Partial Truths: How Fractions Distort Our Thinking

by James C. Zimring

A fast-food chain once tried to compete with McDonald’s quarter-pounder by introducing a third-pound hamburger―only for it to flop when consumers thought a third pound was less than a quarter pound because three is less than four. Separately, a rash of suicides by teenagers who played Dungeons and Dragons caused a panic in parents and the media. They thought D&D was causing teenage suicides―when in fact teenage D&D players committed suicide at a much lower rate than the national average. Errors of this type can be found from antiquity to the present, from the Peloponnesian War to the COVID-19 pandemic. How and why do we keep falling into these traps?

James C. Zimring argues that many of the mistakes that the human mind consistently makes boil down to misperceiving fractions. We see slews of statistics that are essentially fractions, such as percentages, probabilities, frequencies, and rates, and we tend to misinterpret them. Sometimes bad actors manipulate us by cherry-picking data or distorting how information is presented; other times, sloppy communicators inadvertently mislead us. In many cases, we fool ourselves and have only our own minds to blame. Zimring also explores the counterintuitive reason that these flaws might benefit us, demonstrating that individual error can be highly advantageous to problem solving by groups. Blending key scientific research in cognitive psychology with accessible real-life examples, Partial Truths helps readers spot the fallacies lurking in everyday information, from politics to the criminal justice system, from religion to science, from business strategies to New Age culture.

Genres: Nonfiction

256 pages, Hardcover

Published May 10, 2022

Reviews

[David Wineberg · Rating 4 out of 5]

To be human is to be biased, and that is a necessary thing that often needs correcting. That's the essence of James Zimring's Partial Truths. From the trivial to the important, people look at things the wrong way, come to erroneous conclusions, and stand by their ignorance and insist on it for everyone else.

It doesn't sound like such a brilliant species when it's put that way, and Zimring spends a lot of time on Darwin, evolution and natural selection, which themselves are the objects of much misinformation and erroneous implementation. His own conclusion is that natural selection does not favor brilliance; it favors the successful in reproducing. And humans have exceeded all needs and expectations in that domain. Smarts is just a nice-to-have that our brains tend to inhibit.

The book starts off gently, a little too gently for my tastes, as Zimring patiently explains how everyone looks at fractions wrong. He cites the now age-old story of how A&W tried to outdo McDonalds' quarter pounder with a third pounder. It failed, because surveys said people thought 1/3 was smaller than 1/4, because three is smaller than four. Chalk up another win for the school system.

From there, it is on to automobile crashes versus airplane crashes, where everyone thinks it is far safer to drive than to fly. I hope we all know the truth is precisely the opposite (but I'm not betting my own money on it). It was just not what I expected to read.

Then there's an examination of lotteries, where customers focus on winning, looking at the morass through rose-colored glasses. The odds of winning the Powerball are 1 in 292 million. There are typically more than 150 million tickets sold for each drawing, and it can go months without a winner. The way Zimring looks at it, 300 million people a week don't win the lottery. Or, to put it even more dramatically, a customer buying a ticket for every Powerball drawing should expect to win once every 2,807,692 years. But customers never look at it that way. All they see is winners. And they expect to be one one day.

This however, is not the partial truth I (thought I) was here for. Lotteries and quarter pounders are not what I wanted to explore in this era of alternative facts, fake news and civilian slaughter dressed up as an "operation" of "denazification".

It was getting time to throw the book against the far wall and curse myself for agreeing to review it, but Zimring turns it all around. The stories changed from the hoary old to the new, relevant and relatable. And entertaining.

Zimring ventures into the psychic world, where mindreaders continually shock audiences when they spell out some bizarre scenario and an audience member stands and claims it, and demands to know how the psychic knew. Well, the psychic didn't know, but in a large crowd, the odds of someone relating to any situation are quite good. Even if they have to rationalize a lot to make it fit. (It's what readers do with horoscopes every day.)

Another psychic trick is to pretend to almost see the name of a relative of the victim's, beginning with a J, M, S or D. Most Anglo-Saxon names begin with these letters, and if the victim has the average 40 relatives, it's easy for the psychic to hit paydirt. The actual odds that the victim will not have a relative with such a name is one in 276, or .0036 of the time, Zimring says. Psychics play the odds, something everyone should learn.

And then it's the old: in any group of 30, at least two will have the same birthday trick. This is because most people are born in a five month range from fall to spring. The other seven months are hardly represented at all by comparison. Humans are subject to the seasons, including the mating season, the phases of the moon, and the seasons of the year. Spring fever and May weddings are real and standard phenomena for humans. The result is a compressed birthing season.

I loved the quote Zimring obtained from one psychic, (comedian) Mark Edward: "If psychics were real, they wouldn't need to ask even a single question. They would just know." Instead, it is an endless stream of questions and feedback to refine the result and nail the victim with facts s/he could not possibly know.

From psychics he turns to New Age fads, treatments, crystals and beliefs. The innate biases kick in, and otherwise intelligent people hook themselves into supporters and believers. Lying with fractions, which is the common thread throughout the book, is rampant. By making the denominator smaller (by ignoring selected studies, test and reports, or say, segments of the population), the resulting success story becomes inflated and mightily impressive, numerically. Such results must therefore have the force of truth behind them, one would think. Or so people assume. That's how New Age treatments establish themselves as truth.

This, Zimring says, is also the calling card of Big Tobacco, which funded all kinds of studies in order to prove tobacco didn't kill, but only reported the ones that had results it liked. It suppressed the rest, because that was the Big Tobacco deal: we will fund your study, but we retain exclusive rights to publish.

My own favorite example of this lying by omission is the more recent full page ad campaign against recognizing climate change, signed by over 33,000 scientists, an impressive showing. Who were all these scientists who had suddenly spoken up after decades of silence? It turns out they were mostly computer science grads. Out of millions, the sponsors found 33,000 to support their view, never mind that it wasn't their field. But it was their title, and that's all they needed.

My favorite chapter deals with God The Designer. Any time something is amazing, or fits perfectly, or threads a needle through complexity, people like to claim there was an eternal intelligent designer who did that, and that there is no other possible explanation. As Zimring puts it:

"Those who marvel at the fine-tuning of life might as well sit and marvel at how precisely the concrete in a pool fits around the shape of the water in it, and how amazingly perfectly the water fits the pool, even with twists and turns and small cracks. What are the odds of this happening, and not just for one pool , but for every pool! No, given how precisely pools fit the shape of the water and the water fits the pool, clearly an advanced intelligence with astounding engineering and precision designed each body of water and each pool to fit each other perfectly, molecule by molecule and atom by atom. Anything else is be so improbable as to be absurd."

But what it boils down to is the lottery fallacy again - looking at the situation's figures backwards. By examining the size of the universe and how it is populated, Zimring shows how it is all but impossible for an intelligent designer to exist or work its magic at the smallest level of every planet. Just because "we" don't have an answer, an understanding of how things work, does not mean some non-human super being is solely responsible: "Evolution is essentially a self-adjusting fraction that guarantees that life is fine-tuned to itself and its environment."

And Zimring is far from alone in this view. He cites Roger Penrose, a world-renowned physicist, who calculated that the original phase-space volume of the universe would be 1 to the power 10 and then to the power of 123. As Penrose explained: "This now tells us how precise the Creator's aim must have been...One could not possibly even write the number down in full...It would be a 1 followed by 10 to the 123rd power of successive 0s! Even if we were to write a 0 on each separate proton and on each separate neutron in the entire universe - and we could throw in all the other particles as well for good measure - we should still fall far short of writing down the figure needed."

These are the kinds of odds you find against there being an intelligent designer managing the birth of the universe. He calls the argument for an intelligent designer "inference to the best explanation." It is a common tactic, and it obviously works, but it is one that is never testable, let alone provable "unless the number of possible explanations is limited." And when it comes to the universe, there are no such limits, he says.

Similarly, the odds of this being one of several universes, that is, a single universe in the multiverse, suffers the same fate: "While interesting, the multiverse hypothesis remains a weak explanation of the fine-tuned universe." Or, as Zimring most elegantly puts it: "The rules of the world are working exactly as we understand them, but we are mistaking the highly likely for the virtually impossible." In everything from lotteries to intelligent designers, we get the fractions wrong.

Where does an author go after dismissing a Creator and the multiverse? For me the book should have ended there, on a high and bold note, encompassing religion, the whole universe and beyond. But Zimring keeps plowing onward, or more accurately, backward. In a terribly anticlimactic finish, he returns to psychological studies and the manipulation of fractions. He discusses how debiasing may or may not improve things, and how some human subjects walk back their extreme positions when asked for details on how they got there.

Not nearly as interesting as dissing God mathematically.

The human mind operates all kinds of traps and detours that psychologists love to catalog and test. The mind applies biases to everything we see and evaluate. These biases have fed untold thousands of undergrads their beer money, as grad students continue to tease out exactly how and when these biases appear, in endless studies. The biases go by very common and descriptive names: availability bias, confirmation bias, hindsight bias, knew-it-all-along bias, cherry-picking bias, and conformity bias, for example, all of which are examined in the book. They are why two people reading the same text can disagree as to what it says and means. They are why it takes 12 jurors to make a decision, after possibly weeks of argument, diplomatically called deliberation. They are the spice of life, Zimring says. And we probably wouldn't get anywhere without them.

So while Zimring scores with dramatic and proper applications of math, the book is a lot of up and down instead of a powerful build, in my unbiased opinion.

David Wineberg

[Roxana Chirilă · Rating 3 out of 5]

NOTE: NEEDS MORE MATH. I know! It's such a weird note to have. Who wants *more* math in a book, right?!

Well, sometimes, I think most of us do.

Despite its subtitle ("How Fractions Distort Our Thinking") "Partial Truths" contains no math at all, except the little that is necessary to explain what a fraction is.

The core of the book, however, is a discussion of biases, interpreted as misperceived fractions. For example, if you read about a topic, but reject a number of texts simply because they don't align with your point of view, you might come off thinking that a greater proportion of evidence supports your preferred point of view than not.

This is an interesting point of view, but I don't feel it's sufficient in and of itself to make "Partial Truths" a great book. In fact, it sometimes feels like the author wanted to discuss current events (the war in Irak) and phenomena (New Age), and used the opportunity of his ideas to do so. This is alright (we're all allowed to have pet peeves), but these points were expanded to the detriment of the book's supposed topic.

Why I think this needs more math: in defense of dry material

I know many people hated math in school. It's dry. It's abstract. It's a bit like pulling teeth, especially if you do it day after day, in exercises that seem to make no sense and never have anything to do with the real world.

But what math does is speak about underlying rules; while we don't always find knowing those rules useful, they can occasionally come in handy.

If you only sell apples, it's useful to know that if you have 2 apples and get 3 more apples, you're going to have 5 apples. But if you diversify and start selling more types of fruit, 2 + 3 = 5 can be an abstract idea that's very nice to know, because you can apply it to all your fruit.

This holds true for all math. Most of us don't usually need complicated formulas, but every once in a while, they can help us so we don't need to eyeball things and discover if we were right or wrong by trial and error.

A book such as this one has many advantages over a classroom setting. It doesn't need to address all of math; it doesn't need to prepare readers for exams; and it can afford to take things slowly and leisurely in a way teachers can't because they're always pressed for time and results. Thus, to keep things artificially math-free when discussing math feels like the wrong choice.

For example, "Partial Truths" mentions probabilities a few times, but they're never discussed, except to be described as "fractions". I find this odd; probabilities are their own can of worms. They're not immediately intuitive. In fact, if you don't know more about them, they can look like magic.

The odds of getting heads when flipping a coin is 1/2. What are the odds of getting heads at least one time if I flip the coin twice? 3/4. What about the odds of getting heads at least once if I flip the coin thrice? 7/8.

Let's make it more complicated! If you have a trait that appears in 1/300 people, and you have 300 people in the room, what are the odds that at least one person has that trait? A friend who actually remembers math formulas calculated the probability for me: it's about 63%. Slightly less than 2 out of 3 chances. But come on, 1 chance in 300 people, and you have 300 people, what's going on? Why is this so surprisingly low? How is it calculated?

Zimring doesn't discuss this and, because this is a review, neither will I (regrettably). But it's pretty cool, and if you aren't required by exams to actually fiddle with the numbers, the underlying explanation is interesting enough to be worth looking up.

About the book as a whole

There's little here you won't find in other books; there's a high probability (heh) that you'll have heard of cherry picking, confirmation bias and the rest before. Zimring has a slightly new perspective on them (seeing them as fractions), and applies it in various domains. Some details were novel to me, and some observations were interesting, but all in all I feel like the topic has unrealized potential.

Many thanks to NetGalley and Columbia University Press for providing me with a free ARC in exchange for an honest review.

[Jeff · Rating 5 out of 5]

Solid Exposition Of Its Premise. This book is pretty well exactly what its title says it is: an examination of cognitive biases, with fractions as the common access point to all of them. Thought of in this manner, the book is solid, though on its overall points - many political, including repeated attacks on the 45th President of the United States - your mileage will vary considerably. Indeed, on many of the issues Zimring examines, his overall consideration of the issue at hand is actually limited by his devotion to fraction-based thinking, at least within the confines of this text. Ultimately, this book is more a rare take on cognitive biases than anything truly mathematical, and the math here really is simplified such that pretty well anyone capable of reading the text itself can follow the math easily enough. The bibliography clocks in at around 20% of the overall text, which is close enough to the average of similar texts in my experience to be acceptable. Recommended.

[Steve · Rating 5 out of 5]

I enjoyed this book. The book is really about biases but uses fractions as a discussion point. There is not a lot of math in the book; what little there is, is explained very well. The book has a conversational tone and the author shares some personal perspectives, which I usually appreciate. The writing is quite compelling, with some humor and clever wording. The endnotes are also worth reading, however, the endnotes that contain clarifications on the text are mixed in with the endnotes that only contain references and citations. I find this annoying in many books. I much prefer that clarifications or explanations appear as footnotes on the same page as the content. I quickly stopped checking the endnotes but scanned through them at the end and many were worthwhile reading. While the pacing of the book was quite good overall, I found that the book slowed down when the discussion turned to cognitive psychology. Nonetheless, the book was so good overall that it still merits 5 stars. Thank you to Netgalley and Columbia University Press for the advance reader copy.

[J Earl · Rating 5 out of 5]

Partial Truths: How Fractions Distort Our Thinking by James C Zimring is a very accessible look at the many ways we, as humans, undermine our own thinking and how many of these are related to our misuse or misunderstanding of fractions.

First, this is not a book of mathematics, so don't let that keep you away. Many of the ideas addressed have become widely known, or at least widely spoken of, such as confirmation bias. Zimring presents these in plain language and shows how they can be seen as part of our issue with fractions. Whether we only look at the numerator or denominator, whether we keep changing the denominator in order to keep the view we have, even whether we even realize we should be looking at a fraction (or relationship) rather than just a single number.

If you've read some of the scholarship, or even the popular books, on these topics you will recognize many of the broader topics. Whether your reading has been under behavioral economics, social psychology, or any of the other fields that have looked at our cognition, Zimring brings a fresh approach by working through the idea of the fraction. Many are literally about fractions while some are more relationships spoken of as fractions. But then, what are fractions other than relationships?

I would recommend this to anyone who wants to know more about how we all arrive at, and sometimes irrationally maintain, our viewpoints. To the extent that we can become a little less irrational we have to start from understanding what we are doing. This book goes a long way toward clearly illustrating the errors (not all of which are detrimental) in our thinking.

Reviewed from a copy made available by the publisher via NetGalley.

[Frank Hood · Rating 3 out of 5]

A flawed but interesting work.

Zimring's heavily footnoted work (maybe he should have considered a study showing how few people even look up footnotes, much less read the sources referred to) makes a few interesting points.

His most incredible claim is that one company flopped in selling a 1/3 pound burger because people thought it was smaller than a 1/4 pound burger, and that claim is backed up only by a New York Times Magazine article.

He elaborates his arguments with many left-wing anecdotes as if the only politicians he could find distorting the truth were Trump and Bush. Perhaps this is explained by his assertion that, "When given the choice between one-sided arguments versus those that explore both sides of the issue, people tend to consider the one-sided argument as the stronger argument and have more confidence in its conclusions."

He could have used more telling arguments such as:

“CO2 levels will double in 30 years from 400 parts per million to 800 parts per million!”
“CO2 levels may rise from .004% of Earth's atmosphere to .008% of Earth's atmosphere!”

Global warming believers cite the first claim and dismiss the second, and global warming skeptics cite the second and dismiss the first, but they both are saying exactly the same thing!

Despite his flaws of citing arguments that only appeal to one side of the political spectrum, he does have some interesting things to say about human perception of numbers, and he notes some serious problems of current scientific research especially in our era of both self-serving bias and big data methods that make it much easier to make unfounded claims, giving us all plenty of reason to doubt the experts.

He also makes interesting points about humanity evolving as a tribe, so that tolerance for dissenting views is important to the tribe discerning the truth eventually, even if the determination is that the minority view is nuts.

[Maurizio Codogno · Rating 4 out of 5]

Frazioni e scienze cognitive

Se siete tra coloro che hanno paura della matematica, e già alla parola "frazioni" sentono la pelle d'oca, non preoccupatevi: di matematica in questo libro ce n'è davvero poca. La storia delle frazioni è solo il fil rouge del libro, nel senso che ogni tanto Zimring prende il tema di cui sta parlando e dice qualcosa del tipo "per alzare la probabilità che percepiamo basta far crescere il numeratore e calare il denominatore della frazione corrispondente, e per abbassarlo si fa il viceversa": e alla fine butta proprio via l'approccio probabilistico passando a quello frequentista ("su 1000 persone sane ce ne sono 10 che risultano comunque positive a un test per una certa malattia"). Il libro si occupa in realtà di scienze cognitive: detto in maniera terra terra, di come noi sbagliamo a percepire le cose, assegnando probabilità completamente errate agli eventi intorno a noi. La prima parte elenca i modi in cui possiamo sbagliare: errata suddivisione dei casi, confirmation bias (cerchiamo involontariamente le cose che corroborano l'idea che ci siamo fatti a priori), cherry-picking (scegliamo apposta le cose che ci fanno comodo); la seconda parte presenta vari esempi - tutti ben noti a chi si occupa di questi temi - di questi errori all'opera. La terza e ultima parte cambia le carte in tavola ed è la più spiazzante, perché mostra come le percezioni errate possono dare un vantaggio competitivo (basta non arroccarcisi...) Ci sono anche alcune proposte di come poter ovviare al fatto che il nostro cervello è tarato in un modo non probabilistico, ma Zimring è il primo a dire che c'è ancora molto da fare in questo campo. Molto utile secondo me l'avere un riassunto alla fine di ogni capitolo, che può servire al lettore per verificare se ha compreso davvero quanto stava leggendo.

[Scott Humphries · Rating 5 out of 5]

This book contains concise and useful discussions about a number of cognitive and metacognitive issues centered loosely around how humans misperceive fractions, percentages, rates, probabilities, and frequencies. For example, Zimring’s Chapter 10 ("The Hard Sciences") includes one of the better illuminations of the use and abuse of "statistical significance" that I've read. I recommend it.