Approximate results may vary

Part 33 of my ongoing series is coming but I did not get all the code written that I wanted to this week, so it will be delayed. In the meanwhile:

Living in Canada as a child, of course I grew up learning the metric system (with some familiarity with the imperial and US systems of course). If you want to know how many milliliters of liquid a cubic box holds, you just compute the volume of the box in cubic centimeters and you’re done, because a milliliter is by definition the same volume as a cubic centimeter.

Despite having lived in Seattle for over 25 years, I still sometimes do not have an intuitive sense of conversions of US units; for a particular project I knew that I needed an amount of liquid equal to about the volume of a two-inch cube, but the bottle was measured in fluid ounces. Fortunately we have this operation in every browser:

Thanks Bing for those important eight digits after the decimal place there, manufactured from the zero digits after the decimal place of the input. For context, that extra 0043 on the end represents a volume equivalent to roughly the size of a dozen specs of microscopic dust.

But the punchline that made me LOL was “for an approximate result, divide the volume by 1.8046875”, because of course when I am quickly approximating the volume of eight cubic inches in fluid ounces, the natural operation that immediately comes to mind is divide by 1.8046875.

I have some questions.

That was Bing; what does Google do with the same query?

OK, that’s an improvement in that the amount of precision is still unnecessary, but not outright absurd. But the other problems are all the same.

Why division? Maybe it is just me, but I find division by uneven quantities significantly harder mental arithmetic than multiplication; assuming we want the over-precision, would it not be better to say “for an approximate result, multiply the volume by 0.5541125″? 

Why say “to approximate…” and then give an absurd amount of precision in the conversion factor? Approximation is by definition about deliberately not computing an over-precise value.

Surely the right way to say this is “for an approximate result, multiply the volume by 5/9” or even better, “divide by two“. When I saw “divide by 1.8046875” the first thing I thought after “wow that’s so over-precise” was “1.8 is 18/10 is 9/5, so multiply by 5/9“.

I’m going to get there eventually; software can shorten that journey. And I’m going to remember that 5/9ths of a cubic inch is a fluid ounce much more easily than I remember to divide by 1.8046875.

Once you start to see this pattern of over-precision in conversions, it’s like the FedEx arrow: you can’t stop seeing it. Let’s ask the browser how much does an American robin weigh?

64.8 grams. An underweight robin is not 64.5 grams, and not 65.0 grams but 64.8 grams.

To be fair, it looks like this over-precision was the fault of a human author (and their editor) not thinking clearly about how to communicate the fact at hand, rather than bad software this time; if you’re converting “two and a third ounces” to grams it would be perfectly reasonable to round to 65, or even 60. (That third of an ounce is already suspect; surely “two to three ounces” is just fine.) Most odd though is that the computations are not even correct! 2 and 1/3rd ounces is 66.15 grams, and 3 ounces is 85.05 grams, making it rather mysterious where the extra few hundred milligrams went.

I was wondering how many earthworms a robin would have to eat to make up a discrepancy of 0.2 grams. A largish earthworm has got to weigh on the order of a gram, right?

Wow! (For my metric readers out there: 1.5 pounds is 680.388555 grams according to Bing.)

Again: what the heck, Bing? I did not ask for “world’s largest earthworm” or “unusually large earthworms” or even “Australian earthworms”. You know where I live, Bing. (And Google search does no better.)

For some reason I am reminded of Janelle Shane’s “AI Weirdness” tweets; the ones about animal facts are particularly entertaining. These earthworm facts at least have the benefit of being both interesting and correct, but this is hardly the useful result about normal garden-variety annelids that I wanted.

I am also reminded of my favourite animal fact: the hippopotamus can jump higher than a house. It sounds impressive until you remember that houses can’t jump at all.

Obviously all these issues are silly and unimportant, which is why I chose them for my fun-for-Friday blog. And the fact that I can type “8 cubic inches in oz” into my browser or say “how much does a robin weigh?” into a smart speaker and instantly get the answer is already a user interface triumph; I don’t want to minimize that great work. But there is still work to do! Unwarranted extra precision is certainly not the worst sort of fallacious reasoning we see on the internet, but it is one of the most easily mitigated by human-focused software design. I’d love to see improvements to these search functions that show even more attention to what the human user really needs.

31 thoughts on “Approximate results may vary

  1. As part of the COVID-19 mitigations, Disney World is not admitting anyone with a temperature of 100.4 °F or higher. That number sounded oddly specific to me, so I checked, and, yes, it is exactly 38 °C. I suppose that if the medical personnel making the recommendations had published their results in Fahrenheit, then it probably would have been 100 °F / 37.77778 °C.

    • That’s amusing that it is exactly 38°C, but thermometers for detecting fevers are all calibrated to the tenth of a degree F here, and 100.4°F is the published CDC guidance for when someone likely has a serious fever, so it’s not that surprising that Disney is using that guidance.

      However, if you do a little digging on the CDC COVID 19 website you find exactly the sort of thing you are describing; some of the pages say “38 C / 100.4 F” and some of them say “100 F / 37.8 C” as when you have a reportable fever.

      So yeah, it’s a mess.

      • In the context of these two comments, it’s worth pointing out that the “98.6” that we here in the US have all memorized as the “normal” temperature for humans, is _also_ just a converted metric number: 37 C.

        Most of the rants above, I can live with. I agree with them, but I don’t mind extra precision from a web browser/computer UI. And the Bing results are actually quite useful when one is looking to incorporate conversions quick-and-dirty into a program. Yeah, they are useless for a human doing the computations, but it’s a quick way to look up conversion factors for math formulas.

        The thing that really gets me is news reporting where they provide values in both systems, converting from one to get the other, and leaving in pointless digits of precision. For the same reasons you cite above, I don’t need to know the exact number down to the fifth digit. Or often even the second or third digit. For the purposes of the news report, all I really need to know is roughly “how big/long/wide/deep/etc.”. And since the original figure is often an estimate or rounded number anyway, the extra precision is misleading for no good reason.

        • It’s not so much that I object as I thought it was funny, but I take your point.

          News reports are particularly vexing though. Unwarranted precision is bad enough, but it is particularly egregious when paired with crazy approximations. “Hail the size of golf balls (42.67 mm) fell on a Nebraska town today…” sort of thing.

          That of course pales in comparison to genuinely harmful innumeracy of media figures who ought to know better; it boggles the mind that anyone under any circumstances would think that if you had $500M, you could give $1M to 500M people.

          I have a collection of innumerate graphs, charts, maps, and newspaper headlines; maybe I’ll do a fun-for-Friday blog post about them one of these days.

          • True! Excess of decimals of precision aside, I feel like golf-balls as a unit of measurement would have been sufficient unless there is a difference between a metric golf ball and an imperial golf ball…

            But such a series of Friday blog posts would be welcome, I look forward to reading more.

          • That “genuinely harmful innumeracy of media figures” reminds me of an article I saw on global warming many years ago in the Netherlands. The author reported an expected increase in global average temperature of 1°C, noting that this was equivalent to an increase of 34°F. Ever since, I have wondered what would have happened had the reporter’s source quoted a figure of 2°F and the reporter had correspondingly converted that to an increase of -17°C!

          • Re: increase in temperature: indeed, I once saw a sign in an educational display describing a steam engine boiler that was 1000C as being “ten times hotter than a boiling kettle”. Uh, no, that’s not how it works.

        • The annoying thing about body temperature is that it also depends on what point of body exactly you’re measuring at, but it’s almost never said which does bring confusion in international/intercultural environments. Here, for example, it’s traditionally measured inside the armpit, and the normal body temperature in that spot is 36.6 °C. A reportable fever starts at 37 °C and the thermometers have a red mark at this degree (Google images for “mercury thermomter”).

        • Spurious precision in news reports is especially annoying when they are converting currencies, as the conversion factor changes from day to day!

        • HAH!

          I don’t know how many times I have had to say in the last few months “six feet is not magic”. People behave like if they are 6.01 feet away, chance of infection zero, 5.99 feet away, chance of infection 100%. It’s bizarre.

  2. It’s probably a good time to remind ourselves that all good measurements come with an error bar, something that almost nobody who’s not a scientist understands…

      • I would say that this is a pet peeve of mine. Mean, median, and mode are all valid definitions of “average”. It doesn’t matter for this amusing statement, but it does matter in scientific reporting. Scientific articles will use one of these terms, and then a journalist will turn it into average. And then lay people will take average to be the mean, even if the original was median or mode. And then get to a different conclusion.

        • I hear you. A particular peeve of mine is reporting that supposes that “average” and “typical” are the same thing in an economy with higher disparity than existed prior to the French Revolution. The average wealth in the United States is massively skewed by the super-wealthy. Reporting that says, for instance, that the average young American (say, under 35) has a net worth of $75K may be mathematically correct but grossly misrepresents the typical situation. A typical young American has about a $10K net worth, and if they are a college graduate, they are typically saddled with enormous debts.

          • Do your remember the book *Material World*? I calculated that Bill Gates owned about $200 worth of the possessions of the average American family in their photo.

          • I think what the commenter was intending to express by “it doesn’t matter” is that there is nothing important at stake in the statement of this little joke. The fact that the mean and the mode are different is the joke. In contrast, many journalists (or worse, propagandists) confuse “average” with “typical” in a way that does matter because it obscures an important truth.

  3. Counterpoint: If you have too much precision in an answer, it’s easy to round it off to the precision you need. But if you don’t have enough precision, you can’t somehow discover what that missing precision is.

    • Counter-counter-point: if you have *manufactured* precision in an answer — that is, I started with a value accurate to around a tenth of a cubic inch and ended up with an answer accurate to ten billionths of a fluid ounce — it can be similarly hard to know what of that precision is real without having the original value around; and if that original value also has manufactured precision, then you don’t have much of an idea at all what the real error bar is.

      As another commenter wisely pointed out: the error bar is logically part of the value. It is an irony that doubles are both designed for computation of physical quantities, and yet lacking the very feature that would make them more useful for science: tracking error magnitude.

      • Let us hope that in a (far?) future, humanity will eventually take the habit to use numbers with precision included, expressed in a specialized format, as a first-class concept.
        By that time, hopefully everyone will have settled on a common temperature scale, (one that makes sense, i.e. starts at 0 for absolute zero).
        Who knows? Maybe everyone will also finally be using ISO 8601 for dates.

  4. I’m also from Canada but live in the US. I went to engineering school during the metric conversions of the 1970s so I got to learn everything in three unit systems (since most texts used US units). You talk about Imperial units, but then go on to discuss US units. For example, you show Bing’s conversion of 8 in^3 to the wildly precise 4.43290044 US ounces. But, 8 in^3 is 4.61395016 imperial ounces (in Bing precision). I always found it weird that the classic anti-metric saying is “a pint’s a pound the whole world”. US pints (16 US oz) are very different from British (and Canadian) pints (20 Imp. oz), but both countries use the same measure for pounds (about 454 grams, or 453.59237, Bing-style)

    • Ah, good point, yes, I often casually conflate Imperial and US units but you are absolutely correct that they are often quite different.

      I also am reminded of the riddle “which is heavier, a pound of feathers or a pound of gold?” A pound of feathers, obviously.

      • It actually works better with ounces. One rarely quotes avoirdupois and troy pounds.

        My favorite Canadian metric conversion story was when they changed all the road signs from mph to km/hr. I’m from Montreal. Quebec, being Quebec, picked a day and got every contractor in the province out on that day and changed every sign in the province that one day. A short time later, I was driving to Halifax. Nova Scotia didn’t do the change Quebec-style, instead they changed them over a 1 (maybe 2) month period. I got very confused when a passed a sign saying “Halifax 250” and a short time later I passed another that said “Halifax 300”.

        • That reminds me of the story of Högertrafikomläggningen, the 3rd of September 1967, when overnight Sweden changed from driving on the left to driving on the right. That’s a lot of signs to update!

  5. Great post.

    I can’t help but think that the approximation suggestions should be even more coarse if they are going to be helpful. For the cubic inch to fluid ounce example, I think “divide by two” would be sufficient for most approximations _and_ is something people might actually remember. The 9/5 ratio isn’t too bad, especially since it shares the F to C conversion ratio (which is suddenly very interesting…).

    There are times I appreciate accuracy and a “show the full answer” button on these conversion results would be nice if they did simplify the answers, but I think estimation is becoming a lost art in the age of omnipresent calculators and converters.

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  8. Interestingly, I tend to use the word approximately differently than you do. I’m a mathematician, so I use the word approximately to mean “not exactly”, so I might write Pi is approximately 3.14159265358979323846 so that the reader does not believe that I have given the exact value of Pi which would be impossible because it has an infinite number of digits.

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