense for our purposes, because what we want to know is this: What is the probability of getting at least this many multiples of 19 in the data? Consequently, to compute the probability of finding at least as many multiples of 19 as there happen to be in a given data set (e.g., "3 or more"), we have to add the result of y = 3 to the results of y = 4, y = 5, y = 6, and so forth. This is known as the "tail" probability. This can be a lot of work, depending on your particular resources, but you can do it on a spreadsheet with some care. Here is one spreadsheet formula for the tail probability that I use:

Keep in mind that most published presentations of multiples of 19 in the Qur’ân involve the simultaneous application of a variety of mathematical principles, such that it is difficult to assess them in terms of statistical probability. One thing that the statistical method forces us to do is restrict ourselves to extremely few mathematical principles for any one test. If there is a meaningful mathematical structure in the letters or other features of the Qur’ân, we can consequently only look at a finite slice of it at a time using statistics. This means that our results will tend to be very conservative. That is, our results will tend to underestimate the extent of any structure that might be evidenced in the Qur’ânic text. Contrary to what you might think, however, this is actually a positive facet of statistics, because it keeps us from jumping to conclusions, and forces us to proceed on firm ground.