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

USMA — 100 Years Part III

USMA-2002Those Who Don’t Learn From History….

By The Metric Maven

The 21st century arrived on January 1st 2001, and after this temporal milestone passed, NASA looked into their lack of metric practice. The May-June issue of Metric Today quotes an Inspector General report: “NASA’s use of the metric system varies from program to program and from Center to Center.” The article noted that “NASA does not give program/project managers very much guidance for using the metric system.” If there was ever a more perfect example of the importance and need for a measurements coordinator position, with actual authority, I cannot think of one. Pat Naughtin emphasized the need for measurements coordinators, and I’ve written about my own experience with uncoordinated practice in industry.

The eight recommendations from the Inspector General to NASA are just more pablum which will change nothing. Number 6 begins: “NASA should show caution in granting SI waivers to entire programs. Use of SI within a `waivered` project project should be permitted, when appropriate.” SI should be “permitted”? Here is what should have been recommended: 1. NASA must have a mandatory transition to metric-only in two years from this date or face steep penalties. 2. All measurements for all projects must be approved by an office of measurement coordination and must be exclusively in metric, no exceptions, no waivers, no excuses.

Aerospace Week & Space Technology indicated that a law passed to reduce paperwork eliminated the requirement for all agencies to report metric progress. Waivers were no longer required. The magazines suggestion?—“…immediate restoration of the waiver process.” (MT May-June 2001)

The new millennium would not begin favorably for U.S. Metrication. Washington had become even more unresponsive when it concerned metric, which hardly seemed possible.

Metric and inch-pound mistakes continued to produce tragedy and loss. The July-August issue of Metric Today (MT) reported that the crash of Korean Air Cargo Flight 6316 occurred because of a metric-medieval unit confusion. The flight was to travel from Shanghai to Seoul.  The South Korea Ministry of Construction and Transportation determined that the crash was “due to a mix-up in the cockpit on whether the altitude should be measured in meters or feet. According to MT:

The accident took place soon after takeoff in Shanghai and killed 3 crew members, 5 people on the ground, and injured 40 construction personnel when it fell onto a construction site near the airport.

The summary of the report on the incident stated that a Chinese air controller directed the pilots to 1500 [assuming the pilots would know it meant 1500 meters], However, because the international aviation industry commonly measures altitude in feet, it is assumed the pilot concluded the McDonnell Douglas MD-11 was almost 1000 meters too high, so quickly moved to lower the plane. When he realized his error, it was not possible to correct the error in time, and the plane crashed.

Another metric setback occurred when Professional Engineer Thomas R. Warne, P.E. resigned his Utah DOT position, but not before announcing that Utah would revert to inch-pound medieval units for road construction in 12 months. From that point on all work will be done in Ye Olde English units. This reversal appears to have occurred throughout the nation.

The Metric Today issue of May-June 2002 had a face-lift by a graphic artist, courtesy of QSI corporation.

The subject of allowing metric-only packaging in the US continues to be discussed in Metric Today  to this day. No current pro-metric legislation is discussed, as there does not appear to be any. NIST continues to employ a metric coordinator, but their influence appears to be more ceremonial at this point than actual. A focus on metric education is a constant drum beat in Metric Today even as there is no perceptual change to metric in the US making it a virtual experience. All the measures of everyday life are given in Medieval Olde English Units. When metric dimensions do appear in MT, the centimeter pseudo-inch is utilized constantly. The world record eyebrow hair is touted as 9 centimeters (90 millimeters for those with a refined measurement sense) and belongs to Sardar Singh of Amritsar, India. If Metric Today itself has a policy concerning metric usage, the centimeter is still a sacred cow.

In the same issue (MT Sep-Oct-2003) it was noted that Pat Naughtin launched his email newsletter, Metrication Matters, on Jun 9th. Naughtin strongly argued for using millimeters as the common small metric unit. The Nov-Dec Metric Today contains a table with the prefix cluster around unity, and the same number of Medieval Olde English Units for comparison. Which only reinforces a perceived equality between the system and non-system which does not exist.

The issue of implementing a law that allows metric only labeling for goods across the country is a continuing topic. Metric changes in other countries are highlighted. In August of 2004 the California Department of Transportation (CALTRANS) decided to revert back to US units over a two year period.

Professional Engineer Robert Bullard P.E. attempted to begin metric construction in Florida. The local government officials refused to go along with metric drawings. I’ve written about his problems here, and in the end, he’s been compelled by “market forces” to go back to inch-pound designs.

The US hard metric conversion of masonry was overturned (MT Jan-Feb 2005):

The National Concrete Masonry Association (NCMA) announced in a 19 November 2004 press release that Congress has passed the Department of Energy High-End Computing Revitalization Act of 2004, which eliminates the requirement for hard metric concrete units in federal construction plans. The bill, expected to be signed into law by President [George W.] Bush, nixes the mandate for the use of round metric 200 x 200 x 400 mm concrete units, instead of soft-converted 203 x 203 x 406 mm units, in U.S. government construction.

The motivation for this rejection is enigmatic. The manufacturers would be putting less material into each brick, and probably charging the same amount of money.

USMA-Logo-2007A nostalgia for the 1970s metrication period begins to seep into the pages of Metric Today. The only new recurring metric issue is the need for a new definition of the Kilogram. The biggest news is not metric legislation from within the U.S. but pressure from Japan in 2006. Japan insisted that the U.S. Government “…ensure thorough adoption of the metric system in public and private sectors of the United States.” (MT May-Jun 2006). The European Union also began to insist on the U.S. providing metric only trade goods, but nothing would come from it. Metric meetings which had been so prevalent only a couple of decades before, were extinct.

From the time she became USMA president, Lorelle Young tirelessly traveled to Washington to lobby for metrication, and is prominent in Metric Today for constantly promoting metric. Unfortunately it is a never ending task in the U.S., which has reaped only irrational political push-back. One must admire her tenacity and drive in the face of overwhelming reactionary political opposition.

The current era appears, in my view, to be a retreat to the past. In the United States, only minute islands of metric exist within a sea of ancient US measures. New York state had converted to metric for road construction, then announced it would un-convert from metric in late 2006. Metric Today has a considerable number of column-millimeters which feature metric nostalgia, but no federal bills, pro or anti-metric, grace her pages. Why? Because they don’t exist. You will have a better chance of spotting Bigfoot on the House or Senate floor than pro-metric legislation.

NASA announced that its Constellation program was to be metric, and then later retracted that assertion.

Senator Clayborn Pell would die in 2009.

In 2009, metric signs along I-19 in Arizona began to feel the pressure from anti-metric forces. The signs had been placed there in 1979 while the US Metric Board existed. The I-19 Controversy continues and does not appear resolved.

The US Senate confirmed Dr. Patrick Gallagher as Director of NIST in 2009. As longtime readers know, he was the feckless Director who, in 2013, rebuffed the We The People Petition which requested that Congress make metric the exclusive measurement system of the US.

By 2010 The Federal Highway Administration issued a new manual of standards. The 2003 edition had dual-units; the 2009 has only inch-pound.

In 2013 an isolated metric legislative apparition appeared. MT reported:

On 17 January 2013 in the state of Hawaii, Representative Karl Rhoads, who represents Hawaii’s 28th House District, introduced a bill in the state legislature that would make the metric system the official system of measurement in Hawaii by the year 2018.

Why this legislation was introduced is a mystery. It was never passed.

In 2015 Lorelle Young stepped down as President of the USMA. There were immediate changes in Metric Today. The September-October issue contains a new tagline: Advocating the completion of U.S. conversion to the metric system. Had the old tagline remained, it would have informed the reader that 2016 was the USMA’s 100th year. It is hard to see the new tagline as anything other than a denial of the obvious: essentially there has been no beginning of a metric conversion in the United States. Advocating a completion of something which has not begun seems much like advocating that Liberia complete their quest to win the Super Bowl. I’m sure there must be a football team somewhere in Liberia, which means they have begun the process.

The writing found in the November-December Metric Today belies their tagline. USMA VP Paul Trusten has this to say (pp 3-4):

Even now, I feel almost as if I am acting in secret. What began as simple public ignorance has festered into a public prejudice similar to racial prejudice. …

So, with regard to completing US metrication, we have to take what may amount to a historically unique stance as both leaders and protesters. We are NOT underdogs. We are very much a part of the “Establishment,” the people who wield technology on a daily basis.

Martin Morrison, wrote a Metric Training and Education column that month which states:

There is an advantage in having US industry and media take the lead in metric conversion. When the government does things, all the naysayers rev up their anti-metric propaganda, much of which is factually false. When industry and media use metric by their own volition, there is essentially no resistance. Americans just accept it, or don’t even notice it. It just happens. In this context, I am very happy to see the US Metric Association’s new motto: “Advocating the Completion of US Conversion to the Metric System.” That nails it!

No Metric Philosopher has ever said it better. Despite the fact that we’ve had a voluntary metrication program “in place” since at least 1866, and re-emphasized in 1921, industry and the media continue to remain hostile to metric; I don’t see letting them “take the lead” as a viable or even rational position.

A country where the new USMA motto might make sense would be the UK. England is clearly very, very much metric, and should complete their conversion as they are well on their way. In the US, the metric system is but an abstraction that is only discussed when marginal candidates for President of the US mention it as a throw-away line, well into their speech. This metric statement is then used as a quick fear-fix for our media echo-chamber and the outrage junkies that jones for it. There is no metric conversion occurring in the US and (spoiler alert) there is no Santa Claus.

There are however several policies which have been embraced by the USMA which in my opinion do not help metric promotion. They are: 1. Referring to the inch-pound system. What we have in the US is in no way a system and in no way should this mess be equated with SI. Only the metric system is an integrated measurement system. US measures are literally medieval units  2. The encouragement of centimeters 3. The promotion of the prefix cluster around unity (hecto, deci, centi, deca). The absence of an open USMA policy advocating that any new metric legislation must be mandatory and not voluntary. 4. The promotion of metric to medieval conversions as a viable path to metric in the US. 5. The implied assertion that dual-unit dimensioning or packaging is progress toward metrication.

The USMA is celebrating its 100th year of metric promotion. Its centennial. Fredrick Halsey called the original organizers in the early twentieth century and said that he and other anti-metric people in the US had killed the metric system in 1905, and would kill it again in 1921. They were successful. Halsey’s successors have continued to make certain that no metric laws of any type are passed, let alone mandatory metric laws. I wish I could celebrate the fact that the US is so metric in 2016, that the need for the USMA had long passed, and there is no need for it to continue. Congratulations are in order for the USMA’s tenacity in the most hostile anti-metric Frozen Republic on the planet Earth.