Vi and Sal Talk About the Mysteries of Benford's Law

[tmn2727]

pleaaaaase Sal don't bring anyone next time

[tmn2727]

the baddest thing that u did is

[getsuga97]

You were thinking of Luxembourg.

[ceenavanderaa]

Sealand is the country with the smallest population.

[Gyroglle]

Montenegro is a country.

[ceneezer]

Hey guys, I wondered what would happen if this was shifted to another base, for example binary (hmm 100% on 1, how odd :D) and found an amazing thing when shifted to octal ... on 1000 iterations of 2^n 1,2 and 4 had equal (33.3333% odds) of being the first digit - not a single 3,5,6 or 7 I suppose this is the "why" you wanted us to ponder.... though what it means I'm not sure ... maybe counting with our thumbs was a bad idea?

[IHeartViHart]

...At this point it is increasing ten times faster than it was originally, but has exactly ten times as much ground to cover between 10 and 20 as it did between 1 and 2, so it will spend the same time there, then the same amount of time between 20 and 30 as between 2 and 3 etc.

That should give you a good idea of the scale invariance problem, which is what most seem to struggle with.

I advise using Wolfram|Alpha to graph e^(-x) from x = 0 to 5, then (separately) from 5 to 10. ..Look familiar? ;p

[IHeartViHart]

The logarithmic relationship should be true for every base, but the graphs would look different.

The relationship applies to numbers that are generated by most natural systems, and as Desrathedemon states, it's all about exponentiation.

Say a number is increasing a rate which is proportional to its own magnitude.

Starting at one, it will increase slowly, then increase double as fast at 2, then double as fast again at 4 etc until it reaches ten.... (read on in next comment)

[MrAlienAssassin]

Would this still work in a non base-10 numeral system?

[Desrathedemon]

It must have to do with the fact that population, finances, powers of two and Fibonacci numbers grow exponentially. If you look at population, it has a certain doubling time T. The time it takes the population to go from 1 * 10^n to 2 * 10^n will be T. But over a length of time T, the population will go from 2 * 10^n to 4 * 10^n. So the population will have spent the same amount of time having either a 2 or a 3 as a first digit than it did having 1 as a first digit.

[mentox]

I have a similar idea how to explain it for random things like population that counts on it that it's easier to reach over 9xxx then it is for 19xxx because that number is already atleast 2x bigger. and then you'd have to redo the entire 9xxx again before it becomes a 2

[mentox]

everytime you reach a new most significant digit (the 1 here) you have to go through all the numbers you just went through to reach a new most significant number. and since all of those sequences grow faster and faster the 1 stays in front longer then the following numbers

[Lazo00100001]

Sal, Montenegro is a independent country sinse like 2006 :) so you were not wrong.

[NNOTM]

It's incredible if you do it in the binary system: a physical constant ALWAYS begins with the digit 1!!! (Assuming there is no physical constant 0)

[Crystal2193]

OH MY GOD SAL AND VIHART

My two most favorite educational video makers!!!!! AAAAA! *dies a happy geek death*

[andreirocks1992]

6:54 challenge accepted.

1 2 3 4 5 6 7 8 9

7089 4148 2942 2283 1865 1578 1363 1207 1076

[Assimsimchannel]

It is all a question of problability, how certain are you of getting the nunber when you start anything 1. 100%

Whem someone is making any nunber even there is always more tolerance if the number you are reaching for is starts with 1. Lets say the fire man makes an estimate there is 94 thousand people he will problably say there was 100 thousand.

[richmondmaths]

but the digit 1 gets in there first because we give it, literally, primacy in our numbering system. If our cutoff point for a number set is on average at the 50, or 500 or 5,000 level, the ones will already have made their mark and likewise, the 9's and 8's will appear stunted. It's just an unfair system.

[richmondmaths]

It strikes me that the pattern is at least observable when taking limited sets of numbers: Between 1 and 9, there is 1 first digit which is one. However, between 1 and 20, there are 10 first digits which are one. Between 1 and 30, there are still only 10, and so on...

[DeluxeWarPlaya]

I've just died and gone to heaven. My two favourite math videographers in one video, in one place at one time, discussing one idea.

[RogueIrish36]

These represent growth phenomenon. At the extremes a leading 9 represents a very short dwell time in the order of magnitude (growth rate exp(x) no matter what x is will spend very little time in the 9##### range and quickly advance to the 1###### range). This is less technical way of stating below. I work with Zach Galifianakis' cousin and I agree Benford looks like Zach.

[sjwimmel]

I know why Benford's Law works for powers of 'mixing it up'-numbers:

The number line is linear while f(x)=2^x grows faster and faster. This means that every time f(x) arrives at a new decimal (10, 100, 1000, etc.) it will grow slower from 10 to 20 and faster from 20 to 30. The line keeps getting steeper.

In other words, it stays in the 10..20 region longer than in the 20..30 region. This makes the chance of a 1 as a first number more likely.

also: Newcomb looks a bit like Zach Galifianakis.

[stegeo3]

Vi Hart and Sal Khan are both wrestler-sounding names.

[VanMedia]

khan you should use cam studio to record and edit your videos because you can zoom in :D. you could have zoomed in on the graph :D

[sef3gen]

I think it's because numbers themselves are just agreed constructs ie 1 equals 1 because we all agree it does. However, 1 is the first "opportunity" to be assigned a position, 2 is the next "opportunity" to be assigned a position, and once a counted item is assigned a position, it does not hold multiple positions. I wonder if this law holds true with quantum data.

[MrJohnnyd1982]

I agree with Zander....all numbers are derivative of 1. It would work with any base, but it any case, 1 would be the most popular and would give the 1/x graph effect

[rotflmaopmpqxyz]

Would this still work if we were in a different base?

[UltimateRandomN3ss]

It's so weird to hear Vi's voice when it's not at 1000 words per minute

[filmmetoo]

Mr Salman Khan, please add Vi Hart into your full time faculty at Khan Academy.

[Buggy793]

Oh my gosh. Two of the most awesome people on the internet are referencing another awesome math teacher(singing banana). My life is nearly complete.

I have yet to ride a unicorn.

[Zander101084]

I personally think this law makes perfect sense. Correct me if I'm wrong but I think the reason that this happens in real world numbers and not randomly produced numbers, it hints to the fact that the numbers all start from one point and all the random factors kind of allow a certain amount of parts go forth from the standard deviation. I think looking at the first digit of populations is a great analogy. It's obviously more likely to have that smaller amount come up first. idk...

[supmywaxter]

Duke Nukem reference? I think I'm in love -3. Oh and to the previous comment, the Vatican is, in fact, a Country. (Simply Google it, I usually post a link when I call bullshit on someone but you can't on YouTube).

[edwardmeade]

You call that a beard? He's an amateur next to Anthony Trollope: Victorian novelist, champion of the mailbox and first-class beard-grower. Check out his photo on Wikipedia.

[radiator0505]

Do videos with PatrickJMT!

[SimaanFreeloader]

Montenegro is a country. But you were thinking of Monaco. And yes, the Vatican is the smallest (internationally recognized) sovereign state.

[mowmoh]

nvm if it works with prime numbers cant be the human mind, lol

[mowmoh]

PS: Though I believe if you make a chart of "product prices" of all the shops in the world, you will find more 9s than, for example, 2s. That's just customer psychology as well. Shops like to make their prices smaller. And they do especially heart 9s.

[mowmoh]

Sales Rep: "Oh man, I just have to get some more sales, once I'm over 10k sales this month, I'll get a bonus."

[mowmoh]

town council: "in our village live 10k ppl, once we are over that figure we should try to increasing housing opportunities in the neighbor village so that all the villages in our area increase at a similar level"

just a thought.

greetings from germany :)

[valdas0]

She always sounds so stoned...

[danix30001]

Godwin's law is even more mysterious

[Jackson882]

Populations do start with 1..... whenever you're counting things you always start with 1.

[Coerciveutopian]

The vatican is not a country. But Luxembourg is. 

[Ibahz]

Dr. Khan, montenegro is a country.

[drfoxcourt]

Populations, Universal constants, and the math phenom mentioned have a logarithmic distribution.

[SzlampStudios]

As you approach a higher digit, the variablility increases?

[DennisWagenaar]

I prefer the lectures of Sal alone, it seems like you are listening in into somebody else's conversation. Normally Sal's videos feel like he's talking to me but now it feels like he's not talking to me and this draws my attention away from the video. Maybe Vi can be good alone though and maybe I just need to get used to it.

If however they would represent different views or with some kind of other structure it might be better. Now its like having to jump to different thought lines all the time.

[jamesallen74]

You guys have mathematical conversations?  Aweome. :)

[FreemanFighter94]

These "conversations" are pretty neato, but how do they fit in with an actual curriculum?

[ohd34484]

@FreemanFighter94 Do they need to fit into curriculum?

[Axelisten]

Actually, Montenegro is a country.

[KingPtolemyIII]

@Axelisten He was talking about Monaco. Good video BTW.

[TheAustinConlon]

You know what would be awesome? Dr. Drew Pinsky and Sal talking about biology.

[rinwhr]

Guys, Sal is married and has a son lol.

[Skywire1337]

@rinwhr He has a daughter too! :) I'm glad he has time for us !

[griftorifto2]

Wow thats pretty amazing.

[ro33han]

ooooh, sal has your wife seen this?

[killie232]

Montenegro is a country. It has been a country since 2006, I believe.

[Tweetling]

Marry please.

[Raikenbolai]

I wonder if this can be used to detect corrupted statistics. Or would statistics typically be corrupted in accordance with Benfords law?

[PeterCollingridge]

@Raikenbolai I believe Benford's law has been used to demonstrate that some financial data was made-up (although I don't know of any specific cases). When people make up numbers they tend to try to pick numbers starting with an even distribution of digits, so it can be spotted quite easily.

[Raikenbolai]

@PeterCollingridge Groovy. Sounds like a useful tool to remember. At least as a simple flag indicate further investigation is required.

[nafaylodhi]

VI-SAL....this just keeps getting better

[saenmb]

I can't express how much I love the combination of Vi and Sal

[TheGluall]

snakesnakesnakesnakesnake

[artukmi]

montenegro is a country!!

[cinereyes]

i wonder if his office stil his closet cuz then the first minute be quite funny

[MoGaDeX]

That's Duke Nukem's ancestor...

[PaperSpock]

My first approach would be to see how this law acts differently in different bases. In binary, all nonzero numbers would obviously start with 1. I wonder if in base 3, things are skewed even more towards 1 than they are in decimal, slightly more skewed in octal, and less skewed in hexidecimal. in fact, I wonder if, for number systems in base x, if the distribution becomes more even as x approaches infinity.

[tchanabc]

VI HART <3

[jamesjosephclarke]

YAY!!! Vi Hart!!!

[hahs4]

@jamesjosephclarke  whos vi hart?

[jamesjosephclarke]

@hahs4 Just take a search for Vi Hart on youtube, you'll find some of her awesome maths vids in the first results.

[petedeutscher]

My guess would be that it has something to do with Logarithms. Look at Log Paper and the amount of "space" taken up by numbers where Log (x) starts with a 1 is higher in proportion to those that start with 2, and so on. From memory, I'd say that the relative areas would roughly (or if my hunch is correct, exactly) match up with Benford's law.

The million dollar question then becomes *why* and that's a little harder to answer!

[PCGamerPortal]

I think the result can be inferred from modular arithmetic, but I was never too good at modular arithmetic so I will pass out on this one.

[thecubegeek]

See singingbanana's explanation on the same topic.

[werdnativ]

@jsiooa Right, it's a country but I believe he was thinking of Monaco.

[arsenalfooty4]

vi hart!

[OniLinkPlus]

I checked Benford's Law with the sizes of files on my computer (yay Linux terminal). It matched with Benford's Law nearly exactly.

[superdau]

In a little different way this law also holds true for the second digit in numbers.

This law is so consistent that it can be used to help finding cases of tax fraud (at least I heard that). If someone fudges the numbers in a tax return they wont match this distribution anymore. Of course the one faking the numbers can check this, too. So if you don't wont to rise some eyebrows you're limited to certain numbers.

[TheMoses8870]

This is the video from sal that i can't watch.

[AlderDragon]

Sal, since this works for the power of any number, does it also work for any randomly selected set of numbers?

[PoketoMtg]

When a number is beginning at a 8 or 9 it is closely approaching a new 10^n place. Because of that the higher numbers will be jumped over to 1's and 2's

[lilshortypie72]

oh my god

vi hart united with sal khan talk about your educational fandoms crossing

[cosmosgato]

Because you count up.

You are more likely to stay at 1, or 100, then move to 2 (200).

You more likely to stay 2 than move to 3

...

[PCGamerPortal]

@cosmosgato except human populations grow madly and without control

[Dundrio4]

@cosmosgato I second this. Suppose a person starts counting. Once he is is at 10, he will have spent twice as much as time on the digit 1, than any other, only at 19 this will be even. Therefore the graph is skewed like this. However a single person could randomlybe measured at a number like 9,234 and it will be even until he is at 10,000, so it will most likely not be near benford's measurements until you got a large number of people counting.

[Dundrio4]

@Dundrio4 Sorry, at 90 this will be even.

[Yakushii]

Ahhh, I love real world mathematics!

[moguai352]

i felt like this was very hard to listen to.

[Paulorific]

This is amazing. I love everything about this.

[CougarJargon]

whoooo!

[Zlibservacratican]

Kahn and Vi, sittin in a tree...

[xXMathMagicianXx]

@Zlibservacratican Isn't Khan already married?

[Zlibservacratican]

@xXMathMagicianXx Its a joke.

[Zlibservacratican]

Awesome! I love Vi's vids!