Nov 13, 2020

Breaking Benford

In the days after the US Presidential election was called for Joe Biden, many supporters of Donald Trump are crying foul.  Voter fraud. And a key piece of evidence? A century-old quirk of math called Benford’s Law.  We at Radiolab know Benford’s Law well, and have covered it before.  In this political dispatch, Latif and Soren Sherlock their way through the precinct numbers to see if these claims hold up. Spoiler: they don’t. But the reason why is more interesting than you’d expect.

This episode was reported by Latif Nasser. 

Support Radiolab by becoming a member today at Radiolab.org/donate.    

Links: 

Walter Mebane, “Inappropriate Applications of Benford’s Law Regularities to Some Data from the 2020 Presidential Election in the United States”

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[RADIOLAB INTRO]

LATIF NASSER: Well, first, am I imagining that Jad already knows about - 'cause we just talked about it when I was promoting my show, right?

JAD ABUMRAD: About Benford’s Law?

SOREN WHEELER: Just go. Just tell him that. Remind him.

LATIF NASSER: OK, so...

SOREN WHEELER: He knows, but he forgets.

JAD ABUMRAD: Yeah, just...

LATIF NASSER: OK. Well, I think...

Hey. I'm Latif Nasser. This is RADIOLAB. And today, in the week after the election was called for Joe Biden but in this sort of weird middle space, when Donald Trump has refused to concede, our editor Soren Wheeler and I sat Jab down...

(CROSSTALK)

LATIF NASSER: ...Because we wanted to tell him about something from our past that has come back to haunt us.

JAD ABUMRAD: OK.

LATIF NASSER: So a few months ago, you may remember I was telling you about the Netflix show I made.

JAD ABUMRAD: "Connected," right?

LATIF NASSER: Right. And I was telling you about one of the episodes, which is about something that, before my time, RADIOLAB had also separately done an episode about, which was this thing called Benford’s Law.

JAD ABUMRAD: Yes, yes.

LATIF NASSER: This kind of funny, nerdy, little bit mystical, mathematical law that's kind of stupendously broadly applicable?

JAD ABUMRAD: Stupendously broadly applicable.

LATIF NASSER: Yeah.

JAD ABUMRAD: It's one of those.

LATIF NASSER: But so, you know, in my head, I feel like I parked it in the, like, fun, silly, nerdy category of stuff that's more sort of interesting than it is important. But then all of a sudden, over the last week, it became very important.

JAD ABUMRAD: Oh, boy.

(SOUNDBITE OF ARCHIVED RECORDING)

UNIDENTIFIED PERSON #1: Benford’s Law - what is Benford’s Law? And how is it relevant to the 2020 election?

LATIF NASSER: It was all over Twitter, Facebook.

(SOUNDBITE OF ARCHIVED RECORDING)

UNIDENTIFIED PERSON #1: Its mathematics - add clarity and...

SOREN WHEELER: I think some of the first things I saw about it were YouTube videos.

(SOUNDBITE OF YOUTUBE VIDEO)

UNIDENTIFIED PERSON #2: In layman's terms, it's the mathematically and statistically provable point that...

LATIF NASSER: Then it shows up on all these fringe news sites - Newsmax, Boston Herald, Gateway Pundit.

SOREN WHEELER: And then a conservative podcast - people like Dan Bongino start to talk about it.

(SOUNDBITE OF PODCAST, "THE DAN BONGINO SHOW")

DAN BONGINO: What if I told you when you look at some of the voter data at Wisconsin - that Benford’s Law is saying it's not possible?

LATIF NASSER: And...

(SOUNDBITE OF ARCHIVED RECORDING)

SCOTT ADAMS: Benford’s Law is what we're talking about.

LATIF NASSER: ...Scott Adams, the cartoonist - Scott Adams, who's a big Trump supporter - he was talking about it.

(SOUNDBITE OF ARCHIVED RECORDING)

SCOTT ADAMS: Votes don't conform to the natural arc.

LATIF NASSER: There was so much stuff going around online that Twitter actually started flagging mentions of Benford’s Law.

JAD ABUMRAD: Wait. Why?

LATIF NASSER: Because, suddenly, there were just, you know, a lot of people online trying to use Benford’s Law to parse the 2020 election returns to say that the votes for Joe Biden were fraudulent votes, and Benford’s Law proves that.

JAD ABUMRAD: Huh. That's funny because like, we're all waiting - like, you know, there's all this talk about fraud. And everybody's like, OK, show us the fraud. Where's the fraud? You're saying Benford’s Law of all things is what people are using to try and say that there's fraud?

LATIF NASSER: Yeah.

SOREN WHEELER: And then the big hit is that, like, on Twitter, people would be like, by the way, check out RADIOLAB. It's totally true.

LATIF NASSER: Right. So...

SOREN WHEELER: Or, like, go see "Connected." Like, this is the real deal.

LATIF NASSER: Yeah.

JAD ABUMRAD: Wow.

LATIF NASSER: It almost felt like we got pulled into this fight.

JAD ABUMRAD: You never know where your stuff is going to end up. But I assume from what you said about Twitter that there is no there there in terms of Benford proving fraud.

SOREN WHEELER: No, definitely not. I mean, if you talk to the people who know the math, do the math, they say, you know, like, all these guys are wrong.

LATIF NASSER: But interestingly, the way that they're wrong, the way that all these people are sort of fighting about it on Twitter is actually kind of interesting.

JAD ABUMRAD: But can you just remind me what Benford’s Law is? I mean, I have vague memory of this from one of our shows. But, like, what is it again?

SOREN WHEELER: So tell you what - let's play a part of the old piece, and we'll come back to the election battle in just a second. But this was, like, 11 years ago. We did a whole show about numbers. It was actually during your paternity leave with Amil.

JAD ABUMRAD: That's right. This was my paternity leave, yeah.

SOREN WHEELER: So what we did was sort of brought you back into the studio to work you into the piece.

(SOUNDBITE OF ARCHIVED BROADCAST)

JAD ABUMRAD: I'm Jad Abumrad.

ROBERT KRULWICH: And I'm Robert Krulwich.

JAD ABUMRAD: This is RADIOLAB.

ROBERT KRULWICH: We're still talking about numbers.

JAD ABUMRAD: (Vocalizing).

ROBERT KRULWICH: Now we're going to switch (laughter)...

SOREN WHEELER: It was definitely a while ago, and it felt a little bit like a simpler time, but we're going to play it real quick. So the story that we told basically came from an interview that Robert did with a guy named Mark Nigrini, who's a business professor at The College of Jersey.

(SOUNDBITE OF ARCHIVED BROADCAST)

ROBERT KRULWICH: And his favorite story that numbers tell actually starts back in 1938. So imagine an office.

MARK NIGRINI: In Schenectady, N.Y., at the GE research laboratories.

ROBERT KRULWICH: And in that office is a man. And he's sitting at his desk.

MARK NIGRINI: Mr. Frank Benford.

ROBERT KRULWICH: And Mr. Frank Benford is a physicist, who is doing some difficult calculations. And he's hunched over a book, probably, actually, the - one of the most boring books you could imagine.

MARK NIGRINI: Yes.

ROBERT KRULWICH: This is a book of logarithmic tables.

JAD ABUMRAD: What are logarithmic tables?

MARK NIGRINI: So log tables were a very convenient way of doing multiplication in the early part of the last century.

ROBERT KRULWICH: So remember, this is before there were calculators. So if you wanted to multiply something like 145 times 3,564, you could just go to this book and look it up. So each starts with numbers you might want to multiply by - 1 to 100 on the first pages, then 101, 2, up to 200 and 300. And the back of the book is, like, 900s. The further you go, the higher and higher the numbers you use to multiply.

MARK NIGRINI: That's right.

ROBERT KRULWICH: So our Benford fellow - he's sitting there, doing his calculations. And he's looking at the numbers, flipping through the book.

MARK NIGRINI: He's staring at the pages, and...

ROBERT KRULWICH: He notices something kind of weird.

MARK NIGRINI: He noticed that the first few pages were more worn than the last few pages.

ROBERT KRULWICH: Meaning more smudgy and darker and oily, as if he was using the front of the book...

MARK NIGRINI: More than the last few pages.

ROBERT KRULWICH: And he wondered, why is this happening?

MARK NIGRINI: Strange.

ROBERT KRULWICH: I'm not aware of favoring one part of the book over the other. Am I doing something a little odd? Or maybe it's something bigger. And that's when it hit him.

MARK NIGRINI: He thought, maybe, in this world, there are more numbers with low first digits than with high first digits.

ROBERT KRULWICH: What? More numbers that start with one or two than numbers that start with seven, eight or nine.

JAD ABUMRAD: Just because his book is worn?

ROBERT KRULWICH: That's what started him thinking, so here's what he did.

STEVEN STROGATZ: He compiled some tens of thousands of statistics.

ROBERT KRULWICH: That's Steve Strogatz, mathematician at Cornell University.

STEVEN STROGATZ: Just anything he could think of that was numerical - molecular weights of different chemicals, baseball statistics, census data.

MARK NIGRINI: The revenues of all the companies listed on the main stock exchanges in America.

ROBERT KRULWICH: And everywhere he looked, in all these different categories, it seemed, yes, there were more numbers beginning with one and twos than eights and nines.

JAD ABUMRAD: Wait; really?

ROBERT KRULWICH: Oh, yeah.

(SOUNDBITE OF MUSIC)

ROBERT KRULWICH: This has been checked out again and again and again. And it's true.

STEVEN STROGATZ: Size of rivers.

ROBERT KRULWICH: Wow.

MARK NIGRINI: Earthquakes and things like that.

STEVEN STROGATZ: Populations or number of deaths in a war, areas of counties.

MARK NIGRINI: Streamflow data in the United States.

JAD ABUMRAD: What if you were to, say, get all the people in New York together and look at their bank accounts?

MARK NIGRINI: Ah, bank account balances follow Benford's Law nearly perfectly.

ROBERT KRULWICH: Meaning that if you just look in at the amount of money that people have, matter of factly, in all the bank accounts, you'll find they begin with one more often than they begin with two.

MARK NIGRINI: Perfectly, yes.

ROBERT KRULWICH: Yes?

MARK NIGRINI: So actually, they begin with one 30.1% of the time. They'll begin with a two 17.6% of the time. They'll begin with a three 12.5% of the time.

JAD ABUMRAD: That's a big difference.

MARK NIGRINI: And the...

JAD ABUMRAD: Why would three be - I'm sorry. Keep going. Yeah.

MARK NIGRINI: And the poor nine would only occur as a first digit 4.6% of the time, which actually would make the one approximately six times as likely as the nine. And it is quite amazing.

JAD ABUMRAD: What?

ROBERT KRULWICH: That is more than quite amazing. That's deeply suspicious.

STEVEN STROGATZ: I mean, this is crazy, what I'm telling you, and I can't give you good intuition why it's true, but...

ROBERT KRULWICH: Steve and Mark and many, many, many mathematicians will tell you, despite what you may think, there is a preference - a deep preference in the world, it seems - for number sequences that start with ones and then twos and then threes.

SOREN WHEELER: So that was how we did it. There's more, but...

JAD ABUMRAD: I feel like I'm reexperiencing the weirdness of this law again. Is that - at the risk of tracking us off, why is that, that you would get more ones than twos and all that?

LATIF NASSER: Like, this is - yeah - the question I had, too. Is it, like...

SOREN WHEELER: Do you want the RADIOLAB explanation? 'Cause I remember it...

JAD ABUMRAD: (Laughter) Yeah, what was - what did we say back then?

SOREN WHEELER: ...When we did this originally. This was a classic move. It's like, we get to the end of the piece. You ask this very question.

(SOUNDBITE OF ARCHIVED BROADCAST)

JAD ABUMRAD: You still haven't addressed the central mystery here. Why in the world would there be more ones than nines? Shouldn't they be equi...

ROBERT KRULWICH: Equicoincident?

JAD ABUMRAD: Yes.

ROBERT KRULWICH: Well, the answer is actually very complicated and deeply mathematical. The simple answer is...

JAD ABUMRAD: Is there an answer, though?

ROBERT KRULWICH: Yes, there is an answer, and it has to do...

JAD ABUMRAD: Do you understand the answer?

ROBERT KRULWICH: No, just too numeric for me to explain it to you.

(LAUGHTER)

JAD ABUMRAD: OK. All right.

ROBERT KRULWICH: But I will now...

JAD ABUMRAD: (Laughter) Right.

SOREN WHEELER: And then we laugh our way out of the room.

LATIF NASSER: The best...

JAD ABUMRAD: That's funny.

LATIF NASSER: ...Explanation I heard was about - if you imagine a lot of things grow, right? They, like, grow and get bigger, right? And so let's say you're a one - right? - and you want to get to grow to be a two. You have to grow 100% to get to a two. So it takes you a long time to get from being a one to being a two. If you're an eight and you want to grow to become a nine - right? - that's just an eighth of you that needs to grow again, right?

JAD ABUMRAD: Right.

LATIF NASSER: So it's like, you're going to be in that eight zone way shorter than you were in the...

JAD ABUMRAD: Right. That's interesting.

LATIF NASSER: ...Between in the one to two zone. Yeah.

JAD ABUMRAD: I see. So it's like - it's kind of like a - like some kind of a logarithmic situation.

LATIF NASSER: Exactly, exactly.

JAD ABUMRAD: Jumping from one to two is huge, two to four is huge, four to eight is huge. And that's why the hugeness of those jumps means you - I lost the end of that sentence.

SOREN WHEELER: It's sort of why we couldn't in the first - but, you know, the point of the piece - like, even though we didn't explain that, what the piece did do was sort of - and this is the whole reason we did the piece - was go on to talk about how you can actually use Benford's Law in these really pretty consequential ways.

LATIF NASSER: Right.

(SOUNDBITE OF ARCHIVED BROADCAST)

ROBERT KRULWICH: When Mark Nigrini first ran across Benford's Law, he thought...

(SOUNDBITE OF MUSIC)

ROBERT KRULWICH: ...Maybe I can use this law...

(SOUNDBITE OF MUSIC)

ROBERT KRULWICH: ...To bust people.

MARK NIGRINI: For payroll fraud, tax return fraud.

ROBERT KRULWICH: You thought, hey, we can use this to catch a thief?

MARK NIGRINI: That's right.

JAD ABUMRAD: How? How?

ROBERT KRULWICH: Well, Nigrini figured...

SOREN WHEELER: And so, you know, basically, we explain that if books are cooked - like, financial books are cooked - they will not be following Benford's Law. So real books would follow the law. Cooked books don't, so you can spot a thief. And we gave a bunch of examples of that, you know, little stories.

(SOUNDBITE OF ARCHIVED BROADCAST)

DARRELL DORRELL: Ran Benford's, and - boom.

JAD ABUMRAD: Oh, busted.

DARRELL DORRELL: She eventually pled.

ROBERT KRULWICH: Or the guy with a $40 million Ponzi scheme.

DARRELL DORRELL: Ran Benford's, and - boom.

SOREN WHEELER: And then at the end of the piece...

(SOUNDBITE OF ARCHIVED BROADCAST)

ROBERT KRULWICH: Well, almost boom. I mean, Benford's was an element in all these cases. It wasn't the clincher. But still...

DARRELL DORRELL: It was a very compelling argument. And then 10 years from now, it'll be the equivalent of a fingerprint.

(SOUNDBITE OF MUSIC)

SOREN WHEELER: Which is, you know, a strong statement. But, I mean, at the same time, we didn't say anything about elections. And we weren't even thinking about elections.

JAD ABUMRAD: Right.

LATIF NASSER: But when we come back from break, we're going to go right into Benford's Law and election data specifically, which is where the trouble begins.

LATIF NASSER: I'm Latif Nasser. This is RADIOLAB. We're talking about Benford’s Law and the 2020 presidential election.

JAD ABUMRAD: Well, let me ask you. Have any smart statistics, knowledgeable people actually tried to use Benford’s Law to see whether there's election fraud in any election anywhere?

LATIF NASSER: Yeah. Yes. Yeah. And for me in particular - like, I mean, you guys were mostly talking about, like, taxes and forensic accounting, and it was about money and numbers. But in the connected episode I did, I totally talked about elections.

(SOUNDBITE OF ARCHIVED RECORDING)

LATIF NASSER: Well, there is one moment when our millions of personal decisions all get channeled in the same direction, when free will is in full flower - or at least it's supposed to be.

There's a whole segment about - here's how Benford’s Law can potentially relate to elections. And, in fact...

(SOUNDBITE OF ARCHIVED RECORDING)

LATIF NASSER: Walter?

WALTER MEBANE: Oh, hey. How are you, Latif?

LATIF NASSER: Hi. How you doing? (Laughter).

The guy I talked to in that show is the guy that a lot of these people online - it's his work that many of these folks are citing. So I was like...

(SOUNDBITE OF ARCHIVED RECORDING)

WALTER MEBANE: It is now recording, it says.

LATIF NASSER: ...I better call him back up.

(SOUNDBITE OF ARCHIVED RECORDING)

WALTER MEBANE: OK, sure.

LATIF NASSER: Introduce yourself for me. Just tell me who you are in a line.

WALTER MEBANE: Walter Mebane at University of Michigan in Ann Arbor.

LATIF NASSER: So - OK, so what has happened between now and the last time we talked?

WALTER MEBANE: Well, the last time we talked...

LATIF NASSER: And he had seen all of the stuff online. He knew all about it because he'd been getting all these emails in the last week.

(SOUNDBITE OF ARCHIVED RECORDING)

WALTER MEBANE: So the trickle turned into a flood. And my students start to write me and say, hey, you know, this is going on, so...

LATIF NASSER: Saying, hey, look - Benford’s Law and your work, people are using it basically to claim fraud.

(SOUNDBITE OF ARCHIVED RECORDING)

WALTER MEBANE: I was originally pointed to a Twitter post, and then there was another Twitter post, and then I found a YouTube video. And I don't know which one came first.

LATIF NASSER: So, basically, what a lot of people are doing is they're taking these, like, precinct tallies and saying, OK, look - we're entering all the precinct tallies. We're taking the first digits from all the vote tallies for each of the candidates in each of these precincts, and then we're checking how often it starts with one, how often it starts with two and so on.

SOREN WHEELER: Take Chicago, right? They did Chicago, didn't they, with this?

LATIF NASSER: They did, yeah.

SOREN WHEELER: It's not a swingy one. It's a one that Biden won by a lot. But let's just do this real quick. Chicago - how many voting precincts are in Chicago?

LATIF NASSER: There's - I think there's something like hundreds, if not thousands of them.

SOREN WHEELER: Yeah, let's just say a thousand.

LATIF NASSER: OK, sure.

SOREN WHEELER: You just take the votes for Donald Trump in each of those precincts. Here he got 238. Here he got 462. Here he got 521. And then in another column, you take all the totals for - that Joe Biden got in those same precincts - 266 here, 892 here, 326 here. And what it is that's popping up is oftentimes, basically, just a little graphic. Two little curves with the vote counts. It's like, here's Trump's curve. Look at all the ones and a little less twos, a little less three - nice sweeping little curve.

LATIF NASSER: Yeah.

SOREN WHEELER: Oh, here's Biden's. So it's that weird - there's a bunch of sixes in that bumpy curve.

LATIF NASSER: Right, right.

SOREN WHEELER: And they're just like, look at these two. See?

JAD ABUMRAD: I see.

LATIF NASSER: And the way they see it - so this one is for some data from Michigan. It's like, OK, there's less ones than there should be. There's way less twos than there should be. There's more threes than there should be. There's more fours, way more fives, more sixes, more sevens, more eights and less nines. So it's like, it's all over the curve - right? - is the claim.

JAD ABUMRAD: So they're saying that the Biden numbers are...

LATIF NASSER: They look cooked.

JAD ABUMRAD: They don't call according to this pretty Benford curve.

LATIF NASSER: Right.

JAD ABUMRAD: Therefore...

LATIF NASSER: Something's amiss.

JAD ABUMRAD: ...Probably somebody was just, like, keying them in on some...

SOREN WHEELER: Right. Some fake adding - some kind of false, fraudulent adding of numbers.

LATIF NASSER: Yeah.

JAD ABUMRAD: I see.

LATIF NASSER: 'Cause the idea is that random numbers follow this pattern, but human-created numbers don't.

JAD ABUMRAD: This is a way of seeing human meddling.

LATIF NASSER: Right.

JAD ABUMRAD: I see.

LATIF NASSER: That's the idea.

JAD ABUMRAD: That's the claim.

LATIF NASSER: But...

(SOUNDBITE OF ARCHIVED RECORDING)

WALTER MEBANE: All this is completely ridiculous.

LATIF NASSER: Walter was just like, you just can't do this with precincts.

(SOUNDBITE OF ARCHIVED RECORDING)

WALTER MEBANE: 'Cause it's very well known the first digit of precinct vote counts are not useful for diagnosing frauds or anything else.

JENNIFER GOLBECK: So I started jumping into those threads, correcting people and explaining - which didn't win me a lot of friends on Twitter, (laughter) at least among that audience.

LATIF NASSER: So once the thing blew up online, another person I talked to on my show...

(SOUNDBITE OF ARCHIVED RECORDING)

LATIF NASSER: Jennifer Golbeck, professor at the University of Maryland's College of Information Studies.

JENNIFER GOLBECK: As a researcher who studies social...

LATIF NASSER: She's, like, on Twitter, like, fighting, battling with people.

(SOUNDBITE OF ARCHIVED RECORDING)

JENNIFER GOLBECK: Yeah, so...

LATIF NASSER: Are you getting nasty DMs and stuff?

JENNIFER GOLBECK: Yeah. I mean, not the worst that I've gotten, right? More...

LATIF NASSER: Right.

JENNIFER GOLBECK: You know, people cross a threshold where they don't understand where I'm - what I'm saying anymore, so then they just start calling me names, which is OK (laughter).

LATIF NASSER: Right, right.

JENNIFER GOLBECK: I mean, it's for sure not a dumb idea to wonder if we can do this. And part of the attraction for Benford and why people really like it here is that it's super surprising when you first learn about Benford, right? My mind was blown the first time I learned about it. And so on paper...

LATIF NASSER: On RADIOLAB, right?

JENNIFER GOLBECK: Yeah, on RADIOLAB. That's right.

LATIF NASSER: Yeah.

JENNIFER GOLBECK: I mean, it is what got me started doing research in this space - was, like, listening to that RADIOLAB and not being able to stop thinking about it, you know, for weeks.

LATIF NASSER: Yeah.

JENNIFER GOLBECK: There are papers out there that say, look; Benford could be really powerful for doing this, but it's not the way anybody on Twitter was doing it.

LATIF NASSER: The problem is when you're looking at precinct data, these precincts are basically purposefully made just for counting the election data. And the reason that those are helpful is because they're all the same size.

(SOUNDBITE OF ARCHIVED RECORDING)

JENNIFER GOLBECK: Normally, when Benford works, you've got lots of orders of magnitude. So say we're looking at, like, the length of all the rivers in the world. We know that that follows Benford. So there's some really short rivers, and there's some super, you know, continent-long rivers. So there's a lot of changes in the orders of magnitude. You have some...

LATIF NASSER: But the precincts do not have multiple orders of magnitude.

(SOUNDBITE OF ARCHIVED RECORDING)

JENNIFER GOLBECK: Precincts tend to be kind of the same size.

WALTER MEBANE: And the first digit is primarily determined by the size of the precinct. It has nothing to do with anything about the behavior of the voters or the officials or the vote counters. It's just a matter of the design of the precincts.

JAD ABUMRAD: Wait; how so? How would the first number have to do with the size of the precinct?

LATIF NASSER: Well, Walter put it this way.

(SOUNDBITE OF ARCHIVED RECORDING)

WALTER MEBANE: Imagine that all the precincts are about a thousand in size, which is roughly a convenient number, but put whatever you like. And imagine the two candidates get about 50% of the vote.

LATIF NASSER: Which was true in a lot of the, you know, swingy precincts that this data was based on. And if you look at the Trump-Biden totals in those places...

(SOUNDBITE OF ARCHIVED RECORDING)

WALTER MEBANE: Then the first digit everywhere is going to be a four or a five.

SOREN WHEELER: You're going to get 500 to this guy to 500 to this guy...

LATIF NASSER: Right.

SOREN WHEELER: ...To 400 for this guy, 600 to this guy.

JAD ABUMRAD: Yeah.

SOREN WHEELER: You're probably never going to get a one in front of anything unless it's a blowout.

JAD ABUMRAD: So - that's interesting. So they're skewing - because of the way that the precincts are drawn up, you're skewing towards the middle, I would imagine. You're going to get a whole bunch more fours and fives and sixes.

LATIF NASSER: And that's exactly what that chart I was telling you about before that's sort of gone a little bit viral - which is like - that's exactly what it shows. So it says that Michigan data, where it's like, oh, look; the highest numbers are for fours, fives and sixes...

JAD ABUMRAD: Oh, that's funny.

LATIF NASSER: So you're like, oh, so, like, that makes sense that Biden would have gotten fours, fives and sixes, right? And then Jen pointed out, OK, now take another thousand-person precinct, and say this is one where Biden has a really big lead.

(SOUNDBITE OF ARCHIVED RECORDING)

JENNIFER GOLBECK: So if Trump gets 150 votes, then Biden's going to get 850 votes. He's going to get the rest, right? So if Trump follows a Benford’s Law distribution, Biden necessarily will not follow that distribution because he's getting the other part, right?

LATIF NASSER: Oh (laughter). But then shouldn't Biden's votes be the exact - like, the mirror of the curve or something close to it?

JENNIFER GOLBECK: So if it were the situation where we had, like, a thousand people in every precinct, that would totally be true. In reality, there's, like, variation. And so if you look at the plotting of this, what you actually see is that Biden's curve pretty much matches the curve of how many people are in the precinct.

WALTER MEBANE: So that's not Benford’s Law. That's just because the precinct sizes are the way they are.

JENNIFER GOLBECK: Yeah. I mean, I don't think there's any reason to expect anybody's going to look like Benford or not 'cause this first significant digit analysis of Benford in elections doesn't work, period.

LATIF NASSER: Now, there are definitely statisticians out there who say that you cannot Benford-ize (ph) elections at all. But Walter - the thing that's interesting about him is that he thinks - and this is, you know, a lot of the work that he does - that you can look at elections if you look at not the first digit, but at the second digit. This has become his thing. So he's, like, looked at elections in Kenya and Russia and Germany and Turkey and Mexico and Iran, and often, he is using Benford’s Law on the second digit.

JAD ABUMRAD: The first digit is going to be a function of how big the precinct is...

LATIF NASSER: Exactly.

JAD ABUMRAD: ...Which is a human construct. But the second digit - that's going to be more subject to the laws of randomness and whatever - yada, yada, yada. That is not a human construct - not in quite the same way.

LATIF NASSER: Probably, hopefully.

JAD ABUMRAD: Hopefully, right.

LATIF NASSER: So, yeah.

SOREN WHEELER: Super stupid caveat - most mathematician total-freakout geeks will say the thing happening with the second digit is not strictly Benford’s Law. It's just a very Benford-like thing that's a pattern that you can spot.

JAD ABUMRAD: Got it.

LATIF NASSER: Yeah. And he uses phrases like Benford-inflected or Benford-like - or Benford-like - it's like...

(LAUGHTER)

SOREN WHEELER: Flavored.

LATIF NASSER: That kind of thing.

JAD ABUMRAD: I see. I see.

LATIF NASSER: And Walter also makes clear that even if you do look at the second digit...

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WALTER MEBANE: Even then, it's not a perfect signal for fraud.

SOREN WHEELER: He'll do the Benford thing right, but he also has this whole, like, sort of tool belt.

LATIF NASSER: A toolkit, right.

SOREN WHEELER: ...A toolkit - right - that he's made of all these other different statistical things that you bring to bear on it 'cause it's super complicated and backgrounds and expectations, and then you do this with a regression to the - blah, blah, blah. So he's going to bring the whole suite of many complicated I-am-a-super-smart-person tools to bear on this.

LATIF NASSER: Right. And the tools that he's sort of picked up in literally 20 years of studying elections statistically for fraud. But, anyway - OK, so those are kind of, in general, the reasons why he looks at all the stuff that sort of these kind of amateur Benford’s Law analysts online are throwing up. He looks at that and he's like, that's ridiculous. And then so then he was like, OK, I'm - over the weekend, he was like, I'm just going to run this myself.

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WALTER MEBANE: And I produced a working paper, which I think you have the link for.

LATIF NASSER: Yup.

WALTER MEBANE: And it has the diagnostics not only for the second-digit Benford’s Law stuff - lawlike stuff, but the other diagnostics that I've developed over the past decade and a half.

LATIF NASSER: And what did you find in running that stuff, the second-digit stuff?

WALTER MEBANE: There's nothing. I mean, there's no sign of frauds of any kind.

LATIF NASSER: It was like this is a totally ordinary - in fact, it seems pretty exemplary.

WALTER MEBANE: As far as I've seen, there's nothing bad that happened anywhere. This has been miraculous. This election has been so smooth.

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JENNIFER GOLBECK: You can't just plug some numbers into your Excel spreadsheet after you've read a Wikipedia page and do it. But people who feel like, look; I can see the difference, and a bunch of these look the right way and one looks the wrong way, so forget what you're saying about it being complicated; it must be true - those are hard people to convince.

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LATIF NASSER: It's just common sense enough, but it's just obscure and complicated enough that it feels like it's like a piece of evidence that could - you know, it could take on, yeah, sort of a life of its own. It could be this thing that then - you know, there's a certain - admittedly small but segment of the population that's, like, now whenever they talk about the 2020 election, they're going to be like, oh, and Benford’s Law shows that it's all - it was all bogus, you know?

JAD ABUMRAD: Yeah, yeah.

SOREN WHEELER: Yeah. It strikes me as like, is it going to go to court and win the day? No. Is it the kind of thing that maybe next Tuesday Donald Trump would say in a press conference? Yes.

LATIF NASSER: Probably, yes.

SOREN WHEELER: Yes.

(LAUGHTER)

JAD ABUMRAD: Yeah. It's also - like, it's the kind of thing that steps into the breach of a very uncomfortable phenomenon. And we all experienced this - right? - which was election night, you had one result, and then you had this weird, creeping, slow tilt over the next three or four days. And even though we ourselves did a story about how that tilt was coming, the tilt still felt surprising and nonintuitive in some emotional way. And so then that is fertile ground for these Benford-esque ways of explaining the irregularity.

SOREN WHEELER: Yeah, but then, you know, even though the people who actually know what they're talking about say it doesn't explain the irregularity.

JAD ABUMRAD: No, but see - this is the problem with the world we live in, is that there's never been more distrust of leadership and expertise. But you're asking us, in a moment when we're very unlikely to, to trust experts to explain this thing and...

LATIF NASSER: Which is all to say, like, it does feel like it's hard to explain. It's going to be hard for a lot of people to explain, like, why - especially if they're not resorting to experts to help explain it - like, how did it go this way when everybody I talked to, you know, seems to have wanted the opposite to happen?

JAD ABUMRAD: Yeah, totally.

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JAD ABUMRAD: I find myself wanting to do sort of, like, a cost-benefit on Benford's applications in the world.

SOREN WHEELER: Well, the upside would be financials. The upside would be...

LATIF NASSER: Well, not just financials. So there's - yeah, financials is a big one - I mean, tax fraud, like...

JAD ABUMRAD: OK, OK.

LATIF NASSER: But also, another thing is - and this is something that Jen uses in her other research, which is her main research - is how you use Benford's on - in social media. So you can use it to detect bots online. You can see...

JAD ABUMRAD: Really?

LATIF NASSER: And she has done that very thing.

JAD ABUMRAD: How do you...

LATIF NASSER: You can also use it...

JAD ABUMRAD: No, keep going. Keep going. Then I'm going to double back.

LATIF NASSER: No, so you - literally, you count - and this is cool 'cause people can do it at home, too. You count up not - so all the follower numbers of your followers on Twitter, Facebook, whatever. And you can see, based on the followers of your followers, if you just take those first digits, it should follow Benford’s Law.

JAD ABUMRAD: Oh, my God. That's incredible.

LATIF NASSER: And if it doesn't - yeah.

JAD ABUMRAD: That's wild.

LATIF NASSER: And then there's...

JAD ABUMRAD: Is that - so wait, does Twitter use that to prune bots, as they do from time to time?

LATIF NASSER: I don't know if Twitter officially uses it. She - Jen definitely uses it. And then not only that, but also, you can use - and this is another thing in the TV show that we did. I did an interview with former RADIOLAB guest, also my - one of my former college professors, Hany Farid. You can use it to basically sniff out deepfakes, and you can use it to sniff out manipulated media. In an era when it's getting way harder to tell the truth...

JAD ABUMRAD: Really?

LATIF NASSER: You basically look at these embedded number values in a single picture and you can see whether that picture has been edited or how many times it's been saved...

JAD ABUMRAD: Oh, wow.

LATIF NASSER: ...Or kind of things like that. So it's actually - like, I mean, it definitely is a force for chaos and doubt. And it's also sometimes a kind of superhero for truth.

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JAD ABUMRAD: If only we could understand it (laughter).

LATIF NASSER: Right.

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JAD ABUMRAD: All right. Thank you, Latif and Soren, for taking me back into the past and forward into now.

SOREN WHEELER: Of course.

LATIF NASSER: No problem.

JAD ABUMRAD: Definitely go check out "Connected," Latif's show on Netflix. You can find it there. It's amazing. The episode with the Benford stuff is called "Digits."

LATIF NASSER: And also, in addition to that, you can go back into the deep past and listen to the RADIOLAB show called "Numbers," which has not only all that Benford goodness but also a love story, some babies and even a little Johnny Cash. It's a great episode.

JAD ABUMRAD: I'm Jad Abumrad.

LATIF NASSER: And I'm Latif Nasser.

JAD ABUMRAD: Thanks for listening.

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MARK SCHONDORF: This is Mark Schondorf (ph) from Highland Park, Ill. RADIOLAB was created by Jad Abumrad and is edited by Soren Wheeler. Lulu Miller and Latif Nasser are our co-hosts. Dylan Keefe is our director of sound design. Suzie Lechtenberg is our executive producer. Our staff includes Simon Adler, Jeremy Bloom, Becca Bressler, Rachael Cusick, David Gebel, Tracie Hunte, Matt Kielty, Tobin Low, Annie McEwen, Sarah Qari, Arianne Wack, Pat Walters and Molly Webster, with help from Shima Oliaee, Sarah Sandbach and Jonny Moens. Our fact-checker is Michelle Harris.

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