So, in 1889, an actual metal bar, made of iridium and platinum, was constructed to be the official meter and to avoid the ambiguity of the distance to the North Pole. All measuring sticks would then be based on the dimensions of this literally quintessential meter.
But that still didn’t suffice, for this bar could still undergo deterioration. In addition, any slight change in the atmosphere or temperature can change its size, albeit very slightly. These considerations were included in the definition by specifying the pressure and temperature of the bar’s environment, but such precise conditions are very difficult to maintain.
An international group of scientists then constructed more fundamental definitions, first using the wavelength of the emission of a certain isotope of the gas krypton, and finally arriving at our current definition, which involves the distance light travels in a fantastically small, though extremely precisely defined, span of time. In this way, the speed of light and the length of the meter5 are now inextricably and definitionally linked. As our measurements become more precise, the speed of light doesn’t change; instead, the definition of the meter does.
• • •
THE world of measurement involves much more6 than just the meter. If you wish to see how far down the rabbit hole of measurement it is possible to go, I recommend the Encyclopaedia of Scientific Units, Weights, and Measures: Their SI Equivalences and Origins. Compiled by François Cardarelli, a French Canadian chemical engineer, it is truly a wide-ranging document. It has conversion tables for units of measurement throughout history, from Abyssinian units of length to Egyptian units of weight, from the Roman system of distance, which uses a gradus (a single stride when walking) and passus (two strides) to denote distance, to the Assyrio-Chaldean-Persian measurement system. This book is exhaustive.
Interested in moving beyond light-years and parsecs (about three and a quarter light-years) to describe outer space? Then consider the siriusweit, which is equal to five parsecs. Or wondering about the details of the fothers, a British mass for trading lead bullion, or the kannor, a Finnish unit of volume? This book can fulfill your needs.
There are even various discrete units included, such as the perfect ream, which is 516 sheets of paper, and the warp, which is four herrings; it is used by British fishermen and old men at kiddush.
In addition to all this useful and possibly not so useful information, the book includes an intriguing table that shows how each definition of the meter reduced errors in measurement overall: Each successive definition made the meter a bit less uncertain. On the facing page is a chart that displays the table in graphic form.
These data points aren’t just for years. Each redefinition occurred at a very precise date. We know the very day when the meter became as precise as what we have now. Furthermore, the precision of the meter has increased in a regular fashion, and its error has declined in a linear fashion: an exponential decay. Just as scientific prefixes have changed in an exponential fashion, allowing for more precise terminology, so has the way we define measurement itself.
Figure 8. Measurement error of the meter over time. The precision in the definition of the meter has increased, with an exponential decay in error (the error axis is logarithmic) over time. Line shows general downward trend. Data from Cardarelli, Encyclopaedia of Scientific Units, Weights, and Measures: Their SI Equivalences and Origins (Springer, 2004).
The meter’s shift in precision, as well as definition, is not an aberration. Scientists have been driving toward ensuring that metric units in general be based on physical constants of the universe. In addition to the meter’s tie to the light-year, time too is known precisely: A second is defined in terms of the vibration rate of a certain type of cesium atom.
The last basic unit in the metric system to undergo this transition to definition in terms of physical constants is the kilogram. For a long time the official kilogram was defined as the weight of a physical cylinder of platinum and iridium stored in a basement vault outside Paris. Over the past few years, metrologists—the scientists preoccupied with matters of measurement—have been bandying about alternative definitions,7 such as the mass of a sphere of silicon with an exact number of atoms or a precise amount of electromagnetic energy. In October 2011, they finally convened outside Paris at the Twenty-fourth General Conference on Weights and Measures and decided on a definition based on a physical constant as the official description of a kilogram.
Even though our units of measurement have become unbelievably exact, we don’t normally really reach that level of precision. A certain amount of error and uncertainty are baked into our daily lives. Despite the increased precision with which all these units are now defined, we still deal with a certain amount of uncertainty when making measurements. Most people use a fairly basic ruler, despite the recent advances in precision. I can distinctly recall the yardstick my family owned when I was growing up—it was so worn at the ends that it was probably missing nearly an entire inch. Whatever I measured was objectively wrong, but it was close enough for everyday purposes. Similarly, when we exchange money from one currency to another, we don’t mind that these conversions are necessarily approximate, inaccurate by several thousandths of a dollar or euro.
But understanding why we have measurement error, and properly understanding precision, can help us better understand how facts change and how measuring our world can lead to changes in knowledge.
• • •
IN 1980, A. J. Dessler and C. T. Russell published a tongue-in-cheek paper8 in Eos, a journal of the American Geophysical Union. In it, they examined the estimated mass of Pluto’s size over time. We still don’t know Pluto’s mass, at least not exactly. Since it vents gases, astronomers often have trouble telling its size, sometimes viewing its self-generated haze as part of the surface.
Since Pluto’s first sighting, when it was judged to be about the size of the Earth, estimates of its mass have dropped greatly over time. Dessler and Russell explained this by arguing something simple: Pluto itself is actually shrinking. By fitting the curve of Pluto’s diminishing size to a bizarre mathematical function using the irrational number π, they argued that Pluto would vanish in 1984. But don’t worry! According to their function, Pluto would reappear 272 years later (its mass would go from being mathematically imaginary to real again).
Of course, this is ridiculous. A far more reasonable explanation is that our tools improved over time via the Moore’s-like laws that inexorably improve technology, allowing us to resolve Pluto better. While there’s still a certain amount of uncertainty in Pluto’s mass, we now have a much better handle on this fact: There is a clear relationship between what our facts are, increases in technology, and increases in measurement.
When it comes to error, measurement revolves around two terms: precision and accuracy. Any measurement method inherently has these two properties, and it’s important to be aware of them when examining the true value of something. We can understand precision and accuracy through a rather whimsical scenario.
Imagine we are trying to determine the position of a point on a far-off surface by using a laser pointer. We have two different people tasked with trying to locate this point by aiming the laser pointer at it: a young boy who is lazy, and an older man who is very careful in his measurements. The young boy, endowed with the steady hand of youth, is physically capable of pointing the laser exactly at the point on the wall. But he doesn’t want to for very long, because in our rather contrived example, he is inherently lazy, so he always chooses to rest his laser-pointer arm on a nearby surface. In this case, no matter how many times this boy points the laser at the point, it is always lower than the point by a little bit, because he chooses to rest his wrist on a lower surface.
On the other hand, the old man tries his best. He points the laser exactly at the point, but due to his age he has a slight tremor. So the laser is always hovering around the point, within a certain r
ange, but it is rarely exactly where it should be.
In case this hasn’t yet become clear, the old man embodies accuracy and the young boy embodies precision. Precision refers to how consistent one’s measurements are from time to time. If the true length of something is twenty inches, precision refers to how dispersed one’s measurements will be around the true value. If one measurement method always yields values of twenty-five inches, while another measurement method yields values within half an inch of twenty inches, but they are all variable, the former method is more precise, even if its results are wrong.
Accuracy refers to how similar one’s measurements are to the real value. If your measurements are always five inches too high, even if your measurements are very consistent (and therefore are highly precise), you lack accuracy.
Of course, all methods are neither perfectly precise nor perfectly accurate; they are characterized by a mixture of imprecision and inaccuracy. But we can keep on trying to improve our measurement methods. When we do, changes in precision and accuracy affect the facts we know, and sometimes cause a more drastic overhaul in our facts.
• • •
EVERYONE recognizes the periodic table. The gridlike organization of all the known chemical elements contains a wealth of information. Each square itself holds a great many facts. For each element, we get its chemical symbol, its full name, and its atomic number and weight.
What exactly are these last two? Atomic number is simple: It represents the number of protons in the nucleus of an atom of this element and, similarly, the number of electrons that surround the nucleus. Since the number of electrons dictates a large portion of the nature of the interactions of an atom, knowing the atomic number allows a chemist to get a reasonably good understanding of the chemical properties of the element quite rapidly.
The atomic weight, however, is a bit more tricky. When I was younger, in grade school, I learned that the atomic weight is the sum of the number of protons in the nucleus and the number of neutrons in a “normal” nucleus. However, we were also taught that the number of neutrons in each atom can vary. If an element, as defined by the number of protons, can have different numbers of neutrons in its nucleus, these different versions are known as isotopes.
Hydrogen, with its lone proton, is the normal isotope of hydrogen that we think of. However, there’s another version, with one proton and one neutron, known as deuterium. If you make water using deuterium, it’s known as heavy water, because the hydrogen is heavier than normal. Ice cubes of heavy water will actually sink in regular water.
So, really, what the atomic weight describes is something a good deal more complex than what I was told when I was young. The atomic weight is the “average” size of the nucleus, in proportion to the prevalence of all isotopes of that element in nature. So for element X, if there are only two isotopes, we take their relative frequency in the world and weigh these sizes accordingly. In doing so, the atomic weight yields a sort of expected weight of neutrons and protons if you were to take a chunk of that element out of the Earth and the isotopes are all neatly mixed in together.
For a long time, these atomic weights were taken as constant. They were first calculated more than one hundred years ago and propagated in periodic tables around the world, with the occasional updates to account for what was assumed to be more precise measurement. But it turns out that atomic weights vary. Which country a sample is taken from, or even what type of water the element is found in, can give a different isotope mixture.
Now that more precise measurements of the frequency of isotopes are possible, atomic weights are no longer viewed as constant. The Internal Union of Pure and Applied Chemistry recently acknowledged this state of the world, alongside our increased ability to note small variations, and scrapped overly precise atomic weights; it now gives ranges rather than specific numbers, although many periodic tables still lack them.
Through increases in measurement, what were once thought to be infinitely accurate constants are now far fuzzier facts, just like the height of Mount Everest. But measurement’s role is not only in determining amounts or heights. Measurement (and its sibling, error) are important factors in the scientific process in general, whenever we are trying to test whether a hypothesis is true. Scientific knowledge is dependent on measurement.
• • •
IF you ever delve a bit below the surface when reading about a scientific result, you will often bump into the term p-value. P-values are an integral part of determining how new knowledge is created. More important, they give us a way of estimating the possibility of error.
Anytime a scientist tries to discover something new or validate an exciting and novel hypothesis, she tests it against something else. Specifically, our scientist tests it against a version of the world where the hypothesis would not be true. This state of the world, where our intriguing hypothesis is not true and all that we see is exactly just as boring as we pessimistically expect, is known as the null hypothesis. Whether the world conforms to our exciting hypothesis or not can be determined by p-values.
Let’s use an example. Imagine we think that a certain form of a gene—let’s call it L—is more often found in left-handed people than in right-handed people, and is therefore associated with left-handedness. To test this, we gather up one hundred people—fifty left-handers and fifty right-handers—and test them for L.
What do we find? We find that thirty of the fifty left-handers have the genetic marker, while only twenty-two right-handers have it. In the face of this, it seems that we found exactly what we expected: left-handers are more likely to have L than right-handers. But is that really so?
The science of statistics is designed to answer this question by asking it in a more precise fashion: What is the chance that there actually is an equal frequency of left-handers with L and right-handers with L, but we simply happened to get an uneven batch? We know that when flipping a coin ten times, we don’t necessarily get exactly five heads and five tails. The same is true in the null hypothesis scenario for our L experiment.
Enter p-values. Using sophisticated statistical analyses, we can reduce this complicated question to a single number: the p-value. This provides us with the probability that our result, which appears to support our hypothesis, is simply due to chance.
For example, using certain assumptions, we can calculate what the p-value is for the above results: 0.16, or 16 percent. What this means is that there is about a one in six chance9 that this result is simply due to sampling variation (getting a few more L left-handers and a few less L right-handed carriers than we expected, if they are of equal frequency).
On the other hand, imagine if we had gathered a much larger group and still had the same fractions: Out of 500 left-handers, 300 carried L, while out of 500 right-handers, only 220 were carriers for L. If we ran the exact same test, we get a much lower p-value. Now it’s less than 0.0001. This means that there is less than one hundredth of 1 percent chance that the differences are due to chance alone. The larger the sample we get, the better we can test our questions. The smaller the p-value, the more robust our findings.
But to publish a result in a scientific journal, you don’t need a minuscule p-value. In general, you need a p-value less than 0.05 or, sometimes, 0.01. For 0.05, this means that there is a one in twenty probability that the result being reported is in fact not real!
Comic strip writer Randall Munroe illustrated some of the failings of this threshold10 for scientific publication: The comic shows some scientists testing whether jelly beans cause acne. After finding no link, someone recommends they test different colors individually. After going through numerous colors, from salmon to orange, none are found to be related to acne, except for one: The green jelly beans are found to be linked to acne, with a p-value less than 0.05. But how many colors were examined? Twenty. And yet, explaining that this might be due to chance does little to prevent the headline decl
aring jelly beans linked to acne. John Maynard Smith, a renowned evolutionary biologist, once pithily summarized this approach: “Statistics is the science11 that lets you do twenty experiments a year and publish one false result in Nature.”
Our ability to measure the world and extrapolate facts from it is intimately tied to chances of error, and the scientific process is full of different ways that measurement errors can creep in. One of these, where p-values play a role, is when a fact “declines” (and sometimes even vanishes) as several different scientists try to examine the same question.
• • •
IN the late nineteenth and early twentieth centuries, astronomers obsessed over a question that was of great importance to the solar system: the existence of Planet X.
Within a few decades after the discovery of the planet Uranus in 1781—the first planet to be discovered in modern times—a great number of oddities were noticed in its orbit. Specifically, it deviated from its orbital path quite a good deal more than could be explained by chance. Scientists realized that something was affecting its orbit, and this led to the prediction and discovery of the planet Neptune in 1846.
This predictable discovery of a new planet was a great cause for celebration: The power of science and mathematics, and ultimately the human mind, was vindicated in a spectacular way. A component of the cosmos—complete with predicted location and magnitude—was inferred through sheer intellect.
Naturally, this process cried out for repeating. If Neptune too might have orbital irregularities, perhaps this would be indicative of a planet beyond the orbit of Neptune. And so, beginning in the mid-nineteenth century, careful measurements were made in order to predict the location and properties of what would eventually become known as Planet X.
The Half-Life of Facts Page 16
