I recently discovered Benford's Law. I find it very fascinating. I'm wondering what are some of the real life uses of Benford's law. Specific examples would be great.
Statistics – Real Life Usage of Benford’s Law
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Related Solutions
As a rough/somewhat-intuitive explanation of why Benford's Law makes sense, consider it with respect to amounts of money. The amount of time(/effort/work) needed to get from \$1000 to \$2000 (100% increase) is a lot greater than the amount of time needed to get from \$8000 to \$9000 (12.5% increase)--increasing money is usually done in proportion to the money one has. Thought about in the other direction, it should take a fixed amount of time to, say, double one's money, so going from \$1000 to \$2000 takes as long as from \$2000 to \$4000 and \$4000 to \$8000, so the leading digit spends more time at lower values than at higher values. Because the value growth is exponential, the time spent at each leading digit is roughly logarithmic.
Honestly, I do not think there are any non-trivial "real life" applications that would be solved with undergrad level calculus. By "un-deterministic", I assume you mean stochastic and calculating "derivatives" of stochastic processes requires substantially heavier mathematical machinery than standard calculus. Now, if you are referring to the contrived "real world" word problems often found in calculus books, that is a different story. If you really want to use mathematics to solve "interesting" problems (I guess this is more of a matter of taste), I recommend furthering your body of knowledge.
Most "real world" problems are solved via numerical analysis so you might want to consider taking a look at that but even numerical analysis relies on many important concepts from analysis that will not be taught in calculus. Now, as counter-intuitive as it sounds, I'd recommend getting a solid foundation in the theory (especially in the big 3 of linear algebra, real analysis and functional analysis) before you worry about the applications. It will be difficult to solve an original problem without a through understanding of these topics.
I recommend most people start with Sheldon Axler's Linear Algebra Done Right and supplement it with a more computational text like Strang. After that, maybe start with Understanding Analysis by Abbot (again supplement this with a more computational text) then proceed to Baby Rudin or Erwin Kreyszig's Introductory Functional Analysis with Applications. Once you can understand these, you will be much better positioned to formulate and then solve your own problems.
Best Answer
Some of the data generated by Math Stack Exchange itself (and, presumably, by similar sites) ought to approximately follow Benford's law. These would include the distribution of first digits of
This is because Benford's law applies to exponentially growing quantities, and the total number of all of these quantities ought to be growing exponentially. It's only approximate because of artifacts in the way that some of these quantities are determined and because you need several orders of magnitude in order to see Benford's law really kick in.
Anyway, I decided to test this for the first three. The first digits weren't actually that hard to compile because the site allows you to sort the first three from highest to lowest. (Unfortunately, it does not allow you to sort the fourth that way, and I don't feel like wading through 2200+ questions to collect the data.) Here are the results I got on the first three.
Tag Use
(Data collected October 25, 2010.)
Not a bad fit, especially when you consider that there are only three orders of magnitude represented in this measure. There are a disproportionately large number of tags that were created but only used once, which explains the larger frequency of 1 as a first digit.
Votes Per Question
(Data collected October 26, 2010.)
Also not a bad fit, especially since there are currently only two orders of magnitude represented in this measure. Also, I threw out the questions with 0 and negative numbers of votes.
User Reputations
(Data collected October 25 and 26, 2010.)
This is the worst fit of the three, as the frequency of 1 as a first digit is so much larger than the others. However, there are a very large number of users who have never posted a question or an answer and so have a rep of 1. And, for reasons unknown to me, there are also a large number of users who have a reputation of 101, despite never having asked or answered a question.
If you remove 1 as a possible first digit and then rescale the Benford law probabilities to consider only 2 through 9 as possible first digits then the picture looks like the following, which is a much better Benford fit.
Admittedly, none of these data sets would pass Pearson's chi-square test for goodness-of-fit with respect to the Benford probabilities at a reasonable level of significance. However, given some of the artifacts in the data and the fact that there are relatively few orders of magnitude represented, the fit with Benford's law is really not that bad.