Number 1 and Benford's Law - Numberphile

[@anrubio5]

This is why I think Benford's Law holds: I felt like intuitively this should make sense but I couldn't explain why until I drew it out. There is a higher spread between the next instance of the same leading number the higher you go, making it less likely that you will actually have that digit as a leading number. So for instance, if you start with 1 the next instance of that leading number is 10. There is a 9-digit spread. You start with 2. The next instance of that leading number is 20. There is an 18-digit spread. You start with 3. The next instance of that leading number is 30. There is a 27-digit spread. (Or 27 opportunities NOT to hit 3). AND SO ON... You start with 9. The next instance of that leading number is 90. There is an 81-digit spread. Same for the higher values: You start with 1999. The next instance of that leading number is 10,000. There is an 8001-digit spread. You start with 8999. The next instance of that leading number is 80,000. There is 71,001-digit spread. You're again, going to be less likely to get something starting with an "8" than something starting with a "1".

[@StackCanary]

Thanks, this is exactly what I needed to get it to click! For some reason I couldn't intuit any of the explanations in either the video or other comments. Pointing out the (general) spacing between instances of numbers with the same initial digit highlights the sequential nature of numbers, which is apparently what I wasn't thinking of. Another way to look at what you've pointed out is that (generally) to get to the next number with the same leading digit, you must multiply by the base. This means that the smallest number will result in the smallest distance between instances. Now I see why it's so obvious to some people. It seems very obvious from this perspective.

[@patrice1345]

is that due to some manipulation of concept? Since a spread from 1 to 10 primarily deal with 1 digit numbers, whereas 2 to 20 include both 1 digit and 2 digit numbers. Also 10 to 100 only include 2 digit spread, where as 90 to 900 deals more with 3 digit numbers. So can we infer that if we limit the distribution to be only 100-999, the Benford law would not be in effect?

[@bikashagarwal6263]

I guess, the other way to explain this is, that the 1st numeric no is 1 and then 2 and so no... this it self is biased, we tend to start things with 1. Where as if we count backward, the reverse will be true for 9 :) say 9999, 9998,.... the probability of 9 will be highest at 1st and then it will gradually reduce :)

[@bikashagarwal6263]

Or in other words i should say the probabilty to happen somthing (not expexted / strange) in life, for the 1st time will be higher then to happen the same things for the 9th time :)

[@Kaczankuku]

A Ronse Do you think that digit keys of computer keyboards need to repairing due to proportionally usage according to Benford's law?

[@chack321]

This video is about to be a lot more popular.

[@nickniehaus1763]

Lol queue the “here before 1 billion views” comment

[@mccutcheonpe]

Yes, yes it is. Some people know.

[@josephcollier1028]

That’s what brought me here

[@matthiaskufner2283]

So true!

[@Exorine]

dont get too cocky star fox! it wont take too long before papa google and friends start scrubbing the whole internet of any damning arguments...

[@TheWhiteWhale593]

What’s the standard deviation and at what point does a deviation become a mathematical impossibility? Asking for my ballot counting supervisor

[@jasonransdell7055]

“Asking for my ballot counting supervisor.” 🤣🤡🌎 🐸🐸🐸

[@mickeydee3595]

😂😅🤣😂😅🤣😂

[@mosesolsonmd4063]

5.5 standard deviation above the mean for voter turn out in Wisconsin.

[@Anonymous______________]

My understanding the median voter turnout rates float around 63% with the standard deviation of 7.5... oddly enough some of the districts Biden won are five standard deviations ahead of the mean which is statistically impossible.

[@mosesolsonmd4063]

@@Anonymous______________ according to Wikipedia "6σ event corresponds to a chance of about two parts per billion. For illustration, if events are taken to occur daily, this would correspond to an event expected every 1.4 million years"

[@carlarosso46]

Latif Nasser's docuseries Connected brought me here. I was utterly mindblown.

[@ilariaromeo9386]

Same!

[@AnatheStrange]

Same here! I’m in utter shock

[@saadiyousuf9861]

Me too

[@titanagario9166]

Sameeee

[@GalacticGaymer]

Same! Shook.

[@QuotePilgrim]

In binary, the probability of the leading digit being a 1 is 100%.

[@cnoize314]

+QuotePilgrim Yep, and log2(1+1/1) = log2(2) = 1 !

[@euanmcnicholas534]

Can you think of a binary number that doesn't begin with a 1 0010 is the same as 10

[@hctcboy]

+KingHades Na55 Use sig figs Everything here assumes we're not using numbers that begin in 0

[@vaibhav2k13]

+QuotePilgrim What about 0?

[@euanmcnicholas534]

+MK Hammer I realise this aha. I was just explaining why :)

[@dwaltrip77]

Another way of looking at it: Each of the numerals only have an equal shot at being the leading digit if the distribution stops RIGHT before the next power of 10. Some examples of this would be if the distribution was from 1 to 9, or from 1 to 99, or 1 to 999, etc. If there is any extra "spare change" added to the size of the distribution, such as 1 to 99 being increased so that it now goes from 1 to 135, then that means there are 35 extra spots in the distribution that start with 1 (i.e. 100, 101, 102... 134, 135). These new options now make the number 1 a much more likely choice to be the leading digit. Distributions never end nicely right before the next order of magnitude, as the point right between orders of magnitudes is just an arbitrary spot on the number line [1]. This means there is almost always some "spare change". As 1 is the first number, it is the most likely to be part of the "spare change". When we go from one order of magnitude to the next, we go from the 9's to the 1's (99 --> 100, 999 --> 1000, etc). So, if the distribution crosses multiple orders of magnitudes, 1 always has the best shot at being the leading digit. 2 is next in line after we leave the 1's -- 199 goes to 200, 1999 goes to 2000, etc. Thus, 2 has the 2nd best odds of being the leading digit. And then of course the same logic applies for 3, all the way down to 9 being the least likely. This gives us the shape of the graph in the video! [1] If we switch from base 10 to some other base, the points on the number line that mark the crossover from one order of magnitude to the next will all switch! But the number line itself hasn't changed -- we are simply relabeling the positions.

[@dcisme5594]

This. Thank you.

[@Andronamus]

Don't know if I'm the only one who noticed this, but at 0:20, the "L" in "Law" is actually the number 1. Very cheeky :P

[@shadow81818]

I happened to have paused the video just at that spot to let it load, so I saw that. Very cool!

[@katakana1]

I 1ike it that way.

[@Tubecraft1]

Very sharp of you to see that

[@zodfox9321]

The last explanation seemed most intuitive to me.

[@alexhenderson3364]

Benford's Law for numbers. Zipf's Law for words. Patterns are EVERYWHERE.

[@NoConsequenc3]

+Bari Tenor Or they're nowhere and are instead part of the human mind's ability to comprehend things :^)

[@alexhenderson3364]

So we impose our own internal reality on the world around us??? Nice.

[@NoConsequenc3]

Bari Tenor Something like that. We take in information with our very limited senses, and our brain makes sense of it with very limited tools. It would follow that we do not experience reality, but rather that we experience our attempt at understanding information. IDK maybe I'm just bored, forget it

[@huntrofyou]

+Bari Tenor They're not really caused from the same thing.

[@alexhenderson3364]

hunterofyou Oh, of course. I understand that. I just like the fact that there seem to be similar-looking patterns for both areas.

[@luizbobio137]

2020 election will sure introduce this law for a lot of persons.

[@techpriestalex8730]

That's why I'm here

[@GhostFrank11]

Well that’s why I’m here

[@anamarte9859]

Wait so how do I apply this?

[@techpriestalex8730]

@@anamarte9859 you use log (n+1÷n) to get the number, I with my basically understanding if there is ever a number that is high then the total number of 1 then there is something wrong.

[@israr8622]

Same here

[@Sphere723]

I make sense of it by thinking about lining up an infinite set of rulers end-to-end, running my finger over the top, stopping after some random amount of time and then adding up all the 1,2,3's etc. I've passed over. Obviously it all depends on where you stopped on the final ruler, but on that last ruler you've most likely passed 1, and least likely to have passed 9. Data sets which span a bunch of orders of magnitude and are not normally distributed are kinda like this; the numbers are finite and have to stop somewhere. But you have to think of each "ruler" as containing ~10x more data than the one before it (1-9 vs 10-99) . So the "final ruler" you stop on has great influence on the count as it has the most room for data. Within this "final ruler" you'll have a bunch of 1's but few 9's as the 1's with "fill up" first as we assume the data to be relatively smooth and finite. If the 9's where to "fill up" it would mean data spilling over in to the 1's of the next ruler and that becoming the "final ruler". So the 9's never really catch up.

[@clockworkkirlia7475]

Ohhh this is a really clever and intuitive way of looking at it! Thanks!

[@weirdyoda04]

I remember watching this video years ago and now fate has brought me back for a refresher.

[@jhlapp9840]

Same, but I came back for the comments.

[@ragestarfish]

I got a soft spot for scale invariance so I really enjoyed this. One of my favourite numberphile videos.

[@AlenobaLP]

Honestly this comment is 8 years old and I have no clue if you still use this account but omg that just sounds so cute. "I got a soft spot for scale invariance". ☺️🥰😊

[@NoriMori1992]

Steve has a lovely voice and accent.

[@Debbie52S]

Steve is lovely

[@General12th]

You're lovely!

[@meowmeowmoogabenrules4854]

J.J. Shank we are all lovely including you :)

[@pbx7257]

Who’s here after the 2020 Presidential Election? 😂😅

[@veronicasweet4444]

Me, trying to figure this out.

[@WilliamFBogey]

These fuckers need to be dealt with

[@bennypinaula1231]

Me.

[@1234SLUR]

@@WilliamFBogey cope

[@zanvesta2072]

@@1234SLUR biden shills on youtube too x)

[@luminus3d]

7:45 THANK YOU for this explanation! It makes so much sense when you think about it that way

[@numberphile]

great - glad to have helped.

[@DiogenesTheCynic.]

heads up, you about to get more popular :)

[@5gonza541]

To me this was one of the most incredible videos on this channel, I really had no idea about this law. Amazing!

[@darthnick77]

This video is awesome! You explained Benford's law clearly in less than 10 minutes while some Netflix episode can't do it in 45 minutes.

[@2JustJoin]

Souldn't the log scale pattern start at the top in a W type pattern instaed of M, as the first measurement would be one digit of 1, thus 100%, the next being two digits of which 1 is represented by 50%. Not argueing the formula or the maths, just wondering about the log chart.

[@whitechocolatehopkins3903]

Yeah, I noticed that too. Even starting with the raffle example, it should begin at 100%, then decrease towards the 10% range.

[@soumyadatta4514]

Yes, I also thought it would be a saw tooth wave, but mirror image of what Steve have show. I mean sharply increasing then falling with a decay constant, not the opposite. Because the moment you reach an integer of 10, probability rapidly increases and then continue to fall slowly upto next rapid increment.

[@Ofordgabings]

I'd say Benford's law is intuitive. 1 is the default number, it's easiest to reach. It gets harder and harder to reach later numbers, so I'd expect a consistent downward curve, which is what the video shows. Fun stuff.

[@dalek1099]

+Ofordgabings Yes lets suppose that for a distribution the probability of x decreases as x gets larger this is true for several distributions(at least above a certain value) eg.think of countries it is harder to get to 90 million people than 10 million people or 900 million people as opposed to 100 million and thus the figures with the 1 in front always get the largest probability and then 2, then 3.

[@Syberi388]

suddenly relevant again would love to see the analytics for this video some day

[@IoEstasCedonta]

For everyone who's here to push conspiracy theories, skip to 2:30. The counts that "violate" Benford's law come from updates of similar size, so they shouldn't be expected to follow it.

[@patrickderp1044]

why do trumps counts of similar sizes follow benfords law? why do third party counts of similar sizes follow benfords law. the only odd one out, is joe biden, in specific areas, in specific swing states.

[@IoEstasCedonta]

@@patrickderp1044 They don't, really, not the way the nationwide counts do. They're closer to it because they're more spread out, since the wall they come up against is 0 rather than the district population.

[@patrickderp1044]

@@IoEstasCedonta this is accounting for the zero wall. trump has no aberrations in specific counties, third parties have no aberrations in those specific counties. it is only joe biden and the "pause" counties that found extra votes that were totally phony baloney and had no time to fill out downballot hence the house almost flipping back

[@IoEstasCedonta]

@@patrickderp1044 "This is accounting for the zero wall" - that's just it. Benford's law *depends* on the zero wall, to some extent, but in counties where Biden is popular, broken down into districts of similar size, they're bounded by the district sizes instead, so they'll be much more uniform.

[@IoEstasCedonta]

@@patrickderp1044 Anyway, Matt's posted a video on this, where he basically says what I said better than I could. You should watch it.

[@constantinople999]

So happy that I am the owner of this piece of paper! A nice piece of numberphile history from yet another great video

[@branllyr240]

Huh. I remember watching this years ago. Funny how some things come around again. I love this timeline!

[@Slithy]

Wow. This guy is so awesome! I wish i could draw such a large graph with such accuracy. This is so neat.

[@peternguyen1899]

When I fudge numbers on my tax returns, I always keep this theory in mind, and start it mostly with a '1'.

[@MeltedMask]

Keep the greed at bay and nine away.

[@keithrobertson2619]

Something tells me this video is gonna get A bunch more views here in the next couple days lol.

[@x50nathan]

And then get taken down for misinformation 🤣

[@justanormalyoutubeuser3868]

Matt Parker has already debunked that, check his video

[@kiabaha7271]

@3:12 Look at the example article. Do you believe now?

[@007bistromath]

I loved this video when it came out. It better stick around now.

[@cameron7886]

For the election data gathered by county, you have to realize that most counties are cut up into roughly equally populated zones. So immediately that's problematic for Benford's law, because you need a very wide distribution of numbers, and counties are PURPOSEFULLY made to be roughly equal in population size. The election returns for Biden by county weren't orders of magnitudes different from each other, they're within 100-1000. A wider distribution is REQUIRED to trigger Benford's Law. Keep in mind, the law only works, because "leading 1s" return every time you hit a new order of magnitude. It was not triggered for Biden because his performance was consistently within a single order of magnitude. Since this condition is not satisfied, you have to look at the last digit of the county returns, which should be evenly distributed 1-9, and indeed they are for Biden. Donald Trump's returns do follow Benford's law, because his range of performance was larger by county-10s-1000s-spanning two orders of magnitude. This is why his results DO satisfy Benford's law. EDIT: I said counties when I meant districts

[@cameron7886]

@Hyakkaten Trump's victory would defy logic. Luckily he didn't win, so we get to keep logic

[@jpr1011]

I'd love to see a video on uncomputable numbers. You could do a series starting with the integers, then moving on to rationals, reals, complexes, and computables/incomputables

[@basicallyeveryone]

What about KZbin video views and subcriber counts? Will it work or will we detect fraud?

[@magustrigger9195]

You could check by looking at your subscribed list, it shows the channels and sub counts, or go to say pwediepie videos, as he has a massive collection spanning many years and check views or likes or what not easily

[@davidbeckham5579]

Make Benfords Law Great Again!

[@Shutupasecond]

Did you ever forsee this video being an integral part of the 2020 US election?! Haha

[@s108745]

I wanted to see this for myself, so I did this in Excel. In column A, I put all numbers (1 to whatever) and for every number I looked what the first digit was. At 1, I only found one number starting with 1. At 2, I found one number starting with 1 and one starting with 2, and so on. For every number I calculated the percentage (also counting the previous percentages). I did this for the numbers 1 to 9,999. I got a beautiful graph with in the end the following percentages: 1: 24,1547% 2: 18,3273% 3: 14,5466% 4: 11,7363% 5: 9,4973% 6: 7,6356% 7: 6,0420% 8: 4,6489% 9: 3,4112%

[@angusc2042]

Hi Koen, I was just wondering how you made the graph as I am needing to make this for a maths project I am doing.

[@krowa1010]

?? it doesnt work like that man- with a normal distribution like the one you mentioned the probalability of starting from any of the above numbers is equal, so i dont know how you got such a solution

[@s108745]

@@krowa1010 I'm gonna be honest with you. This is a very old comment. I vaguely remember it. I can't really remember the video either. I'm not sure if I calculated something and my explanation is just bad or that I made a mistake. But reading this now, I can only say that you are right. There's clearly not a 25% chance that a random number starts with a 1. Maybe I'll watch the video again. If I find out what I actually meant, I'll let you know. Otherwise, we'll speak eachother in 5 years. ;)

[@TheJohnnyCalifornia]

radiolab's "numbers" episode mentioned this and it also had a segment where it seems that humans are born thinking "logarithmically" and later learn the conventional "counting" method of numbers which could explain why some people think that it would be natural for the digit "1" to be first more often.

[@ineednochannelyoutube5384]

People indeed seem to think in exponents. Thats the only way to grasp existing differences in the scale of the world.

[@Parmenza]

Stand-up Maths just put out a great video explaining this in relation to the 2020 election results

[@Cythil]

The last description is something I have noticed so I knew that 1 tend to hang around a lot more then 9. Simply put: Everything starts with 1 so 1 should be more common. But good video that gives several different perspective on this phenomena.

[@justicesportsman6020]

Seeing this video for a second time makes me feel old. 7 years ago didn't feel that log ago, the video quality says differently.

[@ambro987]

This video will get censored or demonitized

[@CSUChicoInvestorsclub]

I love when you relate your cool math tips to finance! It makes it so much more fun to learn! Tax advice, NASDAQ example, you should do more math/finance stuff! :)

[@Peepholecircus]

@3:12 lol , I know it, you know it, we all know it! right? you know why you're here XD

[@eoghan.5003]

The way that it intuitively made sense to me as soon as he said the law was as follows: As we go up through the numbers we could stop at any point (this is the highest value in our set). No matter what the highest value is, the probability of any value below it starting with 1 can never be beneath 11%. However in most cases it will be above 11%, because we'll have gone through all the 1s but not all the other numbers.

[@crustyferret8956]

I love how this video feels like it’s been shot the morning after a wedding and everyone else in the hotel is still in bed

[@mfundroid1116]

US Election 2020 has forced me to learn a Maths Law that I've never heard of and to be honest, I don't mind. The next two months are gonna be lit.

[@aaronphillips402]

We the party of science and math now. Biden's numbers are breaking this law hard.

[@oldmanjenkins9230]

Yeah but its not random most mail in votes are for him

[@_itzomar1952]

@@oldmanjenkins9230 most, not 100 percent or 90 percent. Conservatives also voted through ballots

[@csarmii]

Cause they have no reason to follow it. Maybe you should understand why the law works as it does first, THEN try to apply it. It only works on data where the numbers are of several ranges of magnitude and are distributed randomly.

[@oldmanjenkins9230]

@@csarmii thank yoi

[@atirix9459]

@@csarmii Yet, the population of different counties follow it quite well, right?

[@ravenasana]

Thanks for explaining this. Saw this idea on Connected but felt unsatisfied with their insinuation that this was some kind of proof of numerical "fate" ... this explanation makes much more sense to me... or at least I think it does 😅😂

[@mattc5937]

Agreed. I saw this on Connected as well and was completely fascinated by the "fate" aspect they presented. I now know they went that route purely for entertainment purposes as this video more clearly explains Benford's law in a much shorter video length. Interestingly the popularity of this video has sky rocketed after our recent US election from people who do not fully understand Benford's law and are using it as "proof" that Trump actually won the US election. Ughhhh!

[@1994ramfan]

Great video! I love how you explained it. I was one of the dummy's like "whaaaatttt noooo wayyy, even distribution"

[@Jotto999]

I'm generally lousy with math, but I can proudly say this made intuitive sense to me. It's easy to imagine that going from a per capita PPP GDP of 10,000 to 20,000 should be vastly more difficult than going from 90,000 to 100,000, because of how much harder it is to double productivity at any level than squeezing out another 11% - which then leads to being back at that numerical wall of having to double, triple, X4-9 again. What's neat is that this effect persists with different base sizes.

[@DestroyerX61]

You can see him in 3:12

[@mdflonline]

We're about to have a civil war over this lol.

[@southafricanizationofsociety20]

“Benford’s Law, Right Flank! Standard Deviation, Left Flank! Charge!!!”

[@magicskyfairy69]

@TheKillSwitch they don't need to fight, cuz if we're being honest, you know the republicans aren't gonna do anything. Despite media claims of "right wing militias", it's the left that actually burns things to the ground and attacks people at random over ideology. A few exceptions on the right? Sure, a couple shooters here and there, but if we're talking about whole mobs burning down cities, that's 100% the left.

[@mdflonline]

@@magicskyfairy69 It might come down to people being attacked regularly in pockets for stating political views, which is no way to live together. We're not far off from that reality already.

[@magicskyfairy69]

@@mdflonline so the right will be quiet. and our kids will be propagandized to join the SJWs. then there will be no more "right". Given the trajectory, there will eventually be communists, and socialists will be the new right wing. The idea of being a libertarian or conservative will just fade away.

[@lambdacode1503]

Somehow it feels relevant now with the election and all but I can't really put my finger on the reason...

[@MisterFanwank]

Same, but I'm sure some deboonker will set me straight by saying I'm a cOnSpIrAcY tHeOrIsT.

[@LJCyrus1]

Benford's law is not useful for predicting election fraud. Benford's law deviations do not prove voter fraud by themselves, further evidence is required.

[@zoeherriot]

@@MisterFanwank well yes, because benford's law does not work well with election results. If you actually apply it to some of Trump's results, it shows fraud. Not because Trump committed fraud, but because the number of votes in various districts were not orders of magnitude different. There are papers on this demonstrating that Benfords law is about as useful as a toin coss for determining potential fraud in election results (i.e. not very).

[@elarsen7823]

What I learned from this video: "No one is nine meters tall." All joking aside, this is an incredibly interesting concept, and I'm glad I discovered the magic of Numberphile. Keep up the good work, guys!

[@thomasramquist645]

elarsen7823 Once this is all locked down by all the teams.... I guess you will ready to launch the next internet!!

[@hengshenglee9635]

2:36 does Benford's law still apply after applying transformations to a population? In the case of human heights, maybe we could take the exponential of a multiple of the numbers? Or any other transformation that could possibly spread the numbers out such that they are in different magnitudes

[@kcccc5682]

It is invariant under scaling (fun to try and prove this). But taking the exponential should make it work for human heights yes 😊

[@calansmith655]

For all of the people commenting about elections, see Matt Parker’s video on the exact topic

[@clockworkkirlia7475]

Yo, everyone scrolling past, do please comment on this to give it a signal boost.

[@robinsonrobinson7503]

@@clockworkkirlia7475 aight

[@aidentaylor5038]

@@clockworkkirlia7475 alright

[@kookykutter69]

@@clockworkkirlia7475alright

[@noincognito1903]

@@kookykutter69 alright

[@austinliu1043]

Does Benford's Law remain true for different number bases? If you took data that conformed to Benford's law in Base 10, and converted it to Base 7, or Base 9, would it still conform to the law? What about higher bases?

[@pmorse0000]

one is still the most frequent number (it is the lowest number so you're gonna hit that if anything, and lower chances up and up. they dont say that but that is best explanation)

[@tryingsmall]

It remains true for all bases. The key to understanding it is that the distribution must be independent of normalization. For example, if you multiply all the numbers in your data set by a constant then the distribution of leading digit probabilities must remain the same.

[@thenexusmaster8175]

1:38... I just want a clip of that

[@vinayseth1114]

7:45 I think the intuitive reasoning is: 1 is the 1st number that pops up after the digits are exhausted for a power of 10 (or any other base no.), so therefore it's going to have the highest probability of popping up. But then again, the counter-intuitive bit comes in given that you don't know how large the distribution is and where it ends.

[@geoffnet1]

This became intuitive the moment I pictured log graph paper. Suppose that each power of 10 is 1 cm high, and you collected a random sample of heights from 0 to n cm and graphed them. The distance from 1*10^m to 2*10^m is about 0.30103 cm (for any m), so it makes sense that there would be about 30.1% of the numbers landing in these sections. Another way to think of it is that 10^x for any x>0 will have a leading 1 if the decimal part of x is between .0 and .30103.

[@Mectojic]

Came back after the powers of 2 video, it's still hard to believe!

[@Rekko82]

Do powers of 2 obey the Benford's law? 1, 2,,4,8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192,16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608,16777216, 33554432, 671008864, 134217728, 268435456, 536870912...when will I get a number that starts with 9?

[@Danilego]

2:00 He made the pi symbol with his forehead

[@datanon3059]

Thanks for teaching me about a new cool math thing Biden. Tell your son I've got his latest bag of rocks ready.

[@ForensicAnalytics]

At 7:10 they talk about forensic accountants using Benford’s Law. I have recently made my Benford's Law Excel software available for free. You can find it by watching my "Forensic Analytics Second Edition, Benford's Law discussion and free Excel software" video on KZbin. A link to the Excel file and the data is given in the video’s Description. A search for “Benford's Law, Part 3 (Analysis in R, no packages), updated code June 2020” will also show how to run Benford’s Law in R, which is valuable for data sets with more than 1,048,576 rows.

[@terapode]

What interesting statement: "We usually don´t hang on with nines"... Great video, Brady.

[@rylandchase488]

Just to be clear for everyone who doesn't understand the law properly. This law only works when you data is over many different orders of magnitude. Because precincts are all relatively the same size, it does not apply.

[@doppelhelixes]

just compare previous years data. That will prove or disprove it. No need for guesswork when we can use facts

[@teresaradsick7314]

Can't you use the final totals instead of individual precincts?

[@spazmobot]

He's adorable.

[@Ultiminati]

Matt Parker made a video about this, about 2020 election and just watch it before checking out any comments below.

[@zylan4967]

That is why I am here as well. It is worrying how many people think it is election fraud based on this.

[@Ultiminati]

@@zylan4967 :/

[@AlexisGelinas]

I work for a broadband cable provider in the US, assisting customers via web chat. I checked the chat session times (in seconds) and sure enough, the distribution followed benfords law almost exactly. This makes a lot of sense because average session time is around 960 seconds. There's a lot more chats in the 1000-2000 second range than there are in the 900-1000 second range

[@chounoki]

At 6:25, how do you get the average of around 30%? What's the calculation behind this average?

[@sabaca304]

+Chounoki *takes note to first watch the whole video and then ask questions*

[@aCloudOfHaze]

SAVE YOUR COUNTRY. THIS IS NOT ACCEPTABLE. RIGGED2020

[@maestbobo]

Are you serious?

[@MisterFanwank]

@@maestbobo We're all serious. It's time for some Election Fraud Punishment.

[@LJCyrus1]

Benford's law deviations do not prove voter fraud by themselves, further evidence is required.

[@zoeherriot]

@@LJCyrus1 They are not even useful for determining fraud due to the way votes are distributed. It's not a relevant test for election results.

[@johnjackson9767]

Huh. Wonder why this could possibly be trending.

[@BarnibusMaximusMusic]

Fine i'll say it. Steve is very hot.

[@looney1023]

OMG Thanks I thought I was the only one...

[@chiajingheng3524]

@@looney1023 samee

[@annabago8621]

Whenever I find myself in a situation like this, I just start scrolling until somebody states the fact so I can like it.

[@portal9935]

loolllll

[@t.2067]

yes

[@subh1]

Simple explanation: Draw the logarithmic scale along an axis. Paint red the regions where the numbers start with 1. The red paint will occupy about 30% of the line. So, the assumption behind the underlying distribution from which the numbers are being drawn is that it should be an uniform distribution over the logarithmic scale. Thus, if f(x) is the actual probability density, we should have f(x) = 1/x. [∵ if y=ln(x), F(y)=1 is distribution over y. Relationship: \int F(y) dy = \int f(x) dx]

[@f1f1s]

My favourite host, dark, relaxed and charming, along with Matt Parker.

[@pencilpen786]

Is this similar to Zipf's law?

[@_vicary]

The election in 2020 may take way longer than before.

[@hamoshytube1853]

@SmoovCat biden was confirmed winner

[@hamoshytube1853]

@SmoovCat literally every time the media says the winner they get it correct

[@BasedBowlCutEnjoyer]

@@hamoshytube1853 George Bush vs John Kerry ring a bell? You're a literal NPC.

[@hamoshytube1853]

@@BasedBowlCutEnjoyer sorry bro i was born on 2006 and i am not american so i dont know what happened in america before 2006

[@conormcmullin4235]

@@hamoshytube1853 then dont make false claims

[@Dray.TheChosen1]

who's here after watching connected on Netflix? 😂😅

[@kissmjae]

me! hahaha🤣

[@ilariaromeo9386]

Me!

[@the_sleepy_engineer]

this explained it so much better and took away all the mystery lol

[@saadiyousuf9861]

Me.

[@AreelKarunungan]

MEEE

[@DaveScottAggie]

As of August, 2020 - the NASDAQ is about 11 thousand. This video has 1306 comments and 11 thousand likes (160 dislikes). The video had 656 thousand views. And Numberphile has 3.4 million subscribers.

[@shedvortex]

I wonder if this could relate to number of corners in in "prime" geometries like platonic solids or even prime numbers. Would be a facinating tie in to geometric music language and fibonnaci sequences.

[@currently7886]

It does work for Fibonacci sequences, I just did the first 100 numbers of Fibonacci and it was tracking pretty close the entire time. I'm sure if I continued it would also be aligned with this Law

[@RhettReisman]

Who's here after Biden Wisconsin results?

[@antbooboo5884]

Not to mention Michigan, Georgia and Pennsylvania.

[@MossadAgentEpstein]

@@antbooboo5884 They say Nevada too as well as NC

[@gotals1]

all swing states....

[@MossadAgentEpstein]

@@gotals1 all states trump was leading in and over night changed

[@rcfiwqx]

@@MossadAgentEpstein Some of them even changed at 4 am in the morning at the exact same time after announcing they they stopped count at 12 am. These ballots man, they just count themselves apparently.

[@douggief1367]

I love how his beard matches his jacket.

[@BillKilmerslayer]

To me, this was always intuitive. You can't have 2 without 1. You can't have 3 without 2, without 1, etc. There will always be more starting units than the ones that follow. All sequences must end, and they will end at an arbitrary point. But they all start with the start, here being 1. Always the same start, but with differing endings, regardless of Base. More of the starting point than any individual terminus.

[@diegouzeda2491]

+Bill Kilmer Yes. This is like saying the leadoff (#1) batter always get more plate appearances during the course of the year...It´s intuitive

[@LakierosJordy]

+Bill Kilmer Thanks. Your comment helped me wrap my head around it. It makes so much more sense now.

[@TheHuesSciTech]

+Bill Kilmer If every country had a new birth every X seconds, where X was a *fixed* number for each country (contrived example I know, but stick with me), Benford's law would *not* hold, even though it's true there that 1 comes before 2, which comes before 3, etc, etc. Benford's law only appears when things travel along exponential trends (i.e., as the population of a country increases, its birthrate increases, so it spends *more time* getting from 1 million to 2 million that it takes to get from 8 million to 9 million, so of course if you look at all the countries in the world, more of them are in the 1xxxx to 2xxxxx phase than the others; it takes a long time to *double* your population). Put another way, your attempted explanation fails to explain why a full 30.1% starts with the digit 1; that number comes from log(2) - log(1); logarithms or exponentials have to be in a proper explanation somewhere.

[@BillKilmerslayer]

Wasn't attempting a proper explanation. Just an (over) simplified, and as stated, intuitive understanding. The above video, and plenty of other articles, give the proper explanation.

[@exoice3582]

+Bill Kilmer well said. wrapped up all the confusion in a couple of sentences.

[@erictaylor5462]

3:00 What if you take people's height in some unit BASED on meters, but something like microns or Angstroms?

[@sjwimmel]

+Eric Taylor That wouldn't work, unfortunately. The problem isn't the absolute size people's height, it's that they are relatively close together. If you would plot people's heights the distances of those points to the x axis would be similar, regardless of the scale of the y-axis. And because the distances are similar, their leading digits would also be similar in any unit of measurement. What might work (though I'm not sure) is to take the differences of people's heights to the mean. That distribution would center around 0 meters, so 0.1m would be more common than 0.2m. Edit: That last part is not true, human height is approximately a normal distribution, which don't follow Benford's Law.

[@jacobb5625]

+Eric Taylor that would just scale it down, there's no change in actual variance of numbers because you're scaling it by a factor/multiple of its own base

[@JacobReynoldson]

Mouldy old video ;-). Glad to see both these tubers improve their delivery over the years

[@marianna2933]

I’m gonna need him to explain the election results now lol

[@justanormalyoutubeuser3868]

Matt Parker has already debunked this fraud claim, check out his video

[@marianna2933]

@@justanormalyoutubeuser3868 thanks, I’ll check it out

[@yokedpredator2127]

So, strictly hypothetically, if there was say an election of some kind, and voter fraud is being disputed, this could accurately tell if voter fraud took place in several states. Strictly hypothetical by the way

[@csarmii]

It couldn't.

[@magustrigger9195]

You could if you took it by county more than by state. More integers to work with so the slope would be less jagged than 50 states

[@grandexandi]

OMG I got even more impressive results! I wanted to put this to the test, so I went around asking everyone I knew what year they were born, and, shockingly, 100% of the answers started with a 1!! I mean, wow! What could be causing that?

[@Bbonno]

I knew of this, and for the first 7 minutes I was wondering how it would work in other bases... and then you answered it. Nice!

[@NeelSandellISAWESOME]

At 2:23 what does he mean by "meta distributions"

[@shaolincheck347]

tfw your vote batches cluster around five

[@CorrectCrusader]

totally not fraud. nothing to see here.

[@currently7886]

Biden right now "WHY MICHIGAN & WISCONSIN!? WHY?!"

[@branllyr240]

Smooth brain on the fixer team, rip democrats.

[@juggernaut316]

oops

[@berserkasaurusrex4233]

Biden skipped two during the debate. He went straight from "Point one" to "and point three". Maybe that's why his fake ballots are throwing off the distribution.

[@alhfgsp]

I wonder how the patterns would look when not using a base-10 number system. consider hexadecimals.

[@pmorse0000]

i think they answer that. no matter what base, 1 is still the most frequent number

[@Epsilonsama]

1 is always 1 no matter what base it is.

[@Glockmog2007]

Ladies and gentlemen, we got him.

[@juggernaut316]

cuff him

[@DaDunge]

The only thing you've got is crappy listening skills the results need to span many orders of magnitude the us election results per voting district span between 100-1000, one order or magnitude which means Benford's law is useless.

[@airblastgoron]

The beautiful thing is that the formula covers that too: the probability for a binary number starting with one is log2(1+1/1) = log2(1+1)=log2(2)=1

[@starwarsjk99]

I find this quite intuitive. Take a number at random, with equal possibility distribution. We let this number be proportional to the magnitude of some unknown quantity. We would need to raise some base to the power of this number to get the exact value of the quantity. When you look at any log scale, the distance between 1X10^n and 2X10^n is always greater than the distance between 9X10^n and 10X10^n, (or 1X10^n+1). Just looking at a log scale makes it apparent that with a truly random magnitude, the value would have a leading one.

[@starwarsjk99]

I wonder why this was discovered so recently. This seems more fundamental than other topics such as calculus. This should have been discovered during the days of logarithms. It is so obvious.

[@thesixthsola]

Should be called Biden’s Law.

[@overbored617]

ohhhhh now I get it..... the smallest digit are the leading digit

[@brissywink]

@@overbored617 nah just fraud.

[@MetalisForever666]

No, he violoates that law. He should go to jail for that.

[@daniell.6463]

Biden's Law - halt voting and add 100k

[@gabrielsyme4180]

Biden doesn’t follow laws

[@trade_iv]

Hmmmm wonder what Detroit and Milwaukee’s voting charts look like

[@khjgfxdhzf]

7:14 huh. . .

[@TheiLame]

the 5:57 interests me, does it mean it really works for different counting systems? i kinda wanna see proof of that :)

[@ja-vishaara]

Then why don't you go and try in hexadecimal, I'll try base three.

[@alexroberts8755]

It wouldn't work in binary, because all numbers begin with a 1 in binary (so technically it would, but it wouldn't be very interesting)

[@breathless792]

Alex Roberts actually technically it does: log(base 2) (1+ 1/1) = 1 (meaning it's probability is certain!

[@ceulgai2817]

breathless792 Actually, that proves that binary is an exception.

[@ceulgai2817]

breathless792 Actually, never mind. I just realized what I'd said. You're right.

[@davidwilkie9551]

Time is prime.., because the connection of information has probability one, infinity/infinity = *, so all identity is some selection of characteristic proportion in reciprocal proportion, which-when the selection process is exponential, leads to an intersection of natural logarithm (including every and all numerical bases in "e", continuous). An actual Mathematician could deduce the probability of the natural occurrence of identities in other number bases, but I'm only committed to comments. Professor John D Barrow has presented a very impressive lecture on this topic. _____ In a manner of (QM-Time mathematical Intuition) speaking.., if this aspect of here-now is the metastable tip of the topological iceberg, the incident cause-effect of e-Pi rationalisation i-reflection modulation, then the natural probability occurrence of potential possibilities in the Universal Quantum Computation is a continuous expression of the number proportions perceived as Benford's Law...