Insignificance

Insignificance2By The Metric Maven

Bulldog Edition

Some years back I became involved in evaluating the results of an experiment that clearly had scientific issues. I assisted two other volunteers, and was mostly there to critique the experimental methods. Early on I asked if the answer was as simple as the data had been cooked. One of the volunteers was a graduate student in mathematics. He looked at me and said “no, it’s fine, I already looked at the data.” I was a bit puzzled and wanted to know why he had such confidence that the data had not been altered. The mathematician said “the data is consistent with Benford’s Law.” I had no idea what that was, and surprised to hear that generally the first significant digit of numerical data is not random. The distribution of 1’s, 2’s, 3’s and so on up to 9 is not uniform. The probability of a one is higher than a two and they all follow a statistical pattern.

My mind had a very difficult time accepting this statement. The mathematician told me to “go look it up” which I did.

The story goes that Frank Benford (1883-1948), while working as an electrical engineer at General Electric, had obtained a well-used copy of a book of logarithms. He noticed that the beginning pages were the most soiled and worn. The idea that people would most often look up logarithms of numbers that begin with number one, and then those with two, and so on up to nine was surprising. Benford wrote up his observation, which is often called Benford’s Law. However Simon Newcomb (1835-1909) had earlier published the same observation in 1881.

I had a hard time accepting this because it meant that the first significant digit of a number is not statistically independent. The mathematical analysis to derive Benford’s law is beyond my expertise.[1] Sven pointed out that Warren Weaver (1874-1978) in his book Lady Luck has a reasonably intuitive explanation of how Benford’s Law comes to be. The relevant section is called The Distribution of Significant Digits, and does not mention Benford directly. Weaver makes this statement: “Although it remained unsuspected or at least unidentified for centuries, this distribution law for first integers is a built-in characteristic of our number system.”

Here is a nice graph from Wikipedia showing the distribution of the first significant digit for numerical data:

Benford-2Thirty percent of the time, the first significant digit of commonly used physical constants found in an elementary physics textbook is one. Census populations follow Benford’s Law, as do income tax data, one-day returns on the Dow-Jones industrial average and Standard and Poors indexes. Benford’s law is often used in forensic accounting to screen for fraud.

At this point, I want you to note something about this plot of Benford’s Law: what is the probability of zero for the first significant digit? Well, there isn’t one. If you add up all the probabilities you end up with 100%, so no probability is assigned to zero for the first significant digit, or should it be called the first insignificant digit?

I had given thought to discussing significant digits in the past, but there are differing views about how to go about determining significant figures in calculations, and so I tended to shy away from any discussion of the topic. Not until a reader took me to task over a statement I made in a blog about the 100th anniversary of the USMA did I decide it was worth some examination:

Also, the Maven writes: “The world record eyebrow hair is touted as 9 centimeters (90 millimeters for those with a refined measurement sense).”
In this case, since the measurement apparently was not to the nearest 0.1 cm, writing it as 90 millimeters would be false precision. (Of course if it were given as, say 9.2 cm, then 92 mm would be better.)
Thus, centimeters should be considered in such circumstances to avoid any indication of false precision; otherwise, centi-, and deci-, deka, and hecto-, should be considered as sort of “informal prefixes”…

While there is a lot of disagreement about how to determine significant digits, the one statement about them which is generally accepted is that adding zeros on the right side of a whole number does not constitute adding significant digits.

Here is a statement from Learn How To Determine Significant Figures:

If no decimal point is present, the rightmost non-zero digit is the least significant figure. In the number 5800, the least significant figure is ‘8’.

Another university website has:

Trailing zeros in a whole number with no decimal shown are NOT significant. Writing just “540” indicates that the zero is NOT significant, and there are only TWO significant figures in this value.

Wikipedia has this to say:

The significant figures of a number are digits that carry meaning contributing to its measurement resolution. This includes all digits except:[1]

Wikipedia in its rules for identifying significant figures states:

In a number without a decimal point, trailing zeros may or may not be significant. More information through additional graphical symbols or explicit information on errors is needed to clarify the significance of trailing zeros.

If a whole number is encountered without any context, the trailing zeros should be assumed as insignificant unless the text specifies otherwise.  Clearly when I pointed out that 9 cm would be better written as 90 mm, I did not conjure up an extra significant digit and imply more measurement resolution. The commentator made an unwarranted assumption about the 9 cm value: “since the measurement apparently was not to the nearest 0.1 cm, writing it as 90 millimeters would be false precision.”  He is acting as a psychic and divining what precision was implied by the person who measured the value and offered it as 9 cm. What was offered up by the commentator is actually an example of false precision. There is no reason to assume the measurement was or was not to the nearest 0.1 cm or 0.05 cm or 0.025 cm. Only a single integer with a single significant figure of 9 is offered. Adding a zero on the end and expressing it in millimeters without providing any additional information altered that fact not one bit.

In this situation the zero is just a final place holder, therefore when an extra zero is added to the end, it does not introduce any increase in implied precision or become a significant figure. The first significant digit is still nine for 90 mm as it was for 9 cm.

“Trapped zeros” are considered significant. In the case of 402, the zero between the 4 and the 2 is significant, but a trailing zero such as that found on 420 is not. Adding an infinite number of trailing zeros to an integer number does not increase the number of significant digits. When I pointed out that Metric Today should change centimeter values to exclusively millimeter values, I only changed 9 cm to 90 mm. I can equivalently write 9 cm as 90 mm, or 90 000 µm or 90 000 000 nm without introducing any extra “implied precision.” Unless I tell you that 90 mm is a value with two significant figures, you should assume the zero is not significant. The centimeter is a coarse enough measurement length, that when implemented for everyday measure, any useful value will have a decimal point, and is more appropriately written in millimeters.

The “implied precision” argument against using millimeters exclusively in everyday life is one that has an appearance of technical relevance, but is no more than an ad hoc truthiness statement. Everyday it is empirically demonstrated as vacuous by those who construct metric buildings in Australia, Bangladesh, Botswana, Cameroon, India, Kenya, Mauritius, New Zealand, Pakistan, South Africa, United Kingdom, and Zimbabwe. It is also theoretically superficial when examined carefully. Adopting knee-jerk contrarianism mantled in truthiness does not contribute to human understanding, it only attempts to squelch it.

Why is this question worthy of an entire blog? Because we probably get more flak on the millimeter vs centimeter question than any other. And the flak comes from metric advocates. Occasionally, it comes from a metric advocate whom we admire. And yet, the argument for keeping the centimeter hanging around like an albatross is always based on a misunderstanding of precision: the notion that that extra zero has some meaning beyond establishing scale.  It doesn’t. Scientists, engineers, and mathematicians are all in accord that it doesn’t.  It really isn’t even a metric question, but it’s only metric advocates that aren’t on the same page here. Odd, that.

[1] Hill, Theodore P., “A Statistical Derivation of the Significant-Digit Law” 1996-03-20 Georgia Institute of Technology

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Who Says!?

Isaac.Asimov01

Isaac Asimov

By The Metric Maven

Isaac Asimov Edition

One Sunday, I was attending my weekly coffee klatch, when one of the participants asked: “Who besides you thinks millimeters should be used instead of centimeters?” I was rather surprised at the question, even though I’m the resident metric advocate. I blurted out “Well, Isaac Asimov does for one, so does the U.S. metric building code, the late Pat Naughtin did, and so did Herbert Arthur Klein in his book The Science of Measurement.” The person who asked the question has a solid scientific background, and what surprised me about the question was the appeal to authority. I have found it very puzzling that when I explain the situation, people do not seem to absorb its meaning, or don’t really think about the simplified symbolic expression.

First, lets start with authority. In his 1983 book The Measure of The Universe, Isaac Asimov has this to say about the centi prefix:

The prefix “centi” (SEN-tih), symbolized as “c,” represents a hundredth of a basic unit, from the Latin “centum” meaning “hundred.” A “centimetre,” therefore, is a hundredth of a metre. The prefix is not commonly used, except in “centimetre,” and its use is falling off even there.

Isaac Asimov has this to say about the milli prefix:

The prefix “milli” (MIL-ih), from the Latin “mille,” meaning “thousand,” is symbolized as “m,” just as “metre” is. A millimetre is therefore symbolized as “mm.” Increasingly “milli-” is replacing “centi-” and “deci-” in use. We are approaching the point where 1 centimetre will routinely be referred to as 10 millimetres, and 1 decimetre as 100 millimetres. This is even more true where these prefixes are used for any basic measure other than “metre.”

There it is, documented with sans serif typeface, Isaac Asimov asserting the utility of the milli prefix over the centi prefix. Asimov also had this to say in his essay “Read Out Your Good Book In Verse” in his 1984 book X Stands for Unknown:

Light wavelengths have traditionally been given in “Angstrom Units,” named in 1905 for the Swedish physicist Anders Jonas Ångström (1814-74), who first used them in 1868. An Angstrom unit is one ten-billionth of a meter, or 1 x 10−10 meters.

Nowadays, however, it is considered bad form to use Angstrom units because they disrupt the regularity of the metric system. It is considered preferable now to use different prefixes for every three orders of  magnitude, with “nano” the accepted prefix for a billionth (10−9) of a unit.

In other words, a “nanometer” is 10−9 meters, so that one nanometer equals 10 Angstrom units. If a particular light wave has a wavelength of 5,000 Angstrom units, it also has a wavelength of 500 nanometers, and it is the latter that should be used.

Again Asimov asserts the importance of using metric prefixes with separations of three orders of magnitude.

Nine years earlier in 1974, Herbert Arthur Klein, in his book The Science of Measurement, wrote this about the metric prefixes:

HAK-PrefixesHerbert Klein sees the atavistic magnifying prefixes deka, hecto, and myria as unnecessary and implies that separations by 1000 are best. He sees only milli as a reducing prefix in good standing, and again argues for reduction by factors of 1000.

Pat Naughtin spent considerable time exploring why millimeters worked so much better than centimeters when implemented in industry. When millimeters were used, the metric transition was quick and almost painless. The introduction of centimeters would delay metric adoption almost indefinitely. He wrote a long discussion of this in 2008 and pleaded for people to use millimeters.

Long time readers know that when the issue was explained to me, and I used millimeters and millimeter instruments in my own engineering work (sans centimeters); I became convinced that the centi prefix, and centimeters are considerable intellectual barriers to metric adoption in the U.S..

After I understood the problem with centimeters, it seemed obvious to use millimeters, but as Isaac Asimov states in his 1971 book The Stars in Their Courses:

One of the pitfalls to communication lies in that little phrase “It’s obvious!” What is obvious to A, alas, is by no means obvious to B and is downright ridiculous to C.

I’m going to do my best to return to my unexamined world view and try to explain the epiphany that struck me at Mach III+. Below is an image from a newspaper film box, probably from the 1970s. The film size is in inches, and it is converted to metric in centimeters.

Newspaper-Film-1970sThe film size is 45.7 x 58.4 centimeters. The number of symbols used is four for each linear dimension. In the everyday world, a measurement with only the precision of a centimeter, is generally too coarse to be of any practical use. The odds that one will measure to an even centimeter are rather low, and so almost all common measures in our world require an unnecessary decimal point and a value for a tenth of a centimeter.

But a tenth of a centimeter is a millimeter. This implies that everyday measurement is generally useful only to a millimeter value. When 45.7 cm and 58.4 cm are written in millimeters, only three symbols are required to express the very same value of length: 457 mm x 584 mm. The mind does not need to stop and perceive the location of a decimal point and parse the decimal number. The number of symbols used is reduced from four to three.

The objection often offered is that one only has to move the decimal point to change from millimeters to centimeters! Pat Naughtin pointed out that often people who work on construction are less familiar with manipulating numbers than scientifically trained professionals. Asking them to slither a decimal point along in any calculations they might do, will only introduce an opportunity for error. In the case of centimeters, the error can be very large because of the unit size chosen.

But, indeed, the proof of the pudding is in the eating, and so it is with the millimeter and the metric system. Pat Naughtin has an extensive discussion (50 pages) about millimeters versus centimeters. His original observation was an empirical one: Industries that used the millimeter had quick and smooth metric transitions, those that chose the centimeter are still in turmoil to this day. Why is this the case? It was analysis after-the-fact that offered clues.

Naughtin makes this observation:

Talking or arguing with people who have not done any measuring with the metric system is quite pointless. But as soon as they experience the simplicity of the metric system for themselves they will then convince themselves that it is the better

Sven had lobbied for the use of millimeters, but it was only when I had all-millimeter rulers and instruments, that I realized their utility, and adopted millimeters exclusively.

I continue to have people who are from “metric countries,” who, with an air of sanctimoniousness say “I’ve never had a problem with centimeters. I use them all the time.” They don’t seem to realize I could just as easily say I’ve used inches (feet, yards, rods, miles) here in the U.S. and I’ve never had a problem. Or stating that “I can use Roman numerals, and have for years,” with the implication that your mind is obviously too small and dim to handle them. Not that they are in fact awkward. It took about 1000 years for people to realize there was a problem with Roman Numerals. They never saw a problem because the were immersed with them.  These denizens of “metric countries,” have an antique metric system usage, that is contemporary with Þe Olde English, and they are fine with the retention of familiarity over simplicity. There is no examination or self-reflection, just a thoughtless assertion. References are offered, reasons explained, and the response appears to be reactionary truthiness, rather than thoughtful introspection.

Certainly Isaac Asimov has demonstrated that he is trustworthy, but I’m sure he would also indicate that a person should never take his word alone. It is always best to understand an idea directly. The question should not be who says?! but why.

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Related essays:

Building a Metric Shed

Metamorphosis and Millimeters