Faster Than a Speeding Bullet
Our New Book this month1 reminded me of a, hopefully, interesting example. It occurs in the following passage, taken from a book about the Warren Commission's report on the assassination of President Kennedy, discussing the Zapruder film of that event:
The total elapsed time from the end of [frame] 312 to the end of [frame] 313 has reached the seemingly infinitesimal figure of 1/18th of a second. …[A] bullet traveling approximately 1,979 miles per second, the speed of the fatal bullet fired by Oswald, theoretically could cover almost ninety miles in that 1/18th of a second.2
When I first read this passage, it made no particular impression on me. However, I later returned to the book looking for specific information―which I never found―and re-reading the passage, I was struck by the speed given for the bullet: almost 2,000 miles a second sounds extremely fast. Since there are 3,600 seconds in an hour, this means a speed of over seven million miles an hour! Of course, bullets are fast, but are they that fast? "Magic bullet", indeed!
Unfortunately, when it comes to speed, there is a dearth of landmark numbers to use for comparison. In particular, I had little idea how fast bullets fly. Most of us are familiar with speeds of less that a hundred miles per hour, but beyond that our experience is limited to airplanes. How fast is a commercial jetliner? Surely, faster than a hundred MPH. Faster than 200 MPH? Faster than 500 MPH?3 Faster than a bullet?
Other than the speeds of automobiles, planes, and bullets, there are also natural phenomena whose speeds might provide landmarks, specifically, sound and light. Off hand, I didn't know exactly what the speed of sound is4, though I did know that it is faster than airliners5. Are bullets faster than sound? Of course, bullets aren't faster than light because nothing is6. I also knew that the speed of light is approximately 186,000 miles per second, but there is an enormous gap between the speed of sound and the speed of light: a vast desert devoid of numerical landmarks.
Now, I would have bet that the speed of the fastest bullet ever fired is not even close to the speed of light, and much closer to the speed of sound. However, 2,000 miles per second is only slightly more than 1% of the speed of light. Nonetheless, the more I thought about it, the harder I found it to believe that a bullet flies anywhere near that fast.
The above is a reconstruction of my thought process after I read the passage for the second time, and began to have some skeptical qualms about what I was reading. Given the lack of landmark numbers, I couldn't easily check the passage for plausibility by checking it against landmarks. However, there is one other technique for checking suspicious numbers that I've previously used7: cross-checking.
Cross-checking is only possible when there are two or more numbers to check against each other, but that's the case here: in addition to the alleged speed of the bullet, we also have the claim that it would cover almost 90 miles in 1/18th of a second. In fact, anything going 1,979 miles per second would travel closer to 110 miles in an eighteenth of a second. While this isn't a huge discrepancy, it's an odd one, and gives further reason to doubt the numbers.
At this point, the plausibility check of these numerical claims came to an end with the conclusion that they are implausible, even highly implausible, but not definitely wrong. It may not sound like it, but this is a successful plausibility check, since such checks will seldom establish that the numbers checked are definitely wrong. Plausibility checks are not replacements for actual research, but preliminary steps to see whether additional research is called for. In this case, such research is definitely in order.
So, how fast do bullets fly? Unsurprisingly, this varies considerably depending on the type of gun and ammunition, from a low of around 400 feet per second for black-powder muskets to ten times as fast for modern rifles8.
More specifically, how fast was the bullet fired by Oswald? Approximately 2,000 feet per second9, which suggests that somehow "miles" was substituted for "feet" in the passage in question. Given that there are 5,280 feet in a mile, this means that the claims were off by a factor of over 5,000. So, instead of travelling around a hundred miles in an 18th of a second, the bullet would have travelled only about a hundred feet.
So, what have I learned from this experience and what do I hope that you learn from it? Here are the bullet points:
- When it comes to high speeds, we quickly run out of landmarks by which to judge claims for plausibility. Our experience with speeds is very limited, and anything over around a hundred MPH is just "really fast".
- The author of the book quoted, together with any editors or fact-checkers employed by the publisher, failed to notice that the speed given should be in feet instead of miles, and was off by an absurdly large amount. So, you can't always count on writers or book publishers to catch even egregious errors.
- Given the lack of landmarks, it's easy for even a skeptical person, such as myself, to not notice when numerical information about speeds is in error, even enormous error.
- A New Book for a New Year, 1/5/2019.
- Richard Warren Lewis, The Scavengers and Critics of the Warren Report (1967), p. 166.
- For the record, the average cruising speed of jet airliners is around 550 MPH. Source: "How Fast Do Commercial Jets Fly? ", Reference, accessed: 1/12/2019.
- To provide another landmark, the speed of sound is around 750 MPH, which would mean that the fatal bullet was travelling at Mach 10,000! Another reason for skepticism. Source: Patricia Barnes-Svarney, editorial director, The New York Public Library Science Desk Reference (1995), pp. 2 & 18.
- The Concorde supersonic transports used to regularly break the sound barrier while flying over the Atlantic ocean between Europe and North America, but they stopped flying fifteen years ago.
- At least, if the theory of relativity is correct.
- Sobriety Check, Part 2, 3/3/2015.
- Patricia Barnes-Svarney & Thomas E. Svarney, The Handy Forensic Science Answer Book (2019), p. 199.
- Colin Evans, A Question of Evidence (2005), p. 149.
A New Book for a New Year: Is That a Big Number?
Numbers are important: getting them wrong has consequences. Numbers give size and shape to our world in all sorts of ways, and we rely on them to inform the decisions we make. It's easy though, when numbers become too large, to become numbed by sheer scale. This is not a book of mind-boggling number facts, or stupefying statistics. This book is all about finding a path through a wilderness of big numbers that we don't grasp as well as we could.1
This new book by author Andrew C. A. Elliott asks an important question. Having not read the book, I don't know whether Elliott makes the point that, as it stands, that question is unanswerable. Except, I suppose, by: It depends.
No number, taken in isolation, is big or small in itself. I'm not referring to the fact that "big" and "small" are vague concepts, though that's true, too. Rather, I mean that whether a number is big, small, or in-between depends on what it quantifies. Seven may seem like a small number, but not if you're talking about human height in feet. Similarly, a million may seem like a large number, but not if you're dealing with the human population of countries.
Of course, some numbers are bigger than others: a million is definitely a larger number than seven. So, whenever you wonder whether a number is big or small, you should ask: relative to what?
One of the uses of numbers is to compare the sizes of sets of things, but we can't do this well if we don't have a sense of the relative sizes of the numbers. Since most of us have little experience with numbers in the millions, billions, and trillions, we tend to have little sense of how such numbers compare2. In the book, Elliott gives five techniques for putting numbers in context3:
- Landmark Numbers: This phrase seems to refer to what I've called "statistical benchmarks", "benchmark figures", or just "benchmarks"4. These are numbers that can be remembered or easily looked up, and which can be used to put other numbers into perspective. For instance, it's useful to know that the current population of the United States is about a third of a billion people5. I now prefer Elliott's terminology, since a landmark helps us to find our way around a landscape; in the same way, a landmark number can help us navigate the numerical landscape.
- Visualisation: You can see what this must be.
- Divide & Conquer: I'm not sure if this is the technique of dividing a big number in order to reduce it to a smaller number that can be more easily "conquered". For instance, the current U.S. national debt is over $21 trillion6, which is way too big for people to grasp. Is that a lot or what? Of course, it would be a lot of money to have on your credit card, but is it a lot for the country to have on its credit card? How can we tell? If you divide the national debt by the landmark number of the current population5, thus giving the per capita debt, the result is close to $65,000. This is an amount within most people's experience, and thus easier to get a sense of its scale. Is that a big number? Since you can now do so for yourself, I'll let you be the judge. In any case, I like the phrase "divide and conquer" for this technique of cutting a big number down to size so that our understanding can conquer it, but I don't know if that's what Elliott means.
- Rates & Ratios: The example of "divide & conquer" that I've just given produces a ratio. In understanding risk and safety, rates are usually more useful than absolute numbers. For instance, far more people die in the shower than do so in skydiving accidents, but this doesn't mean that showering is more dangerous than skydiving since a lot more people shower than jump out of airplanes. To properly compare the safety of these activities, we need to take into account the fact that more people do one than the other. To do so, divide the number dying in each activity by the number of those engaging in it.
- Log Scales: These are not scales for weighing logs―at least, I don't think they are. Rather, what Elliott is referring to are logarithmic scales, as opposed to the more familiar linear scales. Logarithmic scales are used for measuring such things as earthquakes and sound: both the Richter7 and decibel scales8 are logarithmic. That said, I don't know why Elliott thinks that log scales are so useful that they rank with the previous four methods; I guess that I'll have to read the book to find out.
- Andrew C. A. Elliott, Is That a Big Number? (2018), p. 2.
- Douglas Hofstadter, an early writer on innumeracy, referred to this as "number numbness". See Metamagical Themas: Questing for the Essence of Mind and Pattern (1985), Chapter 6.
- Elliott, p. 5.
- See, for instance: Sobriety Check, 2/24/2015. I took this terminology from Joel Best's book, Stat-Spotting: A Field Guide to Identifying Dubious Data (2008), pp. 7-9.
- To be more exact, 324 million, according to Wolfram Alpha; accessed: 1/3/2019.
- According to Wolfram Alpha; accessed: 1/3/2019.
- Bryan Bunch & Jenny Tesar, The Penguin Desk Encyclopedia of Science and Mathematics (2000), under "Richter scale".
- Bunch & Tesar, under "decibel".