Talk:Benford's law/Archive 3

"Benford's Law has been explained in various ways."
"Benford's Law has been explained in various ways."

This is kind of pitiful. Unfortunately, this is what I've seen with many wikipedia articles in general. There are a lot of "experts" who unfortunately don't know much. This is especially the case for many math articles I see, but not limited to that. The basic thrust of the article is totally wrong (case in point the bogus article 0.999999 = 1), or the explanation is muddled and confused. Typical of such math articles is to write down a bunch of impressive looking equations someone copied from a book and doesn't really understand.

I would suggest that wikipedia is more a general encyclopedia for the general reader, not an advanced technical book. It fails on both counts in this case. The explanation is not comprehensible to the general reader, and is quite inadequate for the expert.

This "explained in various ways" reminds me of the famous "five proofs" or whatever number of God's existence. You don't need five proofs, you need just one. The reason five proofs are put forward is kind of desperation. Same here.

If someone who really understands math (or some other real subject beyond "Buffy the Vampire Slayer" plots or such, which is mostly what wikipedia is good for) tries to provide high quality information, teeming multitudes of "experts" shout him or her down. No insult intended, it's just the reality. Would you have 100 random people off the street wielding scalpels to do your brain surgery? No. Did a committee of 100 compose Beethoven's Fifth Symphony? No. Does a random committee of 100 adjudicate Supreme Court decisions? No.

Anyway, off my soap box, back to the problem at hand, here is some help. This is not just a diatribe against wikipedia, but a genuine desire to help the editors who are trying to produce something worthwhile.

Take the numbers 1 to 9. Call them measurements in yards. The first digit happens 1 time for each number. No "Benford's law" so far. Then multiple them all by something, say 3. Aha! You will see "Benford's Law" emerge.

Measurement numbers (distances, times, etc.) are arbitrary, based on the units and the base. Suppose you convert yards to feet, as suggested above. 1 -> 3; 2 -> 6; 3 -> 9; 4 -> 12; 5 -> 15; 6 -> 18; 7 -> 21; 8 -> 24; 9 -> 27. "Benford's law" rear's it's ugly head. If you just look a little, it's obvious that it's easier to "get to 1" than to "get to 9" by multiplying. Big deal. :)

Or suppose you are counting sheep for your insomnia. If the count is in the range 90 to 99 before nodding off, that's 10 slots possible. Moving on to 100-199, that's 100 slots, so ones will tend to "crowd out" the nines. Suppose you go to sleep on sheep 166. Leading 1 wins! But even if you dozed off at sheep 966, leading 1 would STILL win out. Nine has zero chance to "win", can only tie at best, such as 9 or 99 sheep.

"Benford's law" intuitively boils down to the basic facts: *** In incrementing numbers 1 comes before 2, so 1 "has more chances" to be the first digit. *** In multiplying numbers 1 is smaller than 2, so the chance that the product of X with a number starting with 1 results in a number starting with 1 is greater than the chance that the product of X with a number starting with 2 results in results in a number staring with 2.

To show the somewhat arbitrary nature of "Benford's law", suppose you started with 999,999,999 sheep, and counted down, somewhat in the vein of "99 bottles of beer on the wall". :) Then all the numbers would start with 9, assuming your insomnia was not permanent. "Benford's law" would be totally reversed.

I normally do not spend time on wikipedia. It's a waste of time for me. It does pain me to see so much mediocrity. But I do appreciate the editors are trying their best. I wish wikipedia well and hope this comment helps. 71.212.104.23 (talk) 08:25, 22 August 2013 (UTC)


 * Okay, first of all, just because you think 0. 9 = 1 is "bogus" doesn't detract from the fact that it's true. It looks to me like you're blaming the article itself for something that you don't understand. All of the examples you've given ([1...9], counting sheep or backwards from a value] are of arbitrary numbers which wouldn't follow Benford's Law. Things HAVE been explained in various ways and, from what you've said, I can see no problems beyond what you've perceived to be there. JaeDyWolf ~ Baka-San (talk) 10:18, 22 August 2013 (UTC)


 * So, 71.212.104.23, what is the best case for Benford's law? Can prime numbers be shown to follow Benford's? John W. Nicholson (talk) 13:43, 22 August 2013 (UTC)


 * Benford's law is an empirical statement about real-world datasets--a statement which is sometimes but not always true. Since it is not a mathematical theorem, it does not possess a mathematical proof. Different real-world datasets may satisfy Benford's law for different reasons. That's why one explanation is not necessarily enough.


 * (However, even for real mathematical theorems, there is still often a benefit to showing more than one proof, as different proofs may offer different insights and ways of thinking about things. That is why multiple proofs of the same thing can commonly be found in textbooks and courses ... and wikipedia.)


 * 71.212.104.23's arguments for Benford's law are pretty shallow: None of those examples actually follows, or is expected to follow, Benford's law.


 * Of course, the article, like most articles, could be clearer and better and easier to read by non-experts. It would help to point out specific things that are especially confusing. --Steve (talk) 17:28, 10 March 2014 (UTC)


 * It doesn't follow to say that because it's empirical there's no mathematical proof. There must be some explanation, but it hasn't been found.--Jack Upland (talk) 02:57, 24 May 2014 (UTC)


 * Jack -- You can prove a mathematical statement like
 * "If a data-set is generated by process X, then it will follow Benford's law to accuracy Y."
 * For example...
 * "If a data-set is generated by an exponential growth process, terminated at a random time, then it will follow Benford's law to accuracy Y, where Y is blah blah blah (a formula relating to the growth rate, the probability distribution of termination time, etc.)"


 * I can prove that statement, and I can prove 10 other statements just like that. But none of those is a "mathematical proof of Benford's law". Because (1) "Process X" will be different for different data-sets (lengths of rivers, stock-market prices, etc.), (2) To make a rigorous mathematical proof, I have to idealize process X, which means that it won't (strictly speaking) apply to any real-world data-set. For example, bacterial growth under the right conditions is approximately exponential, but not exactly exponential. No real-world data-set is generated by an exactly exponential growth process. And as soon as you start speaking loosely and making approximations, you no longer have a rigorous mathematical proof.
 * The only way to "prove" in a totally rigorous way that the lengths of rivers follow Benford's law to 1% accuracy, is to measure all the rivers, or else to build a rigorously-accurate geological model of river formation and evolution. Who knows, maybe there is some geological process that makes it unusually likely for a river to have a length between 350km and 700km.
 * If you look at the figure to the right, it's very easy to prove that the more broad a probability distribution is (on a log scale), the more closely it follows the Benford's law distribution. But different data-sets are broad on a log-scale to different extents, and for different reasons! :-D --Steve (talk) 06:22, 28 May 2014 (UTC)
 * I think that's ludicrous. Benford's Law is proveable in practice. What's proved impossible is a mathematical explanation of why it's true. There is no explanation of why the lengths of rivers (in metres, yards, or cubits) should obey Benford's Law. Apparently the Law holds even if the figures are inaccurate.--Jack Upland (talk) 11:19, 29 August 2014 (UTC)
 * There is an explanation! It has two parts. The first part of the explanation is a geophysical explanation of why the lengths of rivers has a broad distribution on a log scale. The second part is the mathematical explanation of why anything with a broad distribution on a log scale will always obey Benford's law, as explained in the article, Section 4.1. --Steve (talk) 14:24, 1 September 2014 (UTC)

Rubbish. Singing in the choir is not an interpretation. I can imitate phrases in foreign languages but don't claim to understand them by that fact. Dressing up the phenomenon in mathematical jargon is not an explanation. It is merely reproducing the problem in a logical-numerical manner. If you can't explain it, don't buck-pass to Napier.--Jack Upland (talk) 08:52, 30 November 2014 (UTC)
 * Hi Jack -- Sometimes I see an alleged explanation of something that invokes a math concept I'm not familiar with, let's say the Levi-Civita connection for example. In that circumstance, I have no way to know whether it's actually the explanation I'm looking for, or whether it is merely a restatement of the original question in a more jargon-y way. Both are possible, and I've seen both plenty of times. The only way that I could possibly figure out whether it's a real explanation or not is to put in some effort to learn more about the Levi-Civita connection and build up my intuition about the Levi-Civita connection. After that I can decide whether the conceptual framework of the Levi-Civita connection really does lead me to the clear beautiful explanation I was looking for. And if so, whether that explanation actually requires understanding the Levi-Civita connection, or whether the explanation can be extracted and presented in a way that does not require this background knowledge.


 * The point of this example is: Just because it seems to you that an alleged explanation is not really an explanation but a jargon-y restatement, doesn't mean that that's really the case. Maybe it is a real explanation and you just need to spend a bit more time learning the necessary background. Maybe not. There's no easy way to tell.


 * To me, the explanations in this article do not seem jargon-y at all, and I made the graphic above specifically to help make the explanation accessible to as broad an audience as possible. I am not trying to intimidate anybody. If you cannot understand what the text is saying, you should say more specifically what you find confusing. I'm sure the writing can be improved.


 * If you really understand the text at a technical level and still disagree with it, either because you think that it is not an explanation at all, or because you think that it is unnecessary jargon and it can be described in a simpler way, then I am happy to hear your technical argument to that effect.


 * I don't see how you can ever hope to explain Benford's law without discussing logarithms and probability distributions, since the words "log" and "distribution" are right there in the statement of Benford's law. :-P --Steve (talk) 16:58, 30 November 2014 (UTC)


 * My maths is a bit rusty, but I'm not objecting to the explanation because it's too technical to understand. I came to this page hoping for some explanation of Benford's Law and didn't find one. A description is not an explanation. The article's "explanation" describes the Benford's Law as a logarithmic distribution. But it does not explain why this distribution occurs so widely. I agree with Arno Berger and Ted Hill who said, "The widely known phenomenon called Benford’s Law continues to defy attempts at an easy derivation". I think they understand the maths!--Jack Upland (talk) 05:21, 2 December 2014 (UTC)


 * Not having a simple derivation does not mean that it can't have a simple explanation. There is some cleanup to be done (specifically in the Explanations section), but the general idea is there (in the Mathematical background section).  a13ean (talk) 16:11, 2 December 2014 (UTC)
 * OK, that makes no sense, but dream the dream and live your fantasy.--Jack Upland (talk) 10:15, 15 December 2014 (UTC)

capitalization
The current article text refers to the law as "Benford's law" 28 times and as "Benford's Law" 52 times. I don't know which is correct, but it should be consistent. 2605:6000:EE4A:2900:6250:C93B:E4D4:B4BC (talk) 05:19, 15 January 2015 (UTC)


 * Good catch. The "hobgoblin of little minds" {http://www.goodreads.com/quotes/353571-a-foolish-consistency-is-the-hobgoblin-of-little-minds-adored} is hard to maintain sometimes! The usage should follow the article title. - DavidWBrooks (talk) 13:09, 15 January 2015 (UTC)


 * The current article title is no more definitive than the current text. Both usage and title in this article should follow the standard set by the referenced sources, if they're in general agreement on a styling.  On cursory examination, this standard seems to be "Benford's Law," but I leave it to someone more familiar with the topic to make this determination. 2605:6000:EE4A:2900:6250:C93B:E4D4:B4BC (talk) 19:23, 16 January 2015 (UTC)

Wrong explanation
"Benford's law applies most accurately to data that are distributed smoothly across many orders of magnitude" is just false and should be removed. Consider a variable uniformly distributed between 1 and 10^12; I believe this qualifies as "distributed smoothly across many orders of magnitude." The probability of the first digit of this variable being 1 is exactly 0.1. Moreover, the picture showing the probability in semi-log plot is extremely misleading. The shaded area correspond to the probability only if this graph is P(log(x)) not P(x) graphed in semi-log scale. If it is P(log(x)), and it is expected to be almost uniform so that you can apply the area proportional to width argument, then that means the probability P(x) itself has to be a linearly decreasing function (or a decreasing function that look almost linear at least over one order of magnitude at every point) so that P(log(x)) = P(x) dx/d(log(x)) = x P(x) looks uniform. I noticed that there was a note on the picture explaining that this graph is not a plain probability graphed in semi-log scale, but given the explanation above, you can see how a non-expert audience can be mislead just by reading the section and looking at the picture that a uniform distribution follows Benford's law. Sprlzrd (talk) 16:35, 22 April 2015 (UTC)


 * A variable uniformly destributed between 1 and 10^12 has a 90% chance of being between 10^11 and 10^12. This is not an example of a variable that is "distributed smoothly across many orders of magnitude", it is a variable that is almost entirely restricted to a single order of magnitude. Indeed, it has a 50% chance of being in a mere 0.3-order-of-magnitude window.


 * I have the impression that your complaint about P(log x) is a combination of two things: (1) You acknowledge that it is technically correct (because of Note 8) but think that Note 8 is too easy for readers to miss? (2) As you say, such probability distributions correspond to a P(x) graph that approximately follows 1/x over a certain range ... and you think that such distributions are unexpected and weird? (I'm not quite sure what your point is, sorry.)


 * For (1), it should be easy to fix. We can put references to Note 8 in more places, we can introduce separate labeling for footnotes versus references (like in United_States_dollar and many other articles, see how there is "[6]" vs "[Note 6]"), we can even move Note 8 into the main text. Do you think something like that would help? Or do you have any other suggestions?


 * For (2), such distributions (where P(log x) is slowly-varying over many orders of magnitude, i.e. P(x) kinda follows 1/x over many orders of magnitude) are really common in the world and in science and in math. Just look at any dataset following Benford's law! Many log-normal distributions are in this category for example.


 * If I misunderstand your complaints I'm sorry and I hope you can re-state them to clarify :-D --Steve (talk) 12:17, 23 April 2015 (UTC)


 * Okay, I see. What is written is not the same as what you are trying to say. The variables in my example are uniformly distributed, which is as smooth as any distribution gets, and they are distributed across many orders of magnitude. You are trying to say that the log of your variables is almost uniformly distributed over many orders of magnitude (it doesn't even have to be over many order of magnitude). If that's the case, just say that. I read that sentence for some of my physicist friends and they all interpreted it the way I did.


 * About the graph, I have the same complaint: it shows the probability distribution of the log of the variables, not the semi-log scale of the probability distribution of the variable, and it should be explicitly stated; no one is going to understand the significance of the footnote unless they already know enough about the subject.


 * The fact that anything with uniformly distributed log follows Benford's distribution is correct, and everyone who reads the article should know that, and the picture, if explained properly, does a good job of explaining that. Does it explain the ubiquity of Benford's law? No, it doesn't. And the combination of the first sentence, and the ambiguous explanation of the graph implies it does. I suggest the following: (i) replace the word smooth by uniform, since smooth means something totally different (ii) be very explicit about the distribution of the log has to be uniform not the distribution the variable (we could mention what that means in term of the distribution of the variable itself, P(x) follows 1/x) (iii) be explicit about this doesn't explain why Benford's law is found in real data, since we don't have any reason to expect 1/x distributions everywhere. (iv) remove anything about over many order of magnitude. You can get Benford's distribution for something with uniform log over one decade.Sprlzrd (talk) 21:40, 23 April 2015 (UTC)


 * Thanks, this is really helpful feedback :-D


 * I made some changes along the lines you suggested just now. I agree with "smooth" --> "uniform", and I tried to improve the wording from "log-scale probability distribution" without making it sound too convoluted for non-mathematicians to read. Did I succeed?


 * Your suggestion to "be explicit about this doesn't explain why Benford's law is found in real data, since we don't have any reason to expect 1/x distributions everywhere" has already been taken care of a while ago, it's the last paragraph of the section ... (Random thought: Maybe there should also be a link to Zipf's law? I'm not sure what the exact relation is.)


 * I disagree with your categorical objection to "many order of magnitude". It is generally true that real-world probability distributions that cover many orders of magnitude have smaller discrepancies from Benford's law than probability distributions that are contained within fewer orders of magnitude. This is obvious I think, and explicitly stated by three references in the section.


 * Now, it is true that you can concoct an example (log-uniform in one order of magnitude with sharp cut-offs) that exactly satisfies Benford's law despite being restricted to one order of magnitude. But surely you recognize that this is an artificial and atypical example. But nevertheless I tried to soften the language so that readers don't get the wrong idea. What do you think? --Steve (talk) 12:21, 24 April 2015 (UTC)


 * I liked very much the changes you made. I agree that the data from the cited references seem to show that empirically the broader the distribution the better agreement with Benford's law. I added a little explanation for more mathematical audience in the parenthesis. Is that okay? --Sprlzrd (talk) 18:25, 24 April 2015 (UTC)


 * Sure! But I changed the wording ... it's only the fractional part of the logarithm that is supposed to be uniformly distributed. Do you agree? --Steve (talk) 22:00, 25 April 2015 (UTC)


 * It took me a minute staring at it to understand what that sentence means, but sure, it seems technically correct. It's your call; if you think this adds significant information keep it as it is. It would be nice if you could provide a citation for that sentence. I am happy with the overall changes made in this section; my main objection is definitely resolved :) -Sprlzrd (talk) 14:44, 27 April 2015 (UTC)

Terminal digits in pathology - an irrelevant example
The authors give the following as a non-example for the Benford's law: "The terminal digits in pathology reports violate Benford's law due to rounding, and the fact that terminal digits are never expected to follow Benford's law in the first place." Clearly, this is not false, but it strikes me as irrelevant, and may confuse some readers. One would *never* expect the terminal digit to follow Benford's distribution (it should be fairly uniform for other reasons). So why bring up this specific instance? Jakub Konieczny 20:55, 27 April 2015 (UTC) — Preceding unsigned comment added by Jakub.konieczny (talk • contribs)

Ignored reference
I think this reference could clarify some points and should at least be cited http://www.sciencedirect.com/science/article/pii/S0378437100006336 — Preceding unsigned comment added by Vitelot (talk • contribs) 13:08, 16 July 2015 (UTC)


 * Thanks, I had long been looking for something like that in the literature. I used it as the basis for a new section "Multiplicative fluctuations" . (I know there are other aspects to the paper too, but this is a start.) :-D --Steve (talk) 15:40, 16 July 2015 (UTC)

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The example is wrong
If you look at the tallest 60 buildings in the world, none of them start with a 1 in metres (see the wikipedia page). You can't use Benford on just the tail of the distribution. — Preceding unsigned comment added by 74.64.57.82 (talk) 17:38, 10 March 2016 (UTC)


 * I thought it was referring to "tallest structures" - List_of_tallest_buildings_and_structures_in_the_world, right?


 * I hate to be defending that section, it's not a very good example of Benford's law. I personally think it should be deleted altogether. The figure with populations of all countries is a much better example of Benford's law, and more concise too! As long as we have that figure, I don't see the point in having the "tallest structures" section. --Steve (talk) 01:23, 23 March 2016 (UTC)

Bad Example? - Distributions not Following Benford's Law
I'm not an idiot when it comes to stats, but hardly an expert. This is given as an example of a distribution that would not follow Benford's Law:

"Where numbers are assigned sequentially: e.g. check numbers, invoice numbers"

I think the example is not necessarily wrong, but might require more explanation. Sequentially assigned numbers are generally a good example of Benford's Law in action, e.g. street numbers. All streets have a 100 block, most a 200 and 300 block, and then less and less likely. But those with a 900 block could easily have a 1000 block, and then maybe a 2000 block, and so on. And those are sequentially assigned. I'm not sure the best way to describe the difference between a more limited set of sequential numbers, like check numbers, and more open-ended sets like street numbers, but it seems like a worthwhile distinction to prevent apparent contradictions in the article. Just my handful of change. Sdr (talk) 19:54, 31 July 2016 (UTC)


 * If you take any one street by itself, and look at the house numbers on that street, they will be pretty far from Benford's law. If you look at many different streets as a group, the house numbers will probably be pretty close to Benford's law. Do you agree? If so, the only question is how to reword that bullet point to make it clearer...


 * For example, on my own home street, the numbers go up to ~185, so 1's are very overrepresented and 2's are very underrepresented compared to Benford's law. --Steve (talk) 18:09, 2 August 2016 (UTC)


 * I’m not an expert, but it takes more than a simple monotonic relationship between digit frequencies to be Benford’s Law which has a specific logarithmic probability law. So, it may be that check numbers and street numbers are not examples of Benford’s Law for this reason. Constant314 (talk) 19:06, 2 August 2016 (UTC)


 * An anecdote: A long time ago I took a short course from Richard Hamming.  As an aside on day, he told us that he used to make a little money by walking into a lab and making an even money bet that the next measurement made by anybody in the lab would have a first digit of three or less. Constant314 (talk) 19:19, 2 August 2016 (UTC)


 * The lead does give "street addresses" as an example of Benford's Law. It also says that Benford's Law "tends to be most accurate when values are distributed across multiple orders of magnitude". This is not generally true of street numbers. I'm not sure about cheques, but with invoices, in my experience, the numbers have a set number of digits, e.g. from 0001 to 9999. In this case, I think the initial digits would be evenly distributed, and not follow Benford's Law.--Jack Upland (talk) 22:30, 2 August 2016 (UTC)


 * Where I live, I very commonly come across 1-digit, 2-digit 3-digit, and (in the city) 4-digit street addresses. (Not all on the same street!) This is in Massachusetts, USA. Is it different in other regions?


 * Invoices or checks from a single company will probably not follow Benford's law, but the set of all invoices I receive (which come from many different companies) probably will. I might get a 7-digit invoice number from my phone company, a 3-digit invoice number from my local electrician, etc. --Steve (talk) 14:29, 3 August 2016 (UTC)


 * As before I am not an expert, so this is just my speculation. To be truly Benford, the numbers must be distributed uniformly on a logarithmic scale.  Street addresses, check numbers and invoice numbers are distributed uniformly on a linear scale.  The  frequency of the most significant digit is a logarithmic function of the digit value.  One would expect that changing units would not change the distribution of most significant digits.  For instance, if something measured in meters showed a Benford distribution then converting the measurements to feet should still yield a Benford distribution.
 * Suppose I open a checking account and start with check number 100. When I close the account I’m up to 2699.  Number of checks beginning with one:1100, number beginning with two: 800.  Number checks beginning with three: 100, four:100, five:100, six:100, 7:100, 8:100, 9:100.  It is a non-increasing, but not Benford.
 * The total number of checks written over the lifetime of an account for all closed accounts: maybe Benford.
 * Average sound power I experience in watts per square meter: probably Benford.
 * Average sound power I experience in dBm per square meter: probably not Benford.
 * A manufacturer of opamps produces many types and grades form jelly-bean to ultraprecision. The noise specifications vary by a factor of 1000. The noise of each opamp is measured.  The measurements are probably Benford.
 * Number of bits I uploaded each minute at my URL, not counting minutes with zero bits uploaded: maybe Benford.Constant314 (talk) 16:22, 7 August 2016 (UTC)

Phone numbers as a non-Benford's Law distribution
The article currently gives "the 1974 Vancouver, Canada telephone book" as an example of a distribution that does not obey Benford's Law since "no number [in it] began with the digit 1". However, in the North American Numbering Plan (obeyed by Canada), telephone numbers are never allowed to begin with 1. I feel this example should be removed or, at the very least, this fact should be noted in a footnote. Admiral.Mercurial (talk) 12:47, 30 August 2014 (UTC)
 * I agree that's an error in Raimi (the original paper notes 0 incidence of 1s but doesn't investigate it) and we should probably remove the claim. Protonk (talk) 13:42, 30 August 2014 (UTC)
 * Thanks for catching this. If it's an error in the original paper (and not an introduced Wikipedia error), it would be better to explain the error, with a Wikilink to the North American Numbering Plan. Simply removing the erroneous claim leaves open the possibility of it being re-added,  or of it misleading a reader who follows the footnote to the original paper.  Reify-tech (talk) 13:57, 31 August 2014 (UTC)
 * I'm not sure it's an error. Rather it's a trivial example with an obvious explanation.--Jack Upland (talk) 00:21, 1 September 2014 (UTC)
 * It may be obvious to us here and now, but evidently it wasn't obvious and trivial to the authors of the paper at the time of writing, since they noticed it but offered no explanation for the anomaly. I think it is better to explain it as an example of inadvertent bias, rather than to omit it as "obvious", when it was clearly not obvious to the authors of the paper.  It still remains non-obvious to readers unfamiliar with the North American Numbering Plan.  Reify-tech (talk) 13:29, 1 September 2014 (UTC)
 * I agree with Jack. It is not an error, it is a trivial example with an obvious explanation. The authors of the paper apparently found it too obvious to even mention. But there's nothing wrong with saying it explicitly, so I just added it. :-D --Steve (talk) 14:01, 1 September 2014 (UTC)
 * I must say this example seems really silly: it is like saying that the numbers in the 20s violate Benford's law. GeneCallahan (talk) 16:32, 2 October 2016 (UTC)
 * I agree. It is silly.  Even if the phone numbers were allowed to begin with the digit 1, the distribution would not be Benford because numbers are not distributed logarithmicly. Constant314 (talk) 18:03, 2 October 2016 (UTC)

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Special case of Zipf's Law?
In the introduction section, it states that Benford's law is a special case of Zipf's law. However, the citation is a dead link, and as someone already marked, clarification is needed since the naive interpretation of it being a special case (that the first digit frequencies follow a power law) is obviously false. On the Zipf's law page, some connection to Benford's law is briefly discussed, but I find this still unclear.

The only citation in that part (apart from the same dead link) does not claim that Benford is a special case of Zipf; the closest it gets is: "In this sense, it is interesting to explore also connections with other well known scale invariant features such as the Zipf’s law". Perhaps what is really meant is that Benford's law is a corollary of Zipf's law, if we assume that the probability distribution of numbers follows a power law.

In the meantime, I suggest we simply say (rather vaguely) that it has been argued that this law is related to Zipf's law as they both come from scale invariance.

Gneisss (talk) 21:43, 20 September 2017 (UTC)


 * I agree on all counts and I like your proposed edit. --Steve (talk) 11:59, 21 September 2017 (UTC)

Discussed in book Lady Luck, Warren Weaver, 1963.
Although the name Benford is not mentioned, there are seven or eight pages of discussion of this in chapter XII. The explanation, formula, and diagram given are different from those in the article. The 1961 paper by Pinkham, "On the distribution of first significant digits" is mentioned in a footnote. 92.3.76.2 (talk) —Preceding undated comment added 22:22, 13 February 2019 (UTC)

Discussed in a DSP book
A DSP book claims that Benford's law is followed by distributions with narrow Fourier transform of the PDF, in fact it will hold exactly for PDFs for which the Fourier transform is zero at positive integer frequencies. Note that due to the uncertainty principle, this is exactly the same as saying that the distribution itself is very wide. http://www.dspguide.com/ch34.htm 2A02:6B8:0:845:0:0:1:18 (talk) 11:06, 24 February 2019 (UTC)


 * FYI, there was a discussion about this reference back in 2012 -- see Talk:Benford's law/Archive 2. Do you propose editing the article, and if so, what do you think should change? --Steve (talk) 19:36, 25 February 2019 (UTC)

Actually this is based on logarithms.
I mean, this is correct in the meaning that $$\lg\frac21>\lg\frac32>\lg\frac43>\lg\frac54>\lg\frac65>\lg\frac76>\lg\frac87>\lg\frac98>\lg\frac{10}9$$ Alfa-ketosav (talk) 14:07, 20 October 2019 (UTC)

Dispersion should not be too small
normally never mentioned: the dispersion or variance should be not "to small". A kind of proof in nordisk Matematisk tidskrift from 1965 ( or almost) has that condition included in teh proof. —Preceding unsigned comment added by 130.226.230.8 (talk • contribs) 16:18, 16 May 2008 (UTC)

Benford's Law in Nuclear Physics
Benford's law is applicable for the evaluated nuclear physics quantities. It works reasonably well for large samples (>400) and performs poorly for a small sample (~12). These results have been published in Journal Of Physics G. It is a first application of Benford's law in nuclear physics.

— Preceding unsigned comment added by 24.185.225.158 (talk) 13:14, 25 May 2015 (UTC)

Disparate treatment of Unary and Binary
Binary is shown in the article as being trivially following BL (as it has only one digit capable of having a leading digit). Unary is listed in the article as violating BL (despite it having the same characteristic; having only one leading digit)

Either both or neither are trivially following. — Preceding unsigned comment added by ‎Dusty78 (talk • contribs) 11:43, 16 April 2020 (UTC)

"leading significant digit"
Is "significant" meaningful here? Isn't the leading digit always significant? - Sum mer PhD v2.0 21:36, 3 September 2020 (UTC)


 * I suppose it could be a zero - but that's silly. You're right. - DavidWBrooks (talk) 22:40, 3 September 2020 (UTC)
 * If we're going to assume a zero can be a leading digit, the first infinite number of digits is always a zero. - Sum mer PhD v2.0 02:53, 4 September 2020 (UTC)
 * The word "significant", in this context, means non-zero. The leading significant digit is the first digit one comes to which is not 0. For example, in the number 789 the leading significant digit is 7. In 0.05 the leading significant digit is 5. The leading digit, without the word significant, would mean the given digit on the left. This is poorly defined as it depends on how a number is presented. For example, one could imagine an amount of money being displayed under certain circumstances as £00189.15 in which the leading digit is 0. In the 0.05 example given before the leading digit is again 0. A question about how to present such data might be phrased "how many leading 0s should I show?" Scientists studying something very small might give a quick estimate by saying how many leading 0s a value has. It is rare, however, that leading insignificant digits are discussed or studied, so I think most people would hear "leading digit" and fill in the word "significant" by assumption. It definitely does have meaning though. Awoma (talk) 22:42, 6 November 2020 (UTC)

The word "significant could simply be removed here as a red herring. Benford is a model of exact real numbers, not numbers with some inaccuracy or statistical uncertainty that are rounded, approximated or represented as an interval.73.89.25.252 (talk) 05:32, 12 November 2020 (UTC)
 * Significant does not mean that a number has been rounded or approximated. In the context of leading digits it simply means non-zero. For example, if we took the cost in pounds (or dollars or euros) of all items in a supermarket, this would match Benford's law. Many items would be priced lower than £1, in which case the leading digit would not equal the leading significant digit. A price of £0.45, strictly speaking, has leading digit 0 and leading significant digit 4. Benford makes a statement on the latter. There will be many examples of people leaving out the word significant and letting the reader assume its presence, but I think we are better off avoiding relying on this and just keeping the word significant there. Awoma (talk) 09:31, 12 November 2020 (UTC)

Popular culture
An editor removed the "popular culture" section with three mentions of Benford's Law in movies/TV shows. I have returned them because they inform the reader of an important and unexpected fact: This once-obscure topic is entering the popular realm. That's valuable information, not trivia. - DavidWBrooks (talk) 19:43, 16 September 2020 (UTC)


 * That's original research based on trivia. Yes, the law was mentioned in three TV shows. Is that "unexpected"? Meh. Laws (not those laws...) pop up on police and court shows all the time. How about illnesses in medical shows? Are these indications that RICO or carcinoma are "entering the popular realm"? No. It's an indication that people editing Wikipedia today are watching current TV shows.


 * An indication that something is having an impact on popular culture is independent reliable sources discussing it. Otherwise, various characters in recent TV shows, movies, comic books, novels, operas, magazine ads, video games who lean on an oak tree, have asthma, wear blue, eat French Fries, etc. would be listed in those articles.


 * These are randomly selected instances, selected by the current popularity of the show, the demographics of the show's audience vs. Wikipedia editors' demographics, etc. It's an indiscriminate list.


 * "It is preferable to develop a normal article section with well-written paragraphs that give a logically presented overview (often chronological and/or by medium) of how the subject has been documented, featured, and portrayed in different media and genres, for various purposes and audiences.... The consensus is very clear that a secondary source is required in almost all cases." MOS:POPCULT - Sum mer PhD v2.0 21:10, 16 September 2020 (UTC)


 * Then improve it, don't just toss it out and deprive readers of information, just because you think it's not important enough. Remember - everything in wikipedia is randomly selected; that's what editing is. - DavidWBrooks (talk) 22:25, 16 September 2020 (UTC)


 * Wikipedia is neither an indiscriminate collection of information nor a repository for original ideas.


 * WP:WEIGHT says coverage of information in articles should be based on the prominance of coverage in independent reliable sources. Independent reliable sources on Benford's law, the subject of this article, neither call out those three episodes of recent shows nor discuss that a "once-obscure topic is entering the popular realm". That is your original research.


 * Richard Nixon could list the numerous Academy Award-winning films, Pulitzer Prize-winning books, touring operas, TV shows, Grammy-winning albums, chart-topping songs, popular films, knock-knock jokes and disembodied heads of Nixon in popular culture, demonstrating he has "entered the popular realm". However, sources about Nixon do not discuss them, so they are not in the article. In contrast, Chevy Chase's Saturday Night Live send up of Ford is in Gerald Ford. Why the difference? Chase's skit is discussed in independent reliable sources covering it's impact on Ford (we cite the New York Times, but they aren't alone). That is a popular culture tie to a topic.


 * I can't improve the section by willing into existence sources discussing a 150 year old law in three indiscriminately selected popular TV shows from the past 4 years. There don't seem to be any sources for this original research. We might as well be looking for sources discussing TV shows where someone eats French fries.


 * "Please reorganize this content to explain the subject's impact on popular culture, providing citations to reliable, secondary sources, rather than simply listing appearances." - Sum mer PhD v2.0 22:48, 16 September 2020 (UTC)

I see your wikipedia advice and come back with wp:WikiProject Popular Culture "If material is verifiable, neutral, and well-organized, we feel deleting it is an inappropriate act -- an expression of personal distaste not in keeping with the goal of creating the world's best encyclopedia."

I have argued with purists over "in popular culture" sections for more than a decade. These sections can, of course, become pointless trivia-pits and need to be watched carefully; I have deleted many an item and on occasion an entire section.

But many editors heave them out on sight, as you did, usually because they consider them below the dignity of an encyclopedia, without considering the information content that they indirectly carry to readers.

As to your example, it is obvious to any reader that a president is present in popular culture so listing examples is superfluous - but it is not obvious to any reader that this numerical discovery is present in popular culture and so listing examples can provide information. It is interesting and unexpected that writers of TV dramas would know of Benford's Law; being told that fact can provide insight to readers. Sure that's a judgement call - but judgement is what wikipedia editing involves; otherwise the whole thing could be generated by an algorithm. - DavidWBrooks (talk) 01:27, 17 September 2020 (UTC)


 * Having said that, I have rewritten the section to organize it a bit, add another example. - DavidWBrooks (talk) 01:33, 17 September 2020 (UTC)


 * You have now upped the indiscriminate list of 3 random entries to an indiscriminate list of four entries, tied together by a tiny thread of original research. "Please reorganize this content to explain the subject's impact on popular culture, providing citations to reliable, secondary sources, rather than simply listing appearances."


 * "The use of guns to help stop robberies is sometimes an element in dramas, including: ... (indiscriminate list of gun use in cop shows)."


 * "Richard Nixon is often used as a stand-in for corrupt/hard-nosed/fascist/amoral/focused/racist/whatever government officials in TV shows, movies, operas, songs, etc., including: ... (indiscriminate list)." - Sum mer PhD v2.0 02:37, 17 September 2020 (UTC)

Election Day Edits
We're seeing multiple election day edits that are highly editorial, but no TALK discussion.

I personally tried to access my account but after requesting a password reset have not received any response. I'm not a wikipedian, but the increased politicization and Orwellian attitude of edits have made this place impossibly hostile.

"Who controls the past controls the future: who controls the present controls the past." George Orwell — Preceding unsigned comment added by 173.56.240.14 (talk) 04:12, 7 November 2020 (UTC)
 * Password resets go to the email account you signed up with. Additionally, this article is seeing a spike due to claims of voter fraud.  A quick check of Twitter shows links to this article, which is likely the reason for the traffic.  Wikipedia requires citations for information that is a Reliable source.  --Super Goku V (talk) 23:57, 7 November 2020 (UTC)

Benfords law has always been unreliable when elections are concerned, as surely as Oceania has always been at war with Eastasia. Nothing to see here, move along. 95.202.161.202 (talk) 14:43, 8 November 2020 (UTC)

So, right when Benford's law indicates Biden's campaign might have a mass scale voter fraud, this wikipedia article changed to say Benford's law is wrong?
Seems like a coincidence, doesn't it? A seemingly unchanged, relatively ignored article, about a law that indicated voter fraud time and time again, now discredited?EpicMemeGamer (talk) —Preceding undated comment added 02:59, 7 November 2020 (UTC)

It actually discredited Benford's law being used for election fraud months before the 2020 election, as you can see from this older edit. However, in recent days an anonymous user repeatedly removed that text, causing it to have to be re-added. In fact, the most recent edit added an extra sentence to cast doubt on the study that supposedly "discredited" the application of Benford's law to elections. So by all metrics, Wikipedia got changed in the direction of saying that Benford's law IS applicable to voter fraud. Scoopdaddy (talk) 03:24, 7 November 2020 (UTC)
 * The statement "other experts consider Benford's Law essentially useless as a statistical indicator of election fraud in general." has been part of the article since at least 2013. The only recent addition in that direction was to add a reference. Constant314 (talk) 04:26, 7 November 2020 (UTC)


 * A statement to the effect of "experts" or "other experts", especially lacking citation, is weasel language. 95.202.161.202 (talk) 14:54, 8 November 2020 (UTC)

https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions

Benford's law is not discredited... https://pdfs.semanticscholar.org/e667/b8ad9f58992828ff820ddc8a005de754c5f5.pdf — Preceding unsigned comment added by 2601:82:C201:E6D0:E82A:990B:177F:2B18 (talk) 04:41, 7 November 2020 (UTC)
 * See my post below for why the article you have attached doesn't say it's discredited or not. Particularly, " "On one fundamental point, I think their argument is correct: that the mean of the second significant digits in vote counts is significantly different from 4.187 (The mean expected under Benford's Law) is not evidence that the vote counts are affected by fraud" and concludes with "I do not dispute that 2BL tests need a foundation in some kind of theory to be useful".


 * Likewise, you can refer to https://www.issuelab.org/resources/33905/33905.pdf "The second digit methods are based on the idea that non-anomalous vote counts follow the socalled second-digit Benford’s Law-like (2BL) distribution—a particular distribution of numerical digits occurring from a natural process with each number 0-9 as differentially likely (Pericchi and Torres 2004 Mebane 2006). The last-digit method is based on the idea that unmanipulated vote counts have uniformly distributed 0-9 last digits. The digit-focused methods are controversial as fraud-detection devices (Carter Center 2005; Shikano and Mack 2009; L´opez 2009; Deckert, Myagkov, and Ordeshook 2011; Mebane 2011, 2014), with in particular Mebane (2013a, n.d.) claiming that the second digits ofprecinct vote counts may be produced by normal political processes such as strategic voting and mobilization that do not trace back to illegitimate human manipulation." (Hicken & Mebane 2017) X0n10ox (talk) 05:09, 7 November 2020 (UTC)


 * 2BL is not Benford's Law. It's testing whether Benford's Law can be applied to other digit values. Maybe I've misunderstood, but it seems that you're using remarks against 2BL in attempt to discredit Benford's Law. 2600:1700:49C0:C0A0:24D8:93BF:E654:760D (talk) 06:00, 7 November 2020 (UTC)


 * You seem to be confused, because in the article linked and the article being used as "refuting" the criticism by Dr. Mebane is only talking about 2BL and nothing else. In 2007, Mebane himself says "for testing whether the second digits of vote counts follow a distribution specified by Benford’s law (the 2BL test) (Mebane 2006b)" clarifying that his initial approach to election forensics utilized 2BL tests . If you also read the entirety of the Electronic Forensics Guide which Dr. Mebane co-authored in 2017, you would see that he notes ALL the currently used methods that use Bedford's Law had criticism, and notes his specific criticism for 2BL. . Factored together, Dr. Mebane's research is currently being misused in the article, as it is claiming that his commentary refutes the work of Deckert, but it doesn't at all, and Mebane cites Deckert et al. in his later 2017 work. In the 4 page commentary he released, Dr. Mebane acknowledges the validity of Deckert et al's work, but disagrees with some of the methods they've used. This isn't being accurately represented at all in the article and is indicative that people either had ill-intent in using Dr. Mebane or that they simply haven't read the 4 page commentary they're citing nor have they read any of Dr. Mebane's extensive research and work on the subject. In reality, Dr. Mebane prefers a fourth approach that was a simulation method that was turned into a statistical estimation method by Dr. Mebane. His method provides sophisticated modeling of irregularities and uses data on turnout, valid ballots, number of votes received by each candidate in polling stations to produce estimates of fraud. So either we're talking about 2BL, because that's what Mebane is talking about in his alleged refutation, or Dr. Mebane's work shouldn't be being used in the way that it presently is. X0n10ox (talk) 06:35, 7 November 2020 (UTC)

The Association of Certified Fraud Examiners (ACFE) in a 2018 study explicity identifies the use of Bedford's Law in election fraud world wide: "In recent years, the use of Benford’s Law is finding notoriety in election investigations. Jurisdictions around the world are using it to narrow down potential voter registration fraud and ballot stuffing schemes." Regardless of the views of Mebane or Deckert, the routine use "around the world" of Benford's law in Election Fraud is utterly relevant and needs to be included in this Wiki article. 2600:6C40:7E7F:F963:5DB4:BD41:C1E9:47D5 (talk) 07:49, 7 November 2020 (UTC)JLS


 * That's great and it has nothing at all to do with the conversation at hand, which was regarding people use Dr. Mebane's work out of context. I would note that the link you have provided doesn't provide sources or anything regarding the claim, nor speaks to the methodology. This fact could be added to the article, but it doesn't say anything that the article doesn't already say, which is that Benford's Law is used by some for investigating election fraud and that others have criticized it, the difference being chiefly that the sources involved currently are of an academic and scholarly nature and have listed sources. I'm baffled as to what you think adding the ACFE article from 2018 will add to the article that isn't already there. For that matter, one could make a similar argument that Dr. Mebane's guide for Election Forensics should be cited and the criticism he levies against the digit-testing for the Benford Law (2BL or otherwise), but it doesn't add anything substantive. It just repeats the same thing, some think it can be used for this thing, others think it cannot. For that matter, should we then also add this statement from the University of Washington Center for an Informed Public, which says "Having the distribution of leading digits stray from the expected percentages predicted by Benford’s Law can happen by chance, though it is more common when the law’s assumptions are violated, as they often are with vote tallies. Benford’s Law, and other math-based inquiries, can be used to detect voter fraud, but the vast majority of these violations are not conclusive evidence of fraud." that very specifically talks about how Benford's Law is not conclusive evidence and even talks to this specific election? I don't think it should be, personally, but nor do I think further elaboration is needed on the fact that some people say it works for elections and others contest it. X0n10ox (talk) 08:25, 7 November 2020 (UTC)


 * I looked into it myself! Here's what happened. (Hoping this edit works!)


 * 9 August 2010, this edit shows the first sentence officially added, mentioning the 2009 Iranian presidential elections.


 * 23 March 2011, this edit shows the first revision of the line that eventually was removed, causing the controversy.


 * 23 March 2011, a few hours later, this revision is what ends up being the sentence that is settled on.


 * 5 November 2020, the line is removed.


 * 5 November 2020, it returns just a few minutes later due to unexplained content removal.


 * 5 November 2020, it is removed again.


 * 5 November 2020, after many edits back and forth, the page is finally locked.


 * JBMagination (talk) 20:09, 7 November 2020 (UTC)

Mebane and Election Data
I have concerns with the objectivity of the wording used in mentioning the Mebane(2011) piece. Specifically, the article is not a refutation, it is a 4 page comment on the other paper. It doesn't rise to the standard of an academic research paper, nor does it purport to refute the findings of Deckert, Myagkov, and Ordeshook (2011). Rather, Mebane's commentary is a defense of his previous work which was cited by Deckert, Myhagkov, and Ordeshook in their piece. The first page even concedes "On one fundamental point, I think their argument is correct: that the mean of the second significant digits in vote counts is significantly different from 4.187 (The mean expected under Benford's Law) is not evidence that the vote counts are affected by fraud; but the way they argue for this is unsound". Mebane does not contend that their conclusion is flawed, but that their method of argumentation is incorrect. Currently, the wording in the article creates the appearance that Mebane is saying the findings are completely refuted. On page 270, Mebane further states "The fact that Benford's Law does not supply a purely mathematical argument for applying 2BL to vote counts reinforces Deckert and colleagues' position that applying 2BL to vote counts requires a theoretical basis. Purely mechanical efforts to find such a basis, such as the computational approach used in Mebane (2007), seem not to be sufficient."

Further, "The analysis in Mebane (2007), for example, leaves unexplained large and persistent deviation from expectations in 2000 and 2004 in votes for president cast in Los Angeles, California, that was reported in Mebane (2008). But no reasonable person believes that there was massive election fraud there that no one happened to notice in those years" and also in the conclusion, Mebane notes that he does not mean to assert that the analysis in Mebane and Kalinin (2010) is correct, admits it is preliminary research, admits the research on using 2BL for election fraud isn't greatly peer reviewed, and asks them to clarify the target of "the purely negative contribution they make in the piece discussed here". He then concludes "I do not dispute that 2BL tests need a foundation in some kind of theory to be useful" but says that they do not offer or hint at such a theory or how it could come to be.

You can readily and plainly see, then, that had they actually read the academic source they listed that Mebane doesn't "refute" Deckert, Myhagkov, and Ordeshook but actually agrees with them in several ways. The primary point of this 4 page commentary is seeking clarification from them and offering clarification of his own work that was cited by Deckert and colleagues, while inviting more academic discourse on the subject. — Preceding unsigned comment added by X0n10ox (talk • contribs) 03:28, 7 November 2020 (UTC)


 * A further addition to this, with another source from Mebane furthers the notion that Mebane doesn't refute the results of Deckert and colleagues. See "The digit-focused methods are controversial as fraud-detection devices (Carter Center 2005; Shikano and Mack 2009; L´opez 2009; Deckert, Myagkov, and Ordeshook 2011; Mebane 2011, 2014), with in particular Mebane (2013a, n.d.) claiming that the second digits of precinct vote counts may be produced by normal political processes such as strategic voting and mobilization that do not trace back to illegitimate human manipulation."  — Preceding unsigned comment added by X0n10ox (talk • contribs) 05:20, 7 November 2020 (UTC)


 * 2BL is not Benford's Law. It's testing whether Benford's Law can be applied to other digit values. Maybe I've misunderstood, but it seems that you're using remarks against 2BL in attempt to discredit Benford's Law. 2600:1700:49C0:C0A0:24D8:93BF:E654:760D (talk) 06:00, 7 November 2020 (UTC)


 * I am not trying to "discredit" Benfords Law, I am pointing out the misuse of Dr. Mebane's work presently in the article. The article currently states that Dr. Mebane's 4 page commentary refutes Deckert et al's work, which it absolutely and explicitly does not do, and providing actual evidence from the article quoted as well as later work by Dr. Mebane which shows that he more or less agrees with the conclusion of Deckert et al's work.. X0n10ox (talk) 06:39, 7 November 2020 (UTC)

Can someone include racist and part of the patriarchy after problematic? I have all my citations in order: https://www.seattletimes.com/education-lab/new-course-outlines-prompt-conversations-about-identity-race-in-seattle-classrooms-even-in-math/

https://thejewishvoice.com/2020/08/brooklyn-college-education-prof-claims-math-is-white-supremacist-patriarchy/

Oh yeah..Trust the science. — Preceding unsigned comment added by 2605:A601:AACF:CF00:F42C:10AE:4F5D:778B (talk) 07:26, 7 November 2020 (UTC)


 * For what it's worth, the line which I had contention with that misrepresented Dr. Mebane's work has been rectified. X0n10ox (talk) 06:41, 7 November 2020 (UTC)


 * Am I right in thinking this is cleared up? A quick look over the section and sources in its current form seems like the wording is acceptable, but I may have missed a nuance. Awoma (talk) 09:15, 7 November 2020 (UTC)
 * You are correct. Someone came along and sufficient edited away the problems I pointed out, chiefly that the edit made by Adda'r Yw Yw misstated Dr. Mebane's commentary and stance on the issue at hand (Such as it is that Dr. Mebane did not refute the previously mentioned paper, but provided commentary on it, and agreed with some of its conclusions). X0n10ox (talk) 10:31, 7 November 2020 (UTC)
 * I've expanded this section a bit. It would seem strange to leave the wording of Mebane criticizing the use of Benford's law (though he does note problems), since he himself has pioneered its application in election data. Adda&#39;r Yw (talk) 11:13, 7 November 2020 (UTC)
 * While he did pioneer the usage in the early 2000s in particular of using 2BL tests for election data, his later methodology and research uses a different, more complicated model that includes a multitude of factors beyond what the two sources cited were talking about, which was 2BL tests, this being noted in the USAID Guide to Election Forensics which Dr. Mebane co-authored where Dr. Mebane notes his own criticism of 2BL tests and explains his methodology as a fourth, different option that involves much more data. In particular regarding 2BL, "with in particular Mebane (2013a, n.d.) claiming that the second digits of precinct vote counts may be produced by normal political processes such as strategic voting and mobilization that do not trace back to illegitimate human manipulation" . It's perfectly normal for peoples methodology to evolve and to acknowledge the flaws in one's previous work. Even in the article which you originally used (which was a 4 page commentary and not a refutation), Dr. Mebane criticized the usage of 2BL tests and said it needed more research and a foundational theory which it presently lacked. X0n10ox (talk) 20:37, 7 November 2020 (UTC)

Mebane has just published an analysis of the 2020 US Presidential elections and the claims going around that his methods prove electoral fraud favoring Biden in Milwaukee and elsewhere. . I suggest adding a paragraph at the end of the elections section along the following lines: "Deviations from Benford's Law distribution have been claimed to suggest fraud in the 2020 Presidential election. However, Mebane's 2020 analysis, found 'essentially no evidence that election frauds occurred' in the relevant datasets. " — Preceding unsigned comment added by 69.123.21.171 (talk) 15:58, 9 November 2020 (UTC)