tag:blogger.com,1999:blog-8666091.post6367158048881422834..comments2020-03-13T03:28:45.250+01:00Comments on The Reference Frame: First digit is most likely one: Benford's law is no mysteryLuboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-8666091.post-72648342132057544172011-04-02T15:12:15.644+02:002011-04-02T15:12:15.644+02:00No, I'm not disagreeing necessarily.
What I...No, I'm not disagreeing necessarily. <br /><br />What I mean is, suppose I have a circular scale, as on the dial of a meter, calibrated to the log of the distance on that scale, from 1 to 10. <br /><br />Then the probability of finding a number on that scale from 1 to 2 is 0.30 of that distance, from 1 to 3 is 0.48 of that distance, etc. <br /><br />If this "dial" was used together with an instrument to measure the brightness of a star, say by extinction correlated with intensity of light, then (it seems to me that) not all brightness measurements made with this instrument would have equal probability of being made (even though the brightnesses or distances themselves would be random or equally probable). <br /><br />So I guess this would be a systematic introduction of a bias related to this particular physical method of measurement itself. Hope this isn't too obtuse.Brian G Valentinehttps://www.blogger.com/profile/01523059818774910427noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-64868670684175268662011-04-02T08:25:24.801+02:002011-04-02T08:25:24.801+02:00Dear Brian, the purpose of your comment is complet...Dear Brian, the purpose of your comment is completely impenetrable to me. Do you actually disagree with something we're saying? If you do, it is not clear what it is.<br /><br />The magnitude is defined as a log of the distance - and what? The distance is random and the magnitude is random, too - because it's the same information.<br /><br />The leading digit of the distance does follow Benford's law because the possible values span a large number of orders of magnitude. The magnitude of the star doesn't obey Benford's law exactly because the magnitude is defined as a log in order to make all the possible values of the magnitude "comparable", so most stars are peaked "somewhere" on the log scale.<br /><br />Cheers<br />LMLuboš Motlhttps://www.blogger.com/profile/17487263983247488359noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-42963868741212948152011-04-02T01:36:21.358+02:002011-04-02T01:36:21.358+02:00"Their distances from earth would be 'ran..."Their distances from earth would be 'random'." <br /><br />Um, not quite. Their distances have been calculated using their brightness, which is a log intensity scale. Numbers appearing on that scale (from 1 to 10) would be proportional to the log of the distanceBrian G Valentinehttps://www.blogger.com/profile/01523059818774910427noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-2620637393722005992011-04-01T15:27:45.071+02:002011-04-01T15:27:45.071+02:00Hi Brian. Here is a list of the 300 brightest star...Hi Brian. Here is a list of the 300 brightest stars. Their distances from earth would be 'random'. Simply count the leading '1's in the distance column "dist ly", and you will arrive at approximately 30%. Or easier, count the leading '9's and arrive at about 5%.<br /><br />http://tinyurl.com/brightstarsBemused in Torontohttps://www.blogger.com/profile/06181929060448185152noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-89213714198477111812011-04-01T15:27:01.773+02:002011-04-01T15:27:01.773+02:00Hi Brian. Here is a list of the 300 brightest star...Hi Brian. Here is a list of the 300 brightest stars. Their distances from earth would be 'random'. Simply count the leading '1's in the distance column "dist ly", and you will arrive at approximately 30%. Or easier, count the leading '9's and arrive at about 5%.<br /><br />http://tinyurl.com/brightstarsBemused in Torontohttps://www.blogger.com/profile/06181929060448185152noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-61408099341835308912011-04-01T11:41:06.340+02:002011-04-01T11:41:06.340+02:00Dear Brian, you're wrong. Benford's law as...Dear Brian, you're wrong. Benford's law assumes and implies no other patterns because no other patterns exist, and your claim that the average digit is 4.5 is just wrong is one considers ensembles of digits where the first digits constitute a sizable fraction of the digits.<br /><br />Just try to list the current prices of 1000 stocks and make the statistics how many of them start with 1, 2... or 9, you will see that Benford's law will hold and smaller digits such as 1 are much more frequent than larger ones such as 9.<br /><br />Cheers<br />LMLuboš Motlhttps://www.blogger.com/profile/17487263983247488359noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-75723753628872145632011-04-01T04:22:55.790+02:002011-04-01T04:22:55.790+02:00Sorry, the "Benford law" thing is conten...Sorry, the "Benford law" thing is contentless, because it implies no other, nor makes use of no other, features of a "random" sequence of integers 1 through 9, such as the average of a sum of sequence of integers approaching 4.5 (the average of the inegers 0 through 9). <br /><br />Meaning one could take the word "random" right out of it, and still have the same result if there were no a priori mention of a method of determining the terms of the sequence of integersBrian G Valentinehttps://www.blogger.com/profile/01523059818774910427noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-36952465345389176612011-03-31T20:28:59.430+02:002011-03-31T20:28:59.430+02:00Thanks very much Lubos.
I found in my testing, th...Thanks very much Lubos.<br /><br />I found in my testing, that any product R1 x R2 x R3 say, with R = rnd * 100 say, Benfords pretty nicely. Your suggestion R = e ^ (rnd * 50) does as well, as does (any number) ^ (rnd * even a small number) because they are essentially self-multiplication products I guess.<br /><br />Next time I run into Steve, I'll pass on your hello. He was limping a bit a few days ago, no doubt from his recent matches :-)<br /><br />Thanks again! cheers!Bemused in Torontohttps://www.blogger.com/profile/06181929060448185152noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-81002069331798484592011-03-31T19:31:29.049+02:002011-03-31T19:31:29.049+02:00Dear Doug, the world is a small place! I am Steve&...Dear Doug, the world is a small place! I am Steve's fan, too - not only the Steve as a bright independent researcher but maybe also as a squash player. Squash is hard for me. ;-)<br /><br />Please, send him my best regards if you can. In person, that's quite touching.<br /><br />One needs to be careful to produce Benford-distributed numbers "artificially". Clearly, if you study numbers exp(X) where X is a random integer chosen uniformly from a large enough interval, e.g. 0-50, then exp(X) will satisfy Benford's law.<br /><br />Your webpage about the problem is nicely written and the domain name is impressively relevant, indeed.<br /><br />All the best<br />LubosLuboš Motlhttps://www.blogger.com/profile/17487263983247488359noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-23009202241093647142011-03-31T17:52:58.855+02:002011-03-31T17:52:58.855+02:00I read your blog regularly. Thanks for it. I'l...I read your blog regularly. Thanks for it. I'll say hello to Steve McIntyre for you (Climate Audit). He lives just down the street, and I know he's a fan of yours.<br /><br />I'm not a mathematician nor academic.<br /><br />A few weeks ago I decided to sink my teeth into Benford's Law because I was always slightly disturbed that random numbers generated by a software generator did not Benford, whereas 'natural' datasets of random things did. So I tried to explain the differences between the two kinds of random.<br /><br />I don't know how successful I was, but it sure removed any mystery I once felt for the subject. I wrote it up here www.benfords-law.com.<br /><br />There are a lot of 'ifs' here. If you actually read comments to old posts, and if you like the writeup, and if you think there might be some merit to publishing a link to it (from I don't know where, leave it to you), I'd very much appreciate it. I provide a link to your blog.<br /><br />Thanks for the interesting posts and cheers!<br /><br /><br />Doug BennionBemused in Torontohttps://www.blogger.com/profile/06181929060448185152noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-44449268598547045832010-05-08T04:14:07.694+02:002010-05-08T04:14:07.694+02:00A mathematician colleague of mine
many years ago
e...A mathematician colleague of mine<br />many years ago<br />expressed puzzlement over the predominance of 1's<br />as the leading digit of randomly chosen numbers.<br /><br />My explanation to him was:<br />"Look at the face of a slide rule!"<br /><br />Of course they don't make slide rules any more<br />so I guess that argument no longer applies.Allanhttps://www.blogger.com/profile/04188695437268298621noreply@blogger.com