A Measurement Rant

You’ve all seen this:

1 in = 2.54cm

The only problem is, that’s wrong*. This may seem nit-picky, but understanding why the above is wrong can uncover some clever ways people deceive others (and themselves).

“But everyone knows an inch is 2.54 centimeters,” I hear you say. Hold on a second there, Sparky! Let’s back up. An inch is a measurement. Let’s say I measure a piece of wood and find that it’s 57 inches long. Now I want to know how many centimeters it is. I multiply by 2.54 and discover that the wood is 144.78 centimeters long. So where’s the problem?

The issue is that I measured my stick to the nearest inch, and now through the magic of conversion I claim that I know how long that wood is down to a tenth of a millimeter. The idea that I could get that sort of precision with my tape measure is silly, yet people do this all the time. On road signs you’ll see “Exit 4 mi (6.4 km)”. Is the sign really accurate to 100 meters? That’s a tough assumption to swallow given the first measure is only accurate to the nearest mile.

A rule to remember: when you do a unit conversion, the result is always less precise than your original measurement. Always.

Here’s what your conversion table should read (although this isn’t quite perfect either):

1.00 in = 2.54cm

What’s the difference? Remember that an inch is a measurement. The number of decimal places indicates how exact the measurement is. If I measure something down to a hundredth of an inch, I can (usually) justify claiming precision to three significant figures after I make the conversion. So, if I measure my stick to be 57.0 inches, I can reasonably represent it as 145 centimeters long. The error of a tenth of an inch in the first measure is smaller than the error of one centimeter in the second, so I’m all right. 57.00 inches = 144.8cm.

The people who put things like “1 inch = 2.54cm” into textbooks will claim, “What we actually mean is that exactly one inch equals 2.54 centimeters.” The only problem is, That’s wrong too. First, with measurements there’s no such thing as exact. Every measurement contains error. Always. 1.0000000 inches is not the same as exactly one inch. Second, for almost every comparison of measurements in different systems, the conversion factor itself is not exact. An ounce is not 29.57 milliliters. It’s not 29.5735 ml.

So why does this matter? Ask yourself, how much product is in that bottle or can of your favorite beverage? 12oz or 355ml? In this case, we hope that the more precise measure is applicable. It would be informative if the bottler used 12.0oz rather than just 12; you know the Coca-Cola bottling company knows to great precision how much less than 12oz they can put in a can and still label it 12oz. Technically (though perhaps not legally), 11.50000001 oz could be labeled as 12, but that would not be anywhere close to 355ml.

This sloppiness with units is frightfully common. Even scientific papers with measurements in them sometimes don’t include the margin of error in the measurements – which makes the number pretty meaningless.

Don’t be fooled by false precision! Often it’s harmless, but even subtly it can give the impression that the peple who made the measurements are far more diligent than they actually were. This can give their arguments extra weight, without you even realizing it.

* It turns out I picked a bad example – in 1959 they redefined the inch to make this true. Go figure — the inch is metric now. See the comments below. So, as a unit, the conversion is correct. This has no effect on how you use the conversion in real life. I may go back and change this episode to use a better example.

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15 thoughts on “A Measurement Rant

  1. I’m going to quibble. I totally agree that many measurements are incorrectly converted with false precision. A biggie: what’s a normal body temperature? 98.6 degree Farhenheit, right? No! It’s “about 37 degrees Celcius.” Some genius over converted.

    But, we can /define/ unit conversions. 100 inches really is 254 cm. We’ve defined it that way. It’s an exact ratio. Now, you’ll never find anything that is exactly 100 inches, as you say. No measurement has infinite precision. But whole numbers do have infinite precision.

    That is all to say, 1 in is /not/ inherently a measurement. It is a unit. All measurements come in units (with imprecision), but the units themselves stand unscathed on the Platonic Plane of Forms.

    Also, scientists who publish numbers to measurements without uncertainties listed should be sent back to undergraduate physics/chemistry lab and also shot.

    • … or just laughed out of their field.

      I had forgotten (if I ever knew) that 100 inches was definitially 254cm, so that was actually a bad example. I thought meters came from an estimate of the size of the Earth, but I guess as they refined it they jiggered it to play nicer with inches (though why the bother I don’t know). Wikipedia mentions no such jiggering, but they are surprisingly sparse on the subject.

      I suppose we could figure out what 1/299,792,458 light-seconds is in inches and see where we stand.

    • Aha! It wasn’t that the powers that be jiggered the meter to match up with inches, but the other way around. From WikiAnswers:

      Effective July 1, 1959, the United States and countries of the British Commonwealth defined the length of the international yard to be exactly 0.9144 meters. Consequently, the international inch is defined to be equal to exactly 25.4 millimeters.

      They made the yard metric!

  2. Interesting that you should give as an example a 12-ounce bottle of Coca-Cola. Once upon a time, a very long time ago, I was on a tour of the Coca-Cola canning plant in Portales. There’s one point along the assembly line where underweight cans are kicked off the conveyor belt. Essentially, when you buy that 12-ounce can of Coke, you’re not buying exactly 12 ounces; rather, you’re buying not less than 12 ounces. So you could be getting 355 ml.

    Apparently, this is fairly standard practice in the food industry — if a package says it contains a certain amount of a product, it may not contain exactly that amount, but it will not contain less.

  3. Wanted to sweet it twice. Great post. Isn’t there a meter rod somewhere in Paris that is the international standard?
    inre Jack’s comment: I would be surprised if a journal paper published numbers without uncertainties. I think it is pretty standard peer review practice to highlight such an omission, and would be corrected before publication.

    • I learned in the course of researching Jack’s statement that 254cm = 100 inches by definition, that there have been a series of official meter rods, each more precise than the last. Now those have been abandoned in favor of light in a vacuum.

      Not sure if gravitation effects should be compensated for or whether gravitation should change the definition of a meter, since it’s space that’s distorted. Wikipedia didn’t say anything about that.

  4. Reminds me of how it was I figured out I was not an engineer – I would arrive in class, and one of my classmates would be scurrying around comparing answers on homework – “I got 3.486 but the book says 3.487, and she got 3.488, and he got 3.487 – what did you get? Which one is right?”
    “I got 3.492 and figured that was close enough!”

  5. Keith – women have had this problem for years – in high school, I wore one manufacturer’s size 9. Thirty years later, and I have stuff in my closet in size 6. And no, I haven’t lost weight since then.

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