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April 30th, 2022 (Permalink)

Against Gatekeeping & "Democracy Dies in Darkness"

April 21st, 2022 (Corrected: 4/30/2022) (Permalink)

Credibility Checking, Part 4: Ballpark Estimation

In the previous installment of this series on how to check claims for credibility1, we needed to know how many teenaged drivers there were in the United States in 2007, which is the sort of very specific statistical fact that you are unlikely to be able to find quickly or at all. Rather than giving up, we estimated the number based on a couple of other facts that we knew or were able to look up quickly: the population of the country and the average life span in that year. Thankfully, in order to check a claim for credibility, we seldom if ever need precise numbers, and ballpark estimates will serve.

What is a "ballpark" estimate? It is an estimate that is "in the ballpark", as we say, that is, it's not spot-on but it's close enough. The use of "ballpark" as a name for this type of estimation appears to have come from rough estimates of the number of spectators in a ballpark watching a baseball game2. We frequently don't need an exact number, which is fortunate since an exact number is often not available. In such situations, we can make do with a ballpark estimate, which I define as one that has the same order of magnitude as the number estimated.

What are orders of magnitude? They are tens, hundreds, thousands, tens of thousands, and so on, but also tenths, hundredths, thousandths, and so forth. For an estimate to be "in the ballpark" means that the estimate and the number estimated both belong to the same order of magnitude.

The best way to learn how to successfully guesstimate numbers is to see examples of how to do so, and then try your own hand at it. You may be surprised at how easy it is―and fun!

An Example

What percentage of the American population dies in automobile accidents?3 This is a statistic that you're unlikely to just happen to know. However, you may well know enough to make a good estimate. If you'd like to try answering this question yourself, stop reading here and do so; then come back and read on to see how I estimated it.

What is it that you need to know to make such an estimate? As discussed in the previous part, a percentage is a type of ratio. To figure the ratio, you need two numbers: a numerator and a denominator. In this case, the numerator is the number of Americans who die in car crashes and the denominator is the total number of Americans who die―since everyone dies, this is simply the total population. Since we're doing a guesstimate, we don't need precise statistics for either of these numbers.

As I've mentioned in previous entries4, the population of the United States is a good landmark number to remember, and it's easy: the current population is approximately a third of a billion. So, that's our denominator.

The numerator is trickier: how many people out of that 330 million population die in traffic accidents? I've also mentioned in previous entries that the number of Americans who die in traffic accidents per year tends to be around 30-40K5; let's use the midpoint, 35K. However, this is not our numerator, since that ratio gives the risk of being killed in a car accident in a year. Luckily, the average American lives to be about 80 years old6, so the numerator we want is 35K × 80, which is approximately three million. Now, the math is so simple that you can do it in your head: three million is about 1/100th of 330 million, so the percentage of Americans who die in automobile crashes is about 1%.

Does the Estimate Hold Up?

Now let's compare our estimate with the one given by Weinstein & Adam in Guesstimation3. They actually formulated the question as: "What fraction of American deaths are caused by automobiles?", but any fraction can be easily turned into a percentage1.

So, for the numerator they used 40K Americans "killed on the roads each year", which they multiplied by 75, life expectancy in 20087, to get three million, which represents the total number of Americans who die in a car crash at some age. That's our numerator.

The denominator for the fraction is 300 million―which was the approximate population of America in 2008 when the book was published8. Again, you can do the math in your head, and the result is the same as our estimate above, which is evidence that it is a good estimate.

Another way to test a ballpark estimate is to compare it to your own experience. If one percent of us die in car accidents, what is the chance that you have known someone who died that way. How many people have you known who have died? Obviously, this depends upon how old you are and how many people you have known. Have you known as many as a hundred people who have died? If so, then you probably knew at least one automotive fatality.

I don't know about you, but I have a hard time remembering all of my relatives, friends, and acquaintances who are gone. Still, it doesn't seem to be close to a hundred, but perhaps around half that many. Nonetheless, one close relative of mine died in a car crash, and a friend of a friend also perished that way. So, if anything, it seems as though our estimate may be on the low side.

Finally, how does our estimate compare to official statistics? The Centers for Disease Control and Prevention (CDC) collect statistics on "leading causes of death", among them "Accidents (unintentional injuries)". According to the most recent data, a total of 3,383,729 Americans died in 20209. However, the CDC does not break out the numbers for accidents involving cars.

Thankfully, another alphabet agency, the National Highway Traffic Safety Administration (NHTSA), collects statistics on automotive deaths. According to an NHTSA press release, 38,680 Americans are estimated to have died in automobile accidents in 202010. This, of course, is close to our midpoint estimate of 35K. Again, you can do the math at a glance: 39K is about 1% of 3.4 million; more precisely, it's 1.1%.

Clearly, by these various measures, our estimate did not just hit the dartboard, it hit the bullseye.


One surprise of this exercise is just how many Americans die in automobiles. Why are we not more alarmed by this fact? Why don't we hear more about driving safety? I suspect that at least part of the reason is that we have grandfathered automotive accidents into our risk assessments. It's almost as if such accidents are simply "acts of God" that we have no control over, so we just accept that one-percent of us will die in a car accident, even though most are preventable.

Finally, this exercise shows that by combining what we know with what we can research quickly―using tools such as Wolfram Alpha―and using only simple math, we can make surprisingly accurate estimates. Such estimates are useful, as we have seen in previous installments, for checking the credibility of claims made by activists, advertisers, and politicians.


  1. Credibility Checking, Part 3: Ratios, Rates & Percentages, 3/27/2022.
  2. Webb Garrison, Why You Say It (1992), p. 175.
  3. This example is based on the second question in section 11.1 of Lawrence Weinstein & John A. Adam's, Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin (2008). This is a useful book for practicing ballpark estimates, though it's heavily skewed toward ones involving physics, perhaps because one of the authors is a physicist.
  4. For instance: A trillion here, a trillion there, and pretty soon you're talking about real money., 9/29/2021.
  5. Credibility Checking, Part 2: Divide & Conquer, 2/4/2022.
  6. "What is life expectancy in the United States?", Wolfram Alpha, accessed: 4/19/2022.
  7. "What was life expectancy in the United States in 2008?", Wolfram Alpha, accessed: 4/20/2022.
  8. "What was the population of the United States in 2008?", Wolfram Alpha, accessed: 4/20/2022.
  9. "Deaths and Mortality", Centers for Disease Control and Prevention, accessed: 4/21/2022.
  10. "2020 Fatality Data Show Increased Traffic Fatalities During Pandemic", National Highway Traffic Safety Administration, 6/3/2021.

April 8th, 2022 (Corrected: 4/9/2022) (Permalink)

When the Sleepers Wake

The Agency for Counter-Terrorism (ACT) has discovered the existence of sleeper agents of a foreign power. In addition to discovering the existence of the sleepers, the ACT recovered a secret document outlining the strict rules used by this power in its sleeper agent operations.

Each of the sleeper agents is assigned to two cells―that is, groups of agents that are known to each other. Any two cells have one, but only one, agent in common. In order to protect the sleeper operation, the agents are kept in the dark about the existence or membership of any cells to which they don't belong. In this way, if an agent is discovered by the enemy, only two cells will be in danger of being lost.

In addition, the secret document revealed that there are currently four sleeper cells. However, what ACT would most like to know is just how many sleeper agents they should look for. Can you help? Can you determine from the above information the number of sleeper agents?

Acknowledgment: Thanks to Lawrence Mayes for pointing out an error in the original wording of the puzzle.

April 2nd, 2022 (Permalink)

Infer or Imply?

Imply, Infer. Not interchangeable.1

While researching the previous entry2 in this series on confusible words, I found a news story with the following headline:

Browns refute tanking claims by former coach Hue Jackson3

This is, of course, an example of the mistake of using "refute" to mean "deny" discussed in that entry, but the same article has the following sentence: "Jackson, who is now coaching at Grambling, made several posts on Twitter inferring that he received bonus payments from Browns owner Jimmy Haslam during his two-plus seasons with the team."

To infer is to draw a conclusion based on some kind of evidence; whereas, to imply is to state something that either entails or suggests a conclusion. A claim is usually said to imply something when it does not state it outright but it can be inferred from what is stated. People can imply through the statements they make, because the statements themselves imply, but only people can infer.

Is it Jackson's posts that are supposed to have inferred that he received the payments, or Jackson himself? Jackson's posts may have implied that he received such payments, but they cannot infer it. Moreover, Jackson himself surely knew directly whether he had been paid and would not need to infer it. Instead, either Jackson, his posts, or both implied that he received the payments.

Unlike the mistake of using "refute" to mean "deny", all of the books on usage that I've checked that take a position condemn using "infer" to mean "imply"4. As Bill Bryson writes after giving an example:

According to nearly all authorities, on both sides of the Atlantic, the word there should be implied, not inferred. Imply means to suggest…. Infer means to deduce…. A speaker implies, a listener infers. The distinction is useful and, in careful writing nowadays, expected.5

In the previous entry2, I discussed the fact that some dictionaries are now treating to deny as a standard meaning of "refute", apparently because the word is now used in this sense so often that it's no longer a mistake. However, this doesn't appear to be the case with "infer" used to mean to imply. For instance, the Cambridge Dictionary, which is my usual online source for word meanings, defines "infer" in the traditional way6, and includes a link to a usage note describing the "imply" sense as a "typical error"7.

When I checked it earlier this year, the online Merriam-Webster Dictionary gave "imply" as a second "essential meaning" of "infer"8. However, it labeled it "informal", which I guess was its euphemism for "incorrect". The entry has since been revised to remove "imply", but replaces it with "hint, suggest" as a fourth possible meaning, and the "informal" warning has disappeared9. I don't know whether this is progress or not.

In addition, the entry includes a very confusing "Usage Guide"10: how this brief note is supposed to guide usage is unclear since it gives no guidance. Instead, it claims that the use of "infer" to mean "imply" began centuries ago and continued until some time after World War I. It suggests that the distinction between the two words that developed in the last century was based on a confusion. However, given that Merriam-Webster seems to jump on every linguistic change bandwagon that comes along, why does it resist one that has been around for a century? Even if the usage note is correct that the change was based on a mistake, why should that matter to a descriptivist dictionary, since a lot of linguistic changes are based on mistakes?

All of this shows how capricious these lexicographical decisions are. If the dictionary makers were just frankly prescriptive in their definitions, you could disagree with their recommended usage and argue with it. Instead, the bogus claim that they are just describing the way the language is used is supposed to end all disagreement and prevent you from rejecting bad linguistic advice.

As is the case for many of these confusible pairs, the misuse usually goes only one way, that is, one word is used when the other should be. Typically, "infer" takes the place of "imply", though I recently came across an example going in the opposite direction in a book discussing an allegedly libelous claim: "It would have remained for the court to decide whether any such allusion could have reasonably been implied by any reader of NOW! magazine….11" It must have been the reader who would have inferred the libelous allusion, and the magazine that implied it.

I'm sure that readers of The Fallacy Files are smart enough to infer what my position is on this issue without my having to imply it.


  1. William Strunk, Jr. & E. B. White, The Elements of Style (4th Edition, 1999), p. 49.
  2. Refute or Deny?, 3/2/2022.
  3. Tom Withers, "Browns refute tanking claims by former coach Hue Jackson", Associated Press, 2/2/2022.
  4. Omitting Strunk & White and Bryson, they are:
    • Harry Blamires, The Cassell Guide to Common Errors in English (1997):
      To 'infer" something is a matter of concluding, not of conveying. The speaker implies something, the listener infers something.
    • Michael Dummett, Grammar & Style for Examination Candidates and Others (1993), p. 93:
      Only a person can infer; one statement can imply another, which the speaker also implied by making the first statement. To infer something is to draw a conclusion; to say something from which that conclusion follows or which the speaker means his hearers to take as following is to imply the conclusion, not to infer it.
    • Mignon Fogarty, Grammar Girl's 101 Misused Words You'll Never Confuse Again (2011), under "Imply". This is the most recent book I've checked. Fogarty writes: "The incorrect use of infer to mean imply is so common that in a decade or so it may be considered standard, but for now, careful writers and speakers continue to make a distinction." It's been slightly over a decade since this was written, and if careful users of the language continue to make the distinction it may last for another decade.
    • Robert J. Gula, Precision: A Reference Handbook for Writers (1980), p. 216:
      You imply when you suggest something. You infer when you draw a conclusion from someone else's words.
    • Adrian Room, The Penguin Dictionary of Confusibles (1980):
      In fact the two are indeed often used interchangeably, but strictly speaking: 'imply' means 'express indirectly'…and 'infer' means 'derive by reasoning', 'deduce'. … So if I 'imply' that you are deceitful, I say so indirectly; if I 'infer' that you are deceitful, I gather that you are from what you say or do, or from what I hear about you.
    • Harry Shaw, Dictionary of Problem Words and Expressions (Revised edition, 1987):
      To imply is to suggest a meaning only hinted at, not explicitly stated. To infer is to draw a conclusion from statements, evidence, or circumstances.
  5. Bill Bryson, Bryson's Dictionary of Troublesome Words: A Writer's Guide to Getting it Right (2002).
  6. "Infer", Cambridge Dictionary, accessed: 4/1/2022.
  7. "Imply or infer?", Cambridge Dictionary, accessed: 4/1/2022.
  8. "Infer", Merriam-Webster Dictionary. This is the Internet Archive Wayback Machine's cache of the page for 1/22/2022.
  9. "Infer", Merriam-Webster Dictionary, accessed: 4/1/2022.
  10. "Infer vs. Imply: Usage Guide", Merriam-Webster Dictionary, accessed: 4/1/2022.
  11. Chapman Pincher, The Secret Offensive (1986), p. 45.

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