This issue can be mathematically proven thusly: if an individual chooses a probability distribution that is random, and then chooses a number based on that distribution that is also random, the list of numbers that results from that exercise will fall in line with Benford's Law (Bogomolny, n.d.).
Now that Benford's Law has been addressed and explained from a mathematical standpoint so that the reader has a better idea of not only what the law says but also what it means, it is time to address Zipf's Law as well. Benford's Law is highly important to the field of mathematics, but Zipf's Law also has a great deal of significance and therefore should be explained, addressed, and studied as well.
Originally, the law that Zipf created indicated that, in the corpus of utterances during natural language, the frequency that any word appears is generally, in a rough sense, proportional on an inverse level to the rank that it appears within the frequency table (Li, n.d.). In other words, the word that is used most frequently will appear roughly two times as often as the word that is seen to be the second most frequent, which will then be seen two times as often as the word that appears fourth most frequently, and this trend will continue for the entire list of words. The idea of Zipf's Law relates also to probability distributions and the power law (Li, n.d.; Hill, 1995).
Zipf's Law is not a theoretical law, but is rather an experimental law (Li, n.d.). Issues that take place because of Zipf's Law are commonly called Zipfian distributions. These kinds of distributions are seen in all different types of phenomena, but there are many that say that the Zipfian distributions that take place in real life are somewhat controversial, and that they may not be true Zipfian distributions (Li, n.d.). The easiest way to observe the work of Zipf's Law is to scatterplot the data. When this is done, the axes are log (rank order) and log...
If one is given a set of distributed frequencies that are seen to be Zipfian, and that are sorted from the most common frequency to the least common frequency, the frequency that is seen to be the second most common will appear 1/2 as frequently as the first frequency (Li, n.d.). The frequency that is seen to be the third most common will appear 1/3 as frequently as the first most common frequency. Expanding this all of the way out shows that the frequency that is nth most common will be seen to appear 1/n as frequently as the frequency that is seen to be the first most common (Li, n.d.).
However, there is some lack of precision where this is concerned. Most items have to occur a number of times that is actually an integer (Li, n.d.). In other words, a word that is seen within a document cannot appear in that document 2.5 number of times. It either must appear, in this example, 2 times or three times, since there cannot be an area of the document where only 1/2 of the word appears. Despite the fact that there is some variation and lack of precision, however, when wide ranges are examined and one only desires to have a relatively close approximation, many of the natural phenomena that are seen in this world do obey Zipf's law (Li, n.d.). This is seen to hold true as long as the individual examining the issue is not looking for scientific precision and will accept the slight variation that is seen.
Bibliography
Bogomolny, A. "Benford's Law and Zipf's Law. www.cut-the-knot.org/do_you_know/zipfLaw.shtml.
Hill, T.P. "Base-Invariance Implies Benford's Law." Proc. Amer. Math. Soc. 12, 887-895, 1995.
Li, W. "Zipf's Law." http://linkage.rockefeller.edu/wli/zipf/.
Wikipedia. (2006). Benford's Law. http://en.wikipedia.org/wiki/Benford%27s_law.
