Why do Biden's votes not follow Benford's Law?
ฝัง
- เผยแพร่เมื่อ 9 พ.ย. 2020
- My book is cheap at Waterstones and signed at Maths Gear:
www.waterstones.com/book/humb...
mathsgear.co.uk/products/humb...
Check out Steve Mould's Numberphile video about Benford's Law.
• Number 1 and Benford's...
Buy a signed copy of "How Many Socks Make a Pair?" by Rob Eastaway.
mathsgear.co.uk/products/copy...
There’s more on Mark Nigrini’s work here:
www.nigrini.com/benfords-law/
"Benford's Law and the Detection of Election Fraud" 2011 paper.
www.cambridge.org/core/journa...
And for balance, here is a paper critical of that other paper (but only in the use of a 'second digit' check and they do not dispute the main Benford's Law claims.). pdfs.semanticscholar.org/e667...
And here is a paper by the same author specifically about the 2020 US election results:
www-personal.umich.edu/~wmeban...
Get your Chicago Board of Election Commissioners data here!
chicagoelections.gov/en/elect...
Yep, 2069 precincts. Some would say that's too many.
data.cityofchicago.org/Facili...
If you must, here are links to people using Benford's Law to suggest the Biden votes were fraudulent. Please do no harass or brigade anyone.
github.com/cjph8914/2020_benf...
jonsnewplace.wordpress.com/20...
CORRECTIONS
- Hello loyal viewer. If you are reading this you most likely regularly watch my videos and know that I put corrections here. But the comment section on this video has been, to put it lightly, "wild". I don't think anyone is checking the corrections here! So I'm going to break with tradition and put the corrections in a pinned comment. But in short:
- I should have said I used the Chicago data (instead of a swing state, let's say) because that is what people claiming election fraud were using. I didn't pick it myself to make a point.
- Foolishly I cut a bit of the video where I talk about how Trump's data is also a bad Benford fit but that massive spike of 1s makes it look like a good match. Check out how low 3, 4 and 5 are.
- There has been specific criticism of aspects of that paper I read from, but only the usual back-and-forth of academics. Everyone agrees with the idea that Benford is not a magic tool to detect election fraud (nor is any statistical tool really; they all require careful interpretation).
- As always, let me know if you spot any other mistakes.
Thanks to my Patreon supporters who mean I can spend TWO DAYS trawling through election stats and making plots. I'm meant to be writing a new book you know. So, thanks a lot.
/ standupmaths
As always: thanks to Jane Street who support my channel. They're amazing.
www.janestreet.com/
Filming and editing by Matt Parker
Music by Howard Carter
Design by Simon Wright and Adam Robinson
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
US book: www.penguinrandomhouse.com/bo...
UK book: mathsgear.co.uk/collections/b... - บันเทิง
Not to get political, but what the hell is a number?
They're some kind of Arabic invention.
@@chazk7530 um actually the number zero is an arabic invention you would now this if you didn't have some stupid L I B E R A L education /s
@@chazk7530 oh I was thinking it was some type of antibiotic
It's racist.
@@SquidwardTentacles225 number sounds more like an anesthetic.
If you write the numbers in binary, apparently almost all the numbers start with a 1
But the law is for base 10, not base 2, dear!
Only 0 in binary doesn’t start with a 1. This is irrelevant for the decimal world
More specifically 100% of numbers start with 1 in binary
@@KhoaNguyen-fs6to benfords law works for all bases.
@@matthewhubka6350
No. There's still zero.
Also I suppose you could right any number with a leading trail of zeros.
Im sure the comments will be all perfectly reasonable and coherent discussion on the complete video.
Phrhbfnxlxir bkdkzuxtzvwn bald man doodoo
Lmao sans from undertale number talk
Especially right now everyone got their anger out at the capitol!
Hahaha!
im joe balden, and i approve this message
Fellow data geek here, this was a TEXTBOOK example of how an analyst approaches their work. Bravo, well done!
'Some More News'. He makes the best
Biden-Roasts.
@@slevinchannel7589 you made it political 😐
@@wizzotizzo No,
i didnt. I literally just said something about Biden-Coverage.
See for yourself: the channel i named literally covers BOTH THE GOOD AND THE BAD.
Yes, i said Both as in 'how unbiased news should work'.
@@slevinchannel7589 you made it political 😐
@@conception3509 you're watching a video on this topic gtfo
Let's look at the comments section to see what the experts think
The Dunning-Kruger Effect is a hell of a thing. All the gullible conspiracy theorists who haven't taken a math class in a decade or more are suddenly mathematicians, just like how they're also epidemiologists and economists who clearly know "so much more" than people with degrees in those fields.
😂
@@kane2742 .... Laws don't apply to the left. 😂
You think this guy is an expert? If he were a true unbiased mathematician then he would be arguing that Trumps distribution should follow the same pattern if his theory on the precincts was correct.
@@jajajajajaja867 being unbiased doesn't mean that he agrees with your narrative lol
The way I see it, these things are like metal detectors. They're great at finding points of interest, but you have to start digging to see if it's a coin or a bottlecap.
Perfect analogy
@JRPGFan20000 I was gonna go with unexploded bombs, but sure, I guess a gun kind of works too. XD
I think this is a fair assessment. I don’t see why people are against recounting the election
Yes. It's a red flag detector. A red flag comes up and further investigation must be done.
@JRPGFan20000 or two pretty best friends
I love how you pointed out the importance of context in interpreting data! It's so often overlooked.
Honestly that was the most interesting part about this video
This. So many times i interact with people who don't account for context and just say,"the number don't lie". Of course numbers don't lie, but people can and you have to know the context behind the numbers
@@maxe159 "Senor Joe, the numbers don't lie, and they spell disaster for you at Sacrifice!"
- Scott Steiner
@范德萨阿斯顿发大水发大水发阿斯顿发大水发大水发范德萨我和你吻别我爱你他妈的翔宇我和你吻别元的钱破开该 it's certainly a good question to ask but it's also important to mention specifics. His logic seemed sound to me and if you just say "he could be wrong" without pointing to anything specific, it doesn't hold any weight.
I'd say rather than overlooked, it's often swept under the rug to push an agenda.
1:58 Let's not get distracted that there are 1000π counties in the US
That’s wack
Mmmmmm, a thousand counties with pie.
@@cjwrench07 Gotta catch em all.
I hate how close to reality this statement is.
1kπ
They need to start bringing out maths experts on election coverage, its not like they don't have huge amounts of time.
Mathematicians don't have time for stupid elections
@@poorman-trending nah, more like most people watching the news won’t care abt a mathematician and just arguing over nothing somehow brings in better ratings and views
@@poorman-trending Exactly, and I would guess a mathematician wants nothing to do with politicians for this reason--or maybe they would because they want to show them what the truth actually is?
@@maxwellsequation4887 Perhaps they should, we need them voting.
@@maxwellsequation4887 Pretty sure they do. It just the general public is less likely to listen to a mathematician than a celeb or political figure, thus lower rating for news organization.
Getting some insight behind the votes from a mathematician is refreshing.
Even better then it comes from a Standup Mathematician!
@Stephen Thacker how so? I live in Michigan and a lot of us are flabbergasted.
@Stephen Thacker supply a mathematical proof then
@Stephen Thacker I need proof plz
@Stephen Thacker can you explain further please?
These kinds of misunderstandings are, I think, a subset of a larger problem of people getting 'evidence' confused with 'indicators.' One is often the other, but not necessarily so. The indicator should cause you to look closer, but if you look closer and find no evidence you shouldn't continue to tout the indicator.
Tbf in this case I think the people touting this were just dishonest from the start.
Evidence does not equal proof
@@hedgehog3180 depends. I personally think they were just desperate. I thought from the beginning that it would be close if trump won by any margin in 2020, but fewer in my view felt the same in 2016 leading to many not accepting that election for vert much rhe same reason trumpers didn't want to accept 2020.
It's like many had selective amnesia when it comes to rhe outcomes they wanted.
Yeah. Even The Colorado Supreme Court thinks it was a blowout. No need to interfere this time. [eyeroll]
@@edgunther8136"The absence of evidence is not the evidence of absence."
The takeaway: if you discover an anomoly, you actually have to investigate the source of said anomoly before you can accurately say you know its cause.
Studying the anomalies, rather than the commonalities, might produce some interesting insights.
We know some athletes are enormously better than average. We can ask what role does economic status play into their performance?
I like both W J and Jeff original comments !
The problem is that all places where anomalies are accused, seem to lock up and hide the data to substantiate their “certification.” Sort of like Pfizer asking for, what, 95 years or something before disclosing the study data on the vaccine? Quite equally, and scaringly similar when you think about it.. but you probably aren’t concerned or see the analogy ;)
If I understand what you're saying, I agree. For example, the anomaly in the Trump vote tallies compared to the Biden tallies may, I suspect, be explained in several ways, but most of them having to do with the programming of the voting machines, or how the results are calculated after they're input.
What algorithms could be used to modify (just assuming hypothetically, not claiming they were) the election results? Would they take into consideration the possibility that some of the data coming in are invalid? How would they handle for example, a massive input illegitimate votes for one candidate over the other? Would it try to compensate by manipulating the data for both candidates to resemble expected outcomes, according to what forensic analysts might expect? The idea being to hide the cheating enough to make it look legit.
I'd like to see an analysis like this this one across a larger sample, or better yet, across two larger samples, one of states that reported no "irregularities" compared with a second, of the collection of states that reported substantial irregularities. That would be interesting...
Exactly, another great example of this is his "Perfect Bridge Game" video where he explains away the anomaly of a 1 in 2.2 x10^27 event occurring not just once, but several times.
Thanks for mentioning my name and my work starting at 3:56 :) I did an analysis of the Maricopa County election results and got pretty much the same patterns. Here's an interesting tidbit... At 2:00 you talk about the populations of the 3,141 counties and Benford's Law. At 13:25 you talk about the digits in pi, .... and, of, course, the first four digits of pi are 3141 :) You went full circle or 2πr.
WOAH it’s the man himself
...autograph?
only a mathematician would notice this... nice one, man.
You've clearly done some seriously good maths work but, honestly, those jokes. Call me irrational, but I love it when a tangent turns into a punch-line.
@@ElevatedKustoms link?
County precinct groups are too uniform. Benford described the nature of numbers on a larger scale, so if you cherry pick a city block, or man-made precinct and chart the the leading digits, it will fail as bad any of the full state chart by county assessments of in the swing states for Biden counts. Or, blue state, Trump counts. The selective adherence to the law is suspect.
"AND ELECT ION DATA"
To be honest, if there were someone named "Ion Data" running for any position, I'd be very tempted to elect them based on the name alone.
This reminds me of that time an acquaintance called Justin Case blocked me from Facebook because I wouldn't stop unnecessarily tagging him in my comments.
Elect Ions are a bit of a charged subject.
ION DATA 2024
Science is right and the media isn’t, also humans suck
I would elect ion data if it weren't for all the negative energy around, everywhere in their campaign, up, down....strange....I'll get my coat...
I, for one, welcome our new, presumably robotic, overlord.
I really like that you compared side by side the digit pairs of pi with the last two digits of Biden votes. A very clever way to impartially show the expected variation at that sample size. Without that comparison, people surely would be looking for patterns in the noise, which as we know is a dangerous thing.
If PI is carried to millions of decimal places there are many of what would seem to be improbable strings of numbers such as "1234567890" or "1122334455667788" or "666666666666", but it always breaks out of these patterns.
Would have been hilarious if it actually lined up 100% because that would again mean someone tampered with the data lol.
@@insomnia20422 wym again
@@swinde yup, have 1 million monkeys randomly pressing keys on a type-writer for infinity and you get a quote from shakespear at some point
@@GAxelicI always thought that the saying referred to the Tale of Two Cities just because of that scene from the Simpsons. "It was the best of times, it was the...blurst of times?? You stupid monkey!"
Guy makes complex statistical analysis look like algebra for beginners
Because it really is.
@@davidz2690 Twit off ya spoon
@@davidz2690 Damn, I hope you aren't American. Therapy might help with the insecurities, but it's pretty expensive here. I'm glad you've found a coping method in the interrum.
@@davidz2690 Nice! I'm proud of you son
@@57thorns if it is than it’s not impressive at all that you can do it
Let's check if these data are random!
Statistician: I'll use chi squared test.
Matt Parker: I'll use it as an excuse to put pi in my video.
😂
“These data are”, “this datum is”
@@nelsblair2667 thanks, I have corrected.
69th like
chi_squared gives you one number to test for significance. Matt's idea (a great one) is to give us a picture of what random data actually looks like. I am very impressed. From mere observation, it looks to have the correct mean and s.d. _ a beautiful illustration.
This guy's agenda is nothing political: he's peddling his fantastic book!
He's British lol
@@tangyspy *Australian
@@tangyspy does his nationality affect my statement? I'm making a reference to how he mentions agendas in the video, and how this appears to be a protracted ad for his book 😂
Sure, that also means that, knowing the initial digits, no one has been able to distinguish it from a normal number. The non-randomness would have to be "further down".
Its a good book!
Compared to Bidens normal distribution, Trumps vote counts are best described by a Poisson distribution, which is a pretty sophistacated roundabout way of saying, that Trump just ain't popular in Chigago.
Lets take a look at crime statistics in Chicago while we are at it. Oh. Oh my.
@@CarpetFTW cope lol
@@CarpetFTW What's the argument you're trying to make? Cmon, spit it out, don't just dogwhistle.
@@kiiyll He's just pointing out the fact that statistically, Chicago has high crime. There wasn't an argument, there was a joke.
And if we look at election results from years prior, we'll see a trend emerging. Republicans just ain't popular in Chicago.
Thank You for a non biased look at this without going political.
@@mangonel One might say facts don’t care about your feelings
Also roll tide
@@mangonel sure doesn't feel that way when you are in a class discussing bayesian statistics...
Roll Tide Baby!!!!!
@@mangonel I am going to be an annoying pedant here and say that this is technically correct (the best kind of correct), but in reality people collect the data, choose how to analyze it, select the scale on graphs etc. so there is plenty of room for biases to sneak in
@@mangonel stats sure as hell isn't.
More people need to understand how statistics can mislead you, and how misleading people can make statistics lie to you
@@paperburn Ironically Bill Gates favourite book is on that subject.
@@diesel92kj1 "92% of all statistics are made up on the spot!"
It's a standard government/business/PR tactic: present the statistics' results but never reveal how those statistics were derived. As a simple example look at washing-machine detergent adverts (or any other adverts based on provable results rather than aesthetics): they all claim they're the best, but what is 'the best', & under what conditions?
You can also move the goalposts by adjusting the size of the sample: '9 out of 10 cats prefer it' sounds great, until you realise that only 10 cats were used to test the food (& they were prolly preselected from certain specialised parameters anyway) :)
@@snafu2350 Interesting concept...I didn't know cats used washing machine detergent! LOL (I have to laugh at my own jokes otherwise there would be complete silence. It's a well known fact that 9 out of 10 comedians laugh at their own jokes... Is it just me or is thing going round and round in circles?)
@@snafu2350 It's something I enjoy about UK adverts - they do have some legal requirements about explaining where the result of the survey comes from, sometimes you even see text like "out of 87 people surveyed" when a shampoo is talking about being preferred. Radio ads are even better as you get someone trying to quickly rattle through all the legal disclaimers etc at the end, but they have to do it slow enough that you can still understand them. Still not a perfect system, and you wish the audience were better educated about sampling error etc, but it's nice to still see it
Just took a math stat midterm and one of the trick questions hinged on verifying that the data were random! Very relevant to this video.
Maybe you should revise, because data being random is only relevant when we’re sampling the population so we can make sure it is close to the population, but in this case of election we are looking at the entire population, meaning every single vote. We don’t sample the votes for the election, we count them.
@@anandrai492 yeah, but if the question is "how well does chicago's districts of roughly 100 to 1000 fit Benford's law" the answer is gonna be "not well". Gotta check your data before you try to fit them to something and draw conclusions.
(And the question on the exam was about predicting an election based on a survey and finding a rejection region such that alpha, the chance of a type 1 error, is less than some value bla bla - which only worked in the question if you make sure the survey was random and only then can you apply the CLT and estimate it with a Normal)
@@BenTzionZuckier I took stats and got a B
@@BenTzionZuckier or you can just focus on something else besides the raw count like second order or summation. www.researchgate.net/publication/319526944_Benford's_Law_The_Second-Order_and_Summation_Tests
Surprisingly educational video for what I thought was going to be mindless political clickbait.
the only thing i really learned was use a random number gen for filing false tax returns :)
I did one statistics and probability course in my third semester of engineering. All I have ever been doing now is watching statistic videos.
I hate this subject. But I love it. Help.
Lol so true about stats
Most annoying math to do but the most beautiful math to see
Mood
You and me both!
All my Stats professor ever talked about was gambling!
Same, but biology
"The moral of the story is that everyone has their own agenda they want to push on you.......check out my book Humble Pi!"
Awesome
I mean, at least his agenda is clear, and not harmful to the discussion
That’s why he didn’t do an analysis of Milwaukee.
@@melanieb8746 Whats up with Milwaukee?
What's wrong with advertising his own product in his own video? Is it really worse then all the VPN ads?
This is why statistics is a degree and profession, and not a topic.
That's true for so many current issues.
@Rye Bread lmao that was great
I'll just leave my disagreement here. Plenty of stuff you can discuss without a degree. Just because you might get it wrong doesn't mean you shouldn't discuss it
I believe that Mark Twain had an adage, “You have lies, damn lies and statistics.” Based on this it appears you can use several different number sets to argue whichever point you are trying to prove. I have found that looking at the process to outcome is ultimately the only way to actually prove a thing. But as he said, these are only used to determine if something needs to be investigated. It appears that, in the case of Chicago, that a closer look is needed.
Statistics is definitely a topic though?
I do love how you pointed out the unusualness of the data on both sides, presenting it as strange, then showing how it’s not.
If you had just done this for one side or the other, and left the undisclosed one up to viewer interpretation, it would have been biased, and not an “impartial step back”
I just realized that when you're looking at the last two digits of the Trump votes, you're just getting a Benford's Law distribution again! This makes sense (sort of) because of what Matt said about the precinct vote numbers.
'Some More News'. He makes the best
Biden-Roasts.
@@loturzelrestaurant Did you even reply to the right comment?
@@alienplatypus7712 its a bot
@@ImJustCj That makes sense, hard to tell bots apart from confused boomers sometimes.
@@alienplatypus7712 No boomer would watch "Some More News," but yeah.
But wait this doesn’t fit my biases
It also doesn't fit the facts.
@@pitapocketortwo buddy, you can’t get more factual than this mathematics youtube channel.
In that case reality must be wrong.
@@pitapocketortwo no, he means biases
@Joshua Jason Karl i do lol and this guy agrees with the other educated sources
There's an implicit narrative here that worth making explicit. When it comes to data analytics, the proper question to ask is: why do I have the data that I do? If you simply take your data and analyze it without considering how that data was generated (both collection methodology as well as the phenomenon you wish to understand), you will probably completely misrepresent the actual reality and fail to really understand why you had that data. I suspect this happens *a lot* in practice, especially when companies do data analytics for a myriad of reasons and often have less than stellar data collection methods, let alone failing to consider the real-world process responsible for the data and what, therefore, they should expect to see.
Justin, In Chicago's Graph, Trump's Benford curve shows significantly lower 3's and 4's. That looks like Democrats are THROWING away trump votes in the 300 and 400 and 500 count precincts THUS forcing the 1's in Trumps to be abnormally high.
Second, for Biden, those Blue Democrats are PADDING (adding illegal votes) in the 100's, 200"s precinct counts and making them into 300's, 400's and 500's and 600's. There are no examples in elections that show standard bell curve except Blue Democratic cities which have decades of high-level corruption outside of vote counts. Those cities are complete ghettos with decades of declining population.
@@foundingfathers4462 You didn't watch the video, did you?
@@ThisIsMego nope
@@foundingfathers4462 Ghettos with declining populations? Never been to a big city have you now? News flash! Some people hate Trump! In cities with over a couple hundred thousand people, it’s a different world than most red counties. I suggest you go to a big city with your Trump flag and conspiracy theories and see how many times you get cursed out.
@@ThisIsMego From the ignorant analysis in his comment, I don't think he'd understand the video even if he did watch it. A nice example of the Dunning-Kruger effect.
nothing like sorting youtube comments by recent to lose your faith in humanity
I feel called out
One of the most real comments on youtube.
Thankfully, most of them are just “interesting video, thanks!”
Would be worse though if those comments were found at the top instead.
It seems to go in spells - you get batches of people who watched and understood, then you get ignorant goons posting sequential word vomit as they get triggered by something being debunked, a few complete morons who beg for the ineffectual law to tested on other states (to prove what exactly has never been revealed...), then some people who would be sued if they tried repeating their baseless claims on TV...
Will this video teach people to double check their data and sources prior to spewing nonsense into the internet?
I can only dream.
I'm a retired forensic accountant. I'd be happy to explain forensic accounting to anyone who wants to know about it. I should warn you that there's a great deal of statistics involved and attention to detail is mandatory. As a general rule, the only people interested my explanations are other accountants. After a few minutes everyone else discovers that the topic is less fascinating than they thought.
Do some YT videos ;)
I’d be interested. Took accounting in college from a forensic accountant.
Hey man i was wondering what is a good place to study this on my own, this election got me really interested
I'm starting out in data and I would frankly love to chat about forensic accounting! I've done some reconciliation and QA and analysis, but none of it really involves intense stats and I'd like to do more than just make pivot tables in excel haha
Hey man, I'm actually interested in forensic accounting
I'm going to use this pinned comment to reply to the most-common questions/complaints. It'll be updated infrequently but I am trying to read everyone comments.
- WHY DID YOU ANALYSE DATA FROM CHICAGO NOT A SWING STATE?
Good question. It is because the first Benford-Biden theory I was sent was based on the Chicago precinct vote totals (there's a link way down in the video description). So I analysed the same data they did. I did not choose the Chicago dataset: people claiming election fraud did. You are very welcome to analyse data from elsewhere and show us your working.
- WHY DOES TRUMPS PLOT MATCH BENFORD'S LAW?
It does and it doesn't. I was going to say in the video that Trump's plot is also a bad fit for Benford: there are way too many 1s and then fewer digits from 3 up. That spike of 1s is because Trump got an overwhelming number of vote totals between 10 and 19 and so at a glance it looks like a good fit. I cut that bit though as the video was going to be as long as the election.
- SOME PEOPLE ARE CRITICAL OF THAT PAPER YOU QUOTED
In terms of legitimate criticism, you are probably thinking of Walter Mebane and I have linked to their papers in the video description. But they specifically disagreed with how 'Benford's Law and the Detection of Election Fraud' (2011) treats the 'second digit' check while still agreeing that "It is widely understood that the first digits of precinct vote counts are not useful for trying to diagnose election frauds."
- A STRING OF DISJOINT WORDS AND EMOJIS
I'm not sure how to constructively engage with that, but I'm glad you watched the video and wanted to get involved with the comments. I thought I did a pretty good job of being all-math-no-politics in the video but appreciate people are very passionate about these things. Other commenters: I encourage you to engage constructively wherever possible and 'down thumb' anything untoward.
- IS THAT TRAILBREAKER ON YOUR SHELF?
Yes it is. It is exactly what meets the eye.
That was an enlightening video, Matt. Love from Brazil.
You made math not boring. Congrats.
@@Robbya10 trump's votes had a less robust distribution than Bidens in Chicago though
@@Robbya10 no lol he just got like no votes in Chicago
I think it would have been more interesting to compare 2020 Biden in Chicago to 2016 Hillary in Chicago than 2020 Biden to 2020 Trump if you are going to use just one city instead of national data. Else, we are comparing apples to oranges. Good explanation otherwise on the precinct size rationale.
Love this. You can tell any story you want with data . Digging in an seeing more than 1 aspect of the data is where you start to be able to call out anomalies and ultimately see a holistic view . Well done !
Omg... I know this is an old vid and this comment will never been seen, BUT.... knowing Benford's law, the title piqued my interest. 🤔
Decided to watch and a Biden 2024 campaign ad preluded Matt's video. If I was a conspiracy theorist I'd have gone nuts!! 😂
I get Biden ads. I get Republican ads. I get ads from religious organizations. I get ads from pseudoscientific products. They should know I don't like most of it and they still do it. At some point, they need to just pick people to show stuff to.
Benford's is an acid test, it can be used as an indicator of places to look, it doesn't mean they are not explainable. For forensic accountants alot of the time the evidence is circumstancial and indirect.
this
Good way of putting it.
yea so if you see something unusual it should indicate to investigate, it is not proof in itself.
he literally says this a number of times in the video
The numbers can NOT lie as they are not human.
I saw a documentary on netflix about Benford's law. I was screaming at my screen when they kept on claiming it was kinda "magical" and no one knew how and why it worked.
lol but I guess it's true in a philosophical sense nobody knows why 1+1=2
@@nmarbletoe8210 maths is a game and those are what the rules say, so. The astonishing thing is the ways in which we can use maths to interpret the world
@@nmarbletoe8210 well, it's an axiomatic truth, the tools to prove it with logic aren't hard, just redundant
@@nmarbletoe8210 tachyos.org/godel/1+1=2.html this is literally the formal proof for why 1+1=2
N Marbletoe There is actually a mathematical explanation on why 1+1=2. But you need to go deep and use set theory for an explanation. 1+1=2 is a mathematical statement that can be proven. Even the existence of zero is actually proven under ZFC.
Damnit I did it. I looked down. Never look down.
My own favorite distribution for confirming human behavior is the Zipf distribution, which is what Google used to compute their guesstimated number of "hits" in their searches. Zipf has an advantage over Benford's law in that it is much more directly diagnostic to show what is actually happening in the real world.
I was thinking about Zipf the entire time because of how closely Benford's law's distribution looked like it! Vsauce made a great video on it and Pareto's Law, really makes you think how random randomness actually is.
"While I have you here" - I'm still here Matt, trapped. Please, release me.
😂
The cutest comment I've ever seen
12:15 "There was a spike at 82 because one employee was claiming their breakfast on their way to work every day, which they weren't allowed to do. You can only claim breakfast when you're on the road for work purposes."
*Employee: commuting is for work purposes.*
Someone needs to start their day at the home office then move to a main location on the company dime. (well $0.82) but you get the idea.
@@medleyshift1325 That muffin and drink cost more than $.82, the frequency of the .82 is what called attention to it.
@@JohnDobak it's a play on dime sorry for not being more clear.
For a while I always thought it was crazy how powers of 2 always seemed to start with a 1 when the number of digits goes up. Thought it was pretty cool that you would get “pseudo powers of two” since the lead digit often went 1,2,4. Then one day I realized that it literally HAS to start with a 1 every single time 🤦♂️
Yeah, but the fact that 2^10 is so close to a round base ten number 1000 is a nice coincidence, isn't it? The pattern basically starts at 256, 512, 1024, 2048... And then not so nice.
Oh, I understand now. This means, "The decimal representation of the smallest integer power of two for a given number of digits always starts with a 1." That took me a while.
@@absolutehuman951 4096 and 8192 are fine, but it gets REAL nasty after that
I love what you did with the trump data to show how easy it is for something to look suspicious without more context and why you need to listen to people who know about data rather than just trying to draw your own uninformed conclusions. Love the video!
Where have you been?
The only non-biased people we have left, in this field, disagree with this guy 100%
@@mr.j5919 who are those people?
@@mr.j5919 Please name one person. Literally one math expert who 100% disagrees with this guy.
@@mr.j5919 Name one and give us a source
@@mr.j5919 Where have you been?
How to make a Matt Parker video: explain an interesting math topic and find a way to throw pi in it
It would be irrational not to
funny thing, in the book "Humble Pi", pi only really appears in the title
I'm wondering why he didn't use tau instead.
In videos where it isn't featured, it still is, just as iπ
Gotta make it very clear to tauers he's in the pi camp
The one bit about the random data anomaly due to some employee's breakfast made me laugh out loud. Great video.
Trying to rip off an auditing company sounds like a great idea
@@Stargazer1312 Sounds like a good challenge for some.
You've done a great job here ... and it's really intriguing. Too bad most of the people who need to understand this will not bother.
Yeah it’s sad
Its sounds crazy to me that one person can go to a café for months and always pay the same amount down to the cents... Here we have an inflation of 3% for month...
In my country PS5 costs twice as much as PS4 did back in the day, so now it equals to a monthly wage, haha
@@midge_gender_solek3314 uff that sound rough buddy. I live in Argentina, here we have a 30% tax over the dolar (imposed last year), the PS5 is buyed in dollars so we have to pay (after all the other taxes) arround 70% more that the original price. So here its like 320.000 pesos, and the average salary (per month) its 32.000 so we need arround 10...
Its not a competition, this is awfull i am just complaining... Meanwile the vice president (Cristina Fernandez) gains 1 millon pesos per month...
@@Gaston-Melchiori Tengo un amigo Argentino, y te lo puedo confirmar. Siempre ha querido una Switch desde que salió, pero se tiene que conformar con su amada DS de diez años. Me contó que una Switch allí vale como... 300 000 y algo pesos? No estoy seguro, pero mucho. Y un juego vale como triple de lo que vale en España, que son sesenta euros, un juego triple A, que no es poco.
@@alvaronavarro4895 si lo se, yo soy Argentino y (tadavía) vivo en Argentina. No te sabría decir cuanto sale nada la verdad, en mi trabajo todas las semanas llegan aumentos de 5% y 10% de los proveedores, porque no basta con la inflación infernal, tambien hay que soportar a los empresarios que especulan y por si acaso te arrancan la cabeza con los precios.
@@Gaston-Melchiori yo tambien soy argentino pero vivo en Washington DC. Muy interesante encontrar otros argentinos en los videos de numberphile!! Entiendo el dolor de vivir en argentina en estos momentos, esta toda mi familia en cordoba. Ojala algo cambie pero mas probable que no. Mejor todavia seria poder traerlos a usa pero ya sabemos como es.
2:11 nobody asks “how is benford’s law?” 😔
:(
Our reporters got an exclusive interview with Benford's Law. It told us it's happy that people have gotten so interested in it over the past week, but asked us all could we please learn the conditions under which it does and doesn't apply?
+
This a solid, apparently impartial exploration of the topic. Would be nice if the media had as much respect for its audience as you do yours.
@Gideon U Settle down, "free thinker"
> Would be nice if the media had as much respect for its audience as you do yours.
Realistically, the general media is not going to go into a topic at the mathematical depth a math-focused TH-cam channel is going to. That's simply because the audiences are different.
@@henryptung If they're incapable of or otherwise unwilling to address certain topics they should stop speaking as an authority thereof.
Unfortunately for the media, "respect for its audience" doesn't really bring in the click-bait revenue like oversimplified sensationalism does.
@@Maus5000 lol trusts talking heads.
as soon as you brought up the "trump tower" graph i knew what was going on and laughed out loud. "it's because he didn't get above 2 digits in most precincts! he got 30/40/50's in most places!"
Guys stop commenting about wanting to see the comments then we won’t get to see the comments we really would like to
As someone who lives in America, I find this video very interesting! Thank you for creating!
As someone who doesn't live in America, I too find this video very interesting!
What I've learned from this video: don't have the same breakfast every day
And always roll a die when choosing.
@@nicothoe as a Gamemaster, dice aren't that good at being random either
@@ijemand5672 Why aren't they random enough?
What I haven't learned from this video, because I knew it already: Numbers in tax returns and accounts are not random.
@@ijemand5672 dice are very good at being random.. a 6 sided dice has a uniform distribution to all of its faces, it's pretty good at being random unless its loaded
What I’m taking away from this video is that if you want to fake a five-digit number, just change the third digit and nobody will catch you.
Very well
So what I've learned is that I should use an RNG when doing my taxes. Thanks!
this is the best comment
Tom Scott: "you can't trust me"
Matt Parker: *writes that down*
Weird Idea: they "plotted" it xD
TS: "But you can like me"
- Source: Madeup
I getting tired of seeing the us election banner ad from TH-cam
How do you turn it off?
Ditto. I wish there was an 'I get it; don't show this anymore' button
@@ajmoe YOU CAN'T. THEY NEED TO KEEP BRAINWASHING WEAK MINDS.
I don't see it. Did they stopped doing that or they are doing it now only for US?
@@elenabob4953 I've never seen it, must be US exclusive.
Matt Parker, giving someone the benefit of the doubt and not saying they're wronger than wrong.
My first instinct was that in Chicago, where you'd expect most precincts to be very pro-Biden, you'd expect the first-digits to be higher than 1... If you get 65% of the votes in a city relatively uniformly, any precinct with 500-1500 voters, would be unlikely to yield either 1 vote, 10-19 votes, or 100-199 votes, because the expectations starts above the 200-mark and ends before the 1000-mark.
Also it's ridiculous how low the vote count of Trump voters in Chicago are... If you had proportional representation, I'm sure Trump could find more than 60 person in each precinct who'd vote for him, but they obviously don't see the point in showing up in the state. It's a major dysfunction in American politics. We don't even know who would win with a national popular vote because SO many people don't have a reason to vote anyway.
13:20 Matt: i decided to compare it to the first 2069 digits
Me: there must be a spike at 69 in random numbers chosen by people
...or it's the same as the number of precincts in the Chicago data set.
@@DavidGuild For this case it is. But removing one case from the set doesn't invalidate Kalpit's hypothesis. I'd be surprised if there's NOT a spike there in human chosen numbers. I'd expect to see a few others in there as well.
@@arfyness Might you know any places where I can find an accurate survey of random numbers? I think analyzing it might be fun before I see what others have to say about it.
@@DavidGuild Or Chicago chose to have 2069 districts because it ends in 69.
@@DavidGuild They could have had 2070 precincts instead. But nooooooo, they had to pick a prime number...
As a retired stats prof (hopefully not Dannycode's), I wanted to thank you for clearly explaining your process and the underlying theories. I'm always (yes, still) looking for interesting examples of phenomena. to use.
As a retired Professor, what did you think of the the presenter in this video using the second digit distribution to justify the Benford result in question for Biden, but then using the Benford result for to justify the second digit distribution in question for Trump.
@@fomori2 most underrated comment so far. I'm an accountant, and elections are perfect use for it. I use it as a litmus test for finding irregularities in bulk data.
Here’s proper use of Benford’s Law for elections.
th-cam.com/video/1ald3w9FBmA/w-d-xo.html
@@fomori2 what?
@@JamesWolfpacker Did you learn nothing? Stop spreading misleading information as if you're not literally commenting on a video debunking the snake oil you're selling. Shoo now.
This came up on my feed, and I immediately thought, "You can't use Benford's Law for elections. That doesn't work." But I watched your video anyway, because I thought I'd figure out how to pick apart your math and assumptions.
I was surprised that this was an educational video, and not a propaganda one.
Thank you for a concise, and clearly-explained analysis. :D
I came here for the same reason when it randomly popped up on my feed. I thought this was going to be straight propaganda
It is propaganda; propaganda to use statistics correctly.
My sister got me Humble Pi for Christmas, and I had no idea it was you until I got to the end of this video!
Your sister is awesome
Benford's law gets real spicy in binary!
Indeed. Hence the assumption baked into binary floating-point numbers, that the first bit of the mantissa is always 1 (or else you have 0, which is indicated by a special exponent value); so you get a free space for the sign, since you do not need to store this first bit.
Yeah, since every number except 0 starts with a 1. But It could actually still be useful if you look at the 2nd and 3rd digits
@@kalebbruwer In floating point, you have to throw away one possible value for the exponent, because you can't represent zero otherwise. But you can use this special exponent to indicate other things than zero, such as short integers or more than one kind of "not a number".
@@bluerizlagirl In floating point, the sign and exponent are stored before the mantissa. I was talking about integers where you throw away leading zeroes. I know that's not a thing with computers, but I wasn't talking about computers.
@@kalebbruwer Floating point is basically the same principle as resistor colour codes. And even "zero ohm" links (in a resistor-like package, for machine handling) have a special marking with one black band.
Lessons learned from this video:
Use Benford's law to detect weird data results and then research why the results are the way they are to find out if anything is wrong.
Context people, context matters
Lefties don't want answers, they just want you to accept the results when they favor the regressive left and ignore all discrepencies
Agreed. Statistical models are not definitive last court of appeals. It is as you said, research why they are that way and this Law is still a good rule of thumb. What is good for the gander should be good for the goose, my question in this presentation is but why would Trump's numbers follow Benford's all things being equal it should behave like Biden's too.
BLM= Benfords Law Matters too
@@lpcruz5661 this question is literally answered in the video. Watch the video before commenting. Otherwise, you come across as simply willfully ignorant for asking questions that have already been addressed.
@@LoveJoyPeace4612 I have indeed watched it till the end. I believe my comment is fair. He explains why Biden' does not. Then you can explain to me why Trump follows Benford's? Did he include p-values? He quoted a 2011 paper, well there are recent researches on Benford Law, see this journals.plos.org/plosone/article?id=10.1371/journal.pone.0151235
I applaud you for making a video about politics... about maths.
I was home schooled for high school, and my curriculum was project-based. I didn’t have to take a math class my junior or senior year, but I had been planning a statistics project for if I did do some more math. I thought it would be really interesting to take a set of statistics, probably baseball stats, and then play with the numbers to create different narratives - make it look like one team was cheating, or a ref was being unfair, and then make it look like the opposite. I thought it would be a really great exercise in critical thinking: to get experience with how data can be misrepresented so that when I see those stats in the real world I can more easily understand how it could have been presented to change the narrative. It’s really a shame I never had the time for that project
You could do it now. It'd be fun. :)
Apologies for the necropost but I just wanted to say that the idea of a project-based curriculum sounds SO much better and more engaging (at least for me, lol) than the current most common school system.
As long as people investigate and aim for transparency that's fine by me. No single data point is going to be sufficient.
Scotus has ruled in favor Benford's Law several times including Enron case and it was way more tame then the 4+ deviations we are seeing in some the big cities in swing states
Hacker, you *did* watch this video, right...?
@@Kaoskadosk hope u know Trump's result may have actually been tampered with and that's y I wasn't random anymore... Please do a critical thinking of this, who cheats himself out of an election on a slim chance of winning in court?
@@cozmik_kay i agree, but the examples in the video are bad because no one is contesting Chicago's results.
@@lostalone9320 Yeah, except in this case it's regarding election data, which as this video points out, is pretty pointless to apply Benford's law to.
As an wise accountant once said:
"I only trust statistics I made up myself"
made up or made?
@@ajinasawor "made up" or shuffling your data around until it suits you.
@@Playerdelta08 an accountants job is just making stuff up.
I trust numbers, but the conclusion of those numbers is something that can differ. Take climate change, 415ppm co2 in the air is 0.04% of the atmosphere. It SOUNDS small, but is it really small? No, it covers perfectly in the holes of water vapor, and is actually quite a huge number relatively. A small number, that has a great impact.
It wasn't an accountant who said that, but a propaganda minister. A German. Some Joseph Goebbels, look him up or watch Der Untergang. And he said: write in the papers that Winston Churchill said this.
I clicked on this video just for the comments
God I did too and I regret it because I cant stop now
You should watch the video, its pretty good.
@@timothyjacksondrake4454 yeah it is.
So, in data of inconsistent magnitudes, Benford's law may be useful.
In data of similar magnitudes, the randomness of the last two digits may be more useful. Very interesting
Get a grip... 14:42
@@TheCommono Notice they said may, not is. Like in many of the tax based examples, using that check led to investigating an oddity with an easily discovered explanation. This shows that the method of checking the last two is more useful for similar magnitude numbers, which is a fairly obvious conclusion given the way the math works. In the predictable setting of Trump votes in Chicago (When we know that large cities trend blue), this kind of result should be a given. For unknown outcomes however, This method would be very useful for noticing unexpected oddities, and that is the point.
@@cyber33l I do not understand, why "more" - ?
Regardless of this data, the one's who don't want to verify and look into anomalies have something to hide.
You are referring to Trump's tax returns?
Right! Which is why I don't trust these "scientists" with their round earth theories. They're tired of me asking them to verify the data according to my needs, say it's a waste of money. I know they're got something to hide.
@@garsm2290 The IRS does that automatically, even more so for those with large finances. The fact that Trump was never charged EVER means his tax history and payments were all lawful. Despite it not being public what anyone owes or has paid in taxes, the IRS and the government does and concerning Trump's business of property and building development much of is in New York, you can't simply not report and you can't just tell the city your building doesn't exist so you don't have to pay that years property tax. I guess a bunch of naive college kids who own nothing and assume mostly everything about much of what they think is true wouldn't really understand this yet or bother to think it out.
@@nlsantiesteban Noel, come on. Your claims of the earth not being round is hardly comparable to either holding or not holding the most powerful office in the world for the next four years.
@@nlsantiesteban you mean the round earth that was investigated and debunked by people asking questions and then getting people to prove it?
Let's be honest: most people invoking Benford's Law don't understand Benford's Law. They are using it because it is an argument that supports what they want to believe (ie confirmation bias), even if the argument itself is flawed.
People do this all the time, including you if you are reading this comment. Very easy to spot when someone you disagree with does it, very hard to notice yourself doing it. We should all be a little bit more mentally disciplined about this kind of thing.
What are your thoughts on this? Chicago .....hmmmm! Democrat-run city, in a Democrat-run State, with election workers appointed by Democrat management....hmmmm. Oh, and isn't that one of the cities where Republican scrutineers were banned from being close enough to monitor the counting process, on Dominion vote counter machines (owned by Dianne Feinstein's husband's company) running Hammer and Score-card software controlled by the Democrat Deep State. Hmmmm, I think that might explain the graphical distortions in not following Benford's Law and the lop sided random distribution of POTUS Trump's vote count across the Chicago precincts. Just say'n, that is probably a statistically significant reason to commence a complete audit of the Illinois voting process. Not to mention the rest of the country.
@@jamescarney6894 My thoughts are that this is exactly what I was talking about. You don't want to believe that Donald Trump lost the election, so you bend over backwards trying to prove that 2+2=5. The only way you could believe in this insane conspiracy theory is motivated reasoning.
i disagree
but tbh joe is yo mama and if you know that, then that means i wouldn't have too say joe for you to even relate it too "joe", yo mama.
@darknightoftroy "2000 Mules", I guess you are one of them.
tl;dr:
Violations of Benford's law are an interesting starting point for audits, not a proof of malfeasance or errors.
The TL;DR is violations of Benford's law are meaningless when it comes to elections, so are not a starting point for investigations, even though there are data sets/industries like finance where it can be a reasonable trigger.
Weird that no dates start with 4 or 5
What I learned: Trump is not popular AT ALL in Chicago.
this is the perfect comment, it's funny if u don't get the maths and it's funny if u do too.
How popular is Lori Lightfoot?
@name name why don't you do it, then?
I love how objectively this video is made. No jumping to conclusions, no accusations, no unfounded claims, just mathematics :) and the fact you didn't just answer the question, but started digging deeper into things like the so-called Trump Tower, it really shows why you make such a good teacher!
If only other people think objectively like this
@Johnny Five check Maricopa county, that one has the same problem
@Johnny Five well you can do it yourself, it's really simple, download the data, put it into excel and take a look at it.
@@roycebutler8590
Is it possible the precinct populations there are equally as clumped into an order of magnitude as Chicago's is?
@@leongkinwai9709 I'd be shocked, it's a very red county, and it still wouldn't make sense because that's not really how benfords law works
Google really making sure we don't think wrong. "Robust safeguards help ensure the integrity of election results." right under the vid
Most areas were also heavily divided. You had either lots of small areas where trump would win (which would explain why he follows the law) and Biden won in just about every large metropolitan area with you guessed it, large precincts.
Interesting video.
Did a double take when your comment said Trump "follows the law".
@@LastStar007 Trump does follow the law. Biden is the one who flouts the constitution on a daily basis
@@jypsridic Did you ever take a second to enjoy a joke before you took it seriously?
@@hdjsksoxckjrejwkdld Do you know what the word joke even means?
@Jyps Ridic Look into the mirror and you know what it means.
Been seeing benford's law in discussions about the voter irregularities, and this helped clear things up. I already figured it was mainly a tool that could point to fraud, as you brought up, but now I better understand how it could be better used.
+
"You only get Benford's Law in SOME situations."
So it's Benford's poorly enforced law?
Benford's understaffed police station
Benford's suggestion.
Lets build a wall around it
benford's US police force
In Minnesota they'd defunded enforcement of statistical laws, and now there are outliers running wild.
If we collected data on abuses of Benford's Law, would that distribution follow Benford's Law?
If you think the comments are all civilised, filter by newest first and look again.
Hi
I did that, in February 2022, and for a video about a president, they're still pretty civil.
I have a lot of respect for the fact that you didn't just stop at "the trump one looks suspicious!" and actually explained why both claims are faulty.
That kind of honesty, regardless of where you stand politically, is something we need WAY more of in today's world.
I think that's what Trump is looking for. A validation of EVERY SINGLE LEGALLY cast vote. The only thing I will believe is if they contact every voter to verify their votes.
@@PayNoTax-GetNoVote while that would be great, wouldn’t you say that’s impractical? Should we do that for people who voted in 2016? What if people change their answers over the phone because some guy is haranguingthem? What if people, knowing that race was so close, change their initial vote?
Jesus, get real. He lost, hopefully the investigations complete and assure everyone that, yes, that’s true. Assuming he didn’t lose without evidence is conspiracy thinking
@@iMasterchris Also, the way some people are talking now, if some random 'phones my house and asks about my voting, I'd put the phone down and draw the curtains
Election 2024: We have replaced your votes with a random distribution of Pi.
This would be to obvious
@@paxundpeace9970 which is ok because it was meant to be hilarious
Don't give the dems (who control most of the inner city voting precincts) any ideas..
Over 7k dislikes.... Certain people mad this didn't agree with their conspiracy theories.
Except the exact same jump was seen in the Georgia runoff.
@@MumboJumboZXC cope
@@MumboJumboZXC Cope harder.
@@MumboJumboZXC yo can I have some of that copeium
@@MumboJumboZXC cope harder, snowflake
Thank you so much for uploading this video. It is helping me to get through the pandemic!
Thank you for actually explaining the answer to a question a lot of people have. Much more useful than the little warnings social media companies have that "Election fraud is rare according to the AP" or whatever.
Big tech thinks you are too stupid to handle it
@@fromdarktolight6353 TBH most Trump supporters are too stupid to learn anything
Shut up
"Here's a rubber stamp from our political donors, don't ask questions please."
@@antiantiderivative Ah yes, you’re one of those “open-minded” and “tolerant” lefties.
If my stats classes were this interesting, I wouldn't have hated stats. ;)
If this comment wasn’t so stale, it wouldn’t be stale.
@Mr. Natural Completely agree! But if my Technical Communications minor taught me anything, you should capture your audience from the start. Plus, this particular video isn't presenting anything that a layperson couldn't readily understand. It would be a nice intro to a stats class, IMHO.
@Mr. Natural I think we are just going to have to agree to agree, sir. :)
the derivitives and anti derivitives of "ln" and stuff... ik what you mean, i got a 1 in my ap stats test :(
My son has a Master Degree in Stats
Interesting video. Benford curves are a really odd thing that people cite at times with very little understanding of it. It only works in VERY unique circumstances with VERY specific sets of data.
“Benford’s law isn’t the only tool, there’s another one”
Ah yes, Fenbord’s Law
I like how the "Trump Tower" encroaches on Pi's distribution.
Trump needs to get a slice of pi
the trump tower distribution was very much MP.
pi pi ??/
Heh
Pi is humble, what can I say
Matt: Anyone presenting data will probably have an agenda. And that's ok!
Also Matt: My book talks about this very subject, among others.
Me: And that's ok!
Thats a sad statement. When I do engineering calcs, pushing an agenda is the furthest thing from my mind. My calcs are going to be reviewed by others. Millions of $$ hang on my calcs. Id be on the street if I was pushing a biased outcome. So, dont say “anyone” would push an agenda. Any “professional” doing that is not a professional.
@@Tom-tk3du I would say that your agenda is "you don't want the building to fall down". And that's okay! Not every agenda is evil! Many agendas lead to the person putting forth their best argument for the most accurate outcome as they it
This video is amazing. I love how you show us something might be wrong when in fact that same wrong-ness turns out to be precisely what we should be expecting, because we can look at the data in a different way. I usually don't do statistics, so that's why I find it surprising, but nonetheless I will keep that in mind for the future.
I had no idea about any of this. Thank you, for explaining everything so well.
Lol "problematic at best" is going to be my new motto
It's the subtitle of my life.
@@delphicdescant i'm getting it printed on a tshirt
Tinder bio
If you want an official motto for a family crest:
"Difficilis ad Optimus"
Sadly though even those words now have become politicised
Fantastic video. Textbook example that a light touch is often better than a heavy hand when engaging with such topics. Many youtubers seem to cripple their own perfectly valid points with unnecessary subjective filler.
This is what I've been trying to say about all discourse in America these days. Thank you for putting it into words
Here is a far better video on Benford's Law.
See STEP #1 for the video on Benford's Law
www.foundingfathers.org/Papers/Politics/BenfordsLaw_n_ElectionFraud.aspx
@@foundingfathers4462 singingbanana, nice
yeah, that's a good video, from a great mathematician (also happens to be a friend of Matt Parker)
not sure, why it would be far better though, says pretty much the same thing from what I remember. In particular it also says that the data has to span multiple orders of magnitude, which is why you wouldn't be able to use it in this case
what do you mean "unnecessary subjective filler"?
@@silversilk8438 anything that could make the video sound like it has an agenda or anything that sounds condescending. It’s an easy way to get people to refuse to accept your points. Even if they’re otherwise accurate.
I'll admit it, I came here as a skeptic....but you convinced me. Well done video.
So did i; it makes me reflect on how much of a better place we’d be in if everyone took the other side’s concerns seriously and addressed them instead of trying to discredit them.
Uff, just looking at the downvotes I get the impression that some people really don't like math
some people don't like critical thinking or education at all, unfortunately.