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April 21st, 2020 (Permalink)

New Fallacy: Over-Extrapolation

I've added a new fallacy to the files, which is the first one in quite a while. It's an old one in the sense that I've been aware of it for many years, and have alluded to it a few times in this weblog1. I failed to create an entry for it because it's a statistical fallacy and seemed perhaps too esoteric and technical. However, I've come across more than one recent example that involves the current coronavirus epidemic, so the fallacy has become a timely one. Check it out.

Fallacy: Over-Extrapolation

Update (4/25/2020): Here's an example that I probably would have used in the entry for the above fallacy, but I just came across it:

Dr. Ezekiel Emanuel, chair of the department of medical ethics at the University of Pennsylvania, warned Friday [March 27th, 2020] on MSNBC's "Morning Joe," that based on the current rate of spread, there will likely be 100 million Americans infected by the COVID-19 virus in four weeks. … "Right now, if you look at the numbers, we probably have a million COVID-19 cases in the country. And if this is doubling every three to four days, that means that we'll have 100 million people who have COVID-19 in about four weeks, and that's a frightening thought."2

That is a frightening thought, and I bet many people who heard it were indeed frightened. Emanuel made this claim just four weeks ago yesterday, and there were still less than a million confirmed cases yesterday3. So, Emanuel was off by two orders of magnitude.

This is another example of exponential extrapolation, which is even more dangerous than the linear kind. In the case of epidemics, there is a period of growth near the beginning when the number of cases may indeed be growing exponentially, but such growth cannot continue for long. As the epidemic continues, the doubling time increases. The doubling time for COVID-19 deaths in the United States was around three days on the first of this month, but today it is over two weeks4.

If you do the math on this claim, assuming that there were one million cases on the 27th of last month, doubling every three or four days, then there should have been 128-512 million cases yesterday. I assume that Emanuel must have taken the slower doubling rate of every four days, and then rounded down to the nearest hundred million. Why he included the faster doubling rate of every three days, when such a rate would have hit 100 million in the middle of this month rather than near its end, may be because that was the doubling rate at the time. However, at that rate the number of cases would have exceeded the entire population of the country a few days ago, which should have warned him against extrapolating so far into the future. Perhaps he realized that the rate would slow and tried to be conservative by using the four-day rate.

Note that Emanuel is a medical ethicist: how ethical is it to go on national television and unnecessarily frighten many people? As ethical as shouting "Fire!" in a crowded theater?


  1. See, for instance: New Book: Proofiness, 8/26/2010.
  2. Tim Hains, "Ezekiel Emanuel: U.S. Will Have 100 Million Cases Of COVID-19 In Four Weeks, Doubling Every Four Days", Real Clear Politics, 3/27/2020.
  3. "Cases of Coronavirus Disease (COVID-19) in the U.S.", Centers for Disease Control and Prevention, accessed: 4/25/2020.
  4. "Doubling Time of Coronavirus Deaths and Pandemic Control", Princeton Election Consortium, accessed: 4/25/2020.

April 17th, 2020 (Updated: 4/18/2020) (Corrected: 4/19/2020) (Permalink)

How many ventilators do we need?

Last month, we saw a rather silly example of innumeracy in the news media1. As far as I know, that incident didn't do any harm, except to the dignity of those involved. However, during the current epidemic we are seeing innumeracy that is doing real harm.

For instance, one reason given for some of the most draconian restrictions on public life adopted by the government has been the fear that, unless the spread of the coronavirus is slowed, we will run out of ventilators. Ventilators are machines that are used to breathe for patients who can't breathe on their own, and for those who need them can be life-savers.

A few weeks ago, some in the news media claimed that a projection showed that the United States would need one million ventilators to deal with coronavirus patients, whereas only 160-200 thousand were currently available. The claim seems to have originated with the august New York Times, and spread from there2. However, the projection estimated that a million patients would need ventilators over the course of the epidemic, not that a million machines would be needed. Since a single ventilator may be used by multiple patients over the course of the epidemic, it's possible that the available machines would be sufficient for a million patients. This sort of innumerate misreporting does real harm by unnecessarily frightening people, and may lead to excessively harsh public measures and counter-productive diversions of resources.

According to the Institute for Health Metrics and Evaluation (IHME) projection, we've already passed the peak demand for ventilators on the fourteenth of this month, and on that day less than 17,000 ventilators were needed3. This, of course, is about a tenth of the low-end number of available ventilators given above. So, what went wrong with the original estimate?

A good candidate is the initial estimate that one million patients would need ventilators over the course of the epidemic. As I write this, less than a million total cases of COVID-19 in the United States have been confirmed4, and only a fraction of that million will need ventilator support. The estimate in question came from the following passage in a report by the Society of Critical Care Medicine:

Estimates of hospitalized patients requiring critical care and mechanical ventilation: … A recent AHA webinar on COVID-19 projected that 30% (96 million) of the U.S. population will test positive, with 5% (4.8 million) being hospitalized. Of the hospitalized patients, 40% (1.9 million) would be admitted to the ICU, and 50% of the ICU admissions (960,000) would require ventilatory support. Such projections, however, are gross estimates. Some assumptions underlying these projections are uncertain, and the pacing of a large outbreak would influence whether ICU resources in isolated locations or nationally are severely taxed over many months or quickly overwhelmed over a shorter period. Additionally, COVID-19 patients may remain mechanically ventilated for indeterminate periods of time, with some developing prolonged or chronic critical illness requiring the extended use of ICU beds, ventilators, supplies, and trained clinicians.5

As you can see, the report itself makes it clear that the estimates are uncertain because the underlying assumptions that they are based on are uncertain. The estimate that nearly a million patients would need ventilators was based on the initial estimate that 30% of the American population would become infected, 5% of those infected would be hospitalized, 40% of the hospitalized patients would need intensive care, and half of those needing such care would require a ventilator*. Each of these percentages is uncertain, so with each step of the calculation, uncertainty is piled upon uncertainty. Let's examine the percentages individually:

  1. What percentage of the population will be infected with the virus? We don't know. The estimate of 30% appears to have come from one public health expert in a web seminar for the American Hospital Association conducted on February 26th of this year6. How was the estimate arrived at? I haven't been able to find out. I'm sure the expert gave his best guess, but this was near the beginning of the epidemic, and it's clear now that his ventilator estimate was way off. What I don't understand is why there haven't been attempts to refine those estimates in the month and a half since.
  2. What percentage of those infected will need hospitalization? We should have a good handle on the number of people hospitalized, but since we don't know the answer to the previous question, we don't know what percent it is of those infected. This number has the same problem as the death rate: we can count how many people die or are hospitalized with the virus, but both rates are fractions and we have the numerators but lack the denominator. Where did the estimate that 5% of those infected would be hospitalized come from? I haven't been able to find out.
  3. What percentage of those hospitalized will need intensive care? Finally, we have a statistic that we can do more than guess at: a recent source claims that 16-20% of hospitalized patients are admitted to intensive care7, which is half the original estimate.
  4. What percentage of those in intensive care will need ventilators? Unfortunately, I haven't been able to find any more recent estimate of this number.

Any estimate based on guesses is going to be itself guesswork. Despite this, some news media are still using the original, six-week old estimate, though it's now obviously outdated. For instance, the article that provided the answer for the third question, above, uncritically repeats the original February estimates, even though it was published just today7.

Update (4/18/2020): I've been trying to figure out where in the following graphic the estimate discussed above fits:

Garbage Math

It's not in the graphic, but in the same spirit I think it's:

Garbage × Precise Number/Garbage × Precise Number × Possible Garbage =
Misleadingly Precise Garbage

Source: Randall Munroe, "Garbage Math", XKCD, accessed: 4/18/2020.

*Correction (4/19/2020): This sentence originally began: "The estimate that nearly a million ventilators would be needed…", which is incorrect since the estimate itself was how many patients would need ventilators.


  1. Beyond Innumeracy, 3/7/2020.
  2. Jeryl Bier, "How Misinformation About the U.S. Needing '1 Million Ventilators' Spread", The Dispatch, 4/6/2020.
  3. "COVID-19 Projections", Institute for Health Metrics and Evaluation, 4/17/2020.
  4. "COVID-19 Tracker", Bing, accessed: 4/17/2020.
  5. "United States Resource Availability for COVID-19", The Society of Critical Care Medicine, 3/19/2020.
  6. Lydia Ramsey, "One slide in a leaked presentation for US hospitals reveals that they're preparing for millions of hospitalizations as the outbreak unfolds", Business Insider, 3/6/2020.
  7. Erin Schumaker, "What we know about coronavirus' long-term effects", ABC News, 4/17/2020.

Maths on the Back of an Envelope
April 14th, 2020 (Permalink)

BOTEC1, the Book

Title: Maths on the Back of an Envelope

Subtitle: Clever ways to (roughly) calculate anything

Author: Rob Eastaway

Date: 2019

Quote: Once a statistic…is out there, published in a newspaper, quoted on a website, it becomes "fact", and it can be re-quoted so often that the source may go unquestioned and very quickly be forgotten altogether. It's an important reminder that the majority of statistics published anywhere are estimates, many of them worked out on the equivalent of the back of an envelope. When back-of-envelope calculations produce a very different answer from the one that's been put forward, it doesn't necessarily mean that the estimate is wrong. It means rather that the published figure deserves more scrutiny.2

Comment: I've been cheerleading for many years for the value of "back of the envelope calculations", so I'm excited to see a new book that teaches how to do them. In recent years, I usually speak of "sanity checks" or "plausibility checks" instead of BOTECs, since the purpose of the calculation is to check the plausibility of a number. The phrase "back-of-the-envelope" calculation really refers to estimates that can be used to check if a number is reasonable. When you use a BOTEC to check a number that you meet "published in a newspaper" or "quoted on a website", it doesn't prove that the number is right or wrong. However, if the estimate and the published number diverge by a lot, then "the published figure deserves more scrutiny", as Eastaway puts it, above.

Knowing how to do a BOTEC is a very useful critical thinking skill that is, unfortunately, seldom taught in critical thinking classes or books, though I've seen some signs in recent years that it's catching on. If you're at all wary about learning it because it involves math, and you're scared of math, don't worry! BOTECs seldom require more than the most basic mathematics: adding, subtracting, multiplying, and dividing. If you know how to do these operations, whether in your head, on paper, or with a calculator, you can do a BOTEC. Not only that, but BOTECs are fun! They're rather like puzzles and, unlike most mathematical tasks, there are no right or wrong answers!

Here's how Eastaway describes the book's structure:

I've divided the book into four sections. In the first section, I explore how precise numbers can be misleading, and why it's good not to be entirely dependent on a calculator. The second section includes the arithmetical techniques and the other knowledge that is an essential foundation if you want to embark on back-of-envelope calculations. This includes a refresher on how to do arithmetic that you may not have needed to practise since you left primary school, as well as short cuts that you probably never encountered there. The rest of the book shows how to use these techniques to tackle problems, from everyday conversions, to more serious issues like helping the environment. And at the end, there is a collection of so-called Fermi questions: quirky and esoteric challenges to come up with a reasonable answer based on very little hard data.3

Rob Eastaway is the author of some other books on math, some on puzzles, and one on cricket, though I'm sorry to say I haven't read any4. This book appears to have been published in the United Kingdom5, and I don't know whether or when it will appear in an American edition, but I think you can still get copies here in the states, though they may take longer to arrive. What better way to spend your quarantine time than learning how to do back-of-the-envelope calculations? While you wait for the book to arrive, you can try your hand at several such problems from earlier entries in this weblog, see:


  1. Back Of The Envelope Calculation.
  2. "Prologue", emphasis in the original.
  3. "Prologue".
  4. Except for the one on cricket: I'm glad I haven't read that.
  5. Why else use the plural "maths" in its title? For one thing, "maths" is hard to say, and it sounds like you're lisping. In American English, "math" is short for "mathematics", and "cricket" is a type of insect, not a sport.

April 12th, 2020 (Permalink)

Don't Put All Your Easter Eggs in One Basket

The Easter Bunny was hopping down the bunny trail on Easter morning with a basket full of colorful eggs. E.B. met a boy and gave him half of the eggs in the basket plus half an egg. Then, E.B. met a girl and gave her half of the remaining eggs in the basket plus half an egg. Then, E.B. came across another boy and gave him half of what remained in the basket together with half an egg. Finally, E.B. met another girl and gave her half of what was left in the basket plus half an egg. E.B.'s basket was now empty and not a single egg had been broken!

How many Easter eggs were in the Easter Bunny's basket at the start of Easter morning?

Richard Jewell
April 6th, 2020 (Permalink)

Movie Review:
Richard Jewell

In addition to reading, another thing you can do in these times of anti-social distancing is to stay home and watch a movie. The recent film Richard Jewell is out on DVD, and perhaps available from some streaming services. I mentioned it previously in connection with the media frenzy and misreporting surrounding the case that it dramatizes1.

This is only the second movie review I've ever written here2, and the first for a fiction film, though this one is "based on a true story". I suppose that most people are now familiar with at least the broad outlines of the Jewell case, so I won't bother with the usual plot synopsis that you find in movie reviews. If you're unfamiliar with the case, check out my previous entry on it1.

My interest in this movie arose out of an interest in the Jewell case, and the light it sheds on the news media. In this case, most of the damage done by media malpractice was to Jewell and his mother. Currently, millions of people are suffering due not to the new virus, but to the media's misreporting of it. However, that's a subject for a later time.

Unfortunately, Richard Jewell is really two movies in one, and they mix like oil and water. The movie is at its best in portraying the "good guys"―Jewell, his mother, and their lawyer―but the "bad guys"―FBI agents and a local reporter―are portrayed as cartoonish movie villains.

The worst treatment is reserved for the local reporter, Kathy Scruggs. I didn't believe the portrayal for a second, but I don't think this is the fault of the actress who played her, but that of the screenplay. To call it a "caricature" would be unfair to caricatures. Caricatures are exaggerated but they're still supposed to look like the person portrayed, whereas this character doesn't look like any actual person. Instead, she is a sort of cross between a film noir femme fatale and the female reporter from His Girl Friday. Of course, I never met the real Scruggs, so I suppose it's possible that she was really like this, but I find it difficult to believe.

I can forgive the unrealistic portrayal of the FBI agent characters because they were given phony names. What's unforgiveable is that Kathy Scruggs was a real person whose real name is used in the movie. Since Scruggs died, she cannot defend herself against this defamation, and the movie ends up doing to her what it is seemingly exposing as having been done to Jewell. Would it have been alright if the media had waited until Jewell was dead before smearing him? Presumably, the FBI agents in the film are given fictional names, because the real-life agents are still alive and would sue for libel if they were named, whereas Scruggs can't sue. The makers of this movie sure don't deserve any awards for courage.

The only excuse that I can see for the portrayal of Scruggs would be if it were well-established that she acted in real-life the way the character does in the movie. However, based on everything I've read, there's no evidence that she seduced an FBI agent in order to get her scoop on Jewell3.

Another thing that bothered me about the movie is the frequent foul language, mostly from the FBI agents but some from the Scruggs character, but then I'm old-fashioned. As far as I can tell, this movie earned its R-rating entirely on the basis of language: there is no nudity, and only one mildly sexual scene when the Scruggs character seduces the FBI agent; the only violence is when the bomb explodes, and you see a few people covered in phony blood. It could've easily been a PG-13 or even PG, if only the language had been toned down. I doubt that FBI agents talk like the fictional ones in this movie, but then I don't know any real ones. However, if you're like me and hearing the F-word in every other sentence bugs you, then you may want to avoid it.

I've been very critical of this film, which deserves it, but despite all that I recommend it, at least if you can stomach the language. Paul Walter Hauser starts out portraying Richard Jewell as a buffoon, but by the end of the movie he has become a believable and sympathetic person. Kathy Bates' performance as Jewell's mother is both believable and moving from beginning to end. The other actors are generally fine, even when their characters are written as cartoonish, cussing baddies.

There's nothing wrong with this movie that a better script wouldn't have fixed. If only the FBI agents were portrayed as flawed human beings instead of comic book villains, and the script had not taken a cheap shot at a dead woman, this would have been a great film instead of a merely good one. If you do decide to see it, however, keep in mind that it's not history, it's Hollywood.


  1. See: Richard Jewell and "The Voice of God", 12/31/2019
  2. For the first review, see: Movie Review: Best of Enemies, 2/23/2016
  3. Julie Miller, "The Richard Jewell Controversy―And the Complicated Truth About Kathy Scruggs", Vanity Fair, 12/13/2019

Recommended Reading
April 2nd, 2020 (Permalink)


If you're trapped inside during this double pandemic of coronavirus and fear of it, you might as well do some reading. I usually do these "Recommended Reading" lists at the end of the month but, like last month, I'm moving this one up. Hopefully, by the end of this month, you won't be stuck inside your house or apartment, so you won't need this list so much. Even now, I suggest reading something completely unrelated, such as the rest of The Fallacy Files! However, if you must keep reading about the virus, then at least read some articles that will put it into perspective.

Note that I do not necessarily agree with or endorse every claim made in the following readings. Moreover, I am definitely not trying to strike a balance between pro-panic and anti-panic articles: if you want to read scare-mongering stories, just go to any mainstream media site and you can get your fill.

Here's an article on why you shouldn't obsess about the current virus, and how to stop doing so:

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