Keywords: measure, measurement, ISO, SI, units, derived units, metrology

I’ve had various discussions about science where people have made statements like, “That can’t be measured” and “Science can’t tell us anything about that.” Since I have yet to be presented with a well-defined concept that science is not capable of analyzing (vs. necessarily already having analyzed) I feel such statements need to be addressed. This is the second in a series about metrology – the science of measurement.

In *Metrology is not about the weather. Part I – How to weigh a potato*, I introduced the most important thing to know about a measurement: there is always an error in both accuracy and precision. I then outlined the traceability of measurement standards from the International Organization for Standardization (ISO) to a produce scale used to weight a potato. The final section discussed how the formalization of measurement error is also a formalization of my three philosophical foundations discussed in *Am I a figment of your imagination?* In this installment, I will further explore the ISO standards and introduce the formal definition of a measure, which allows expansion beyond ISO based measurements. This will establish a foundation for further blogs on the relation between measurement and science.

**SI Base Units and Their Derivatives**

All physical measurements in the modern world derive from the International System of Units (SI). (The ISO coordinates the SI standard.) These are a set of seven physical units such as the second and the meter. I’ve copied the detailed but concise review of the basics from Wikipedia below, but first, here are some key points:

- There are
*seven base units*: second, metre, kilogram, ampere, kelvin, mole, and candela. - The base units are used to define
*derived units*which are combinations of the base units. (Some of these have standard names.) - The SI defines
*prefixes*, such as the somewhat familiar “mega“, “giga”, “milli”, and “nano.” These allow easy description of orders of magnitude. Is it a billion of or a billionth of?

From Wikipedia:

“The International System of Units (SI, abbreviated from the French *Système international (d’unités)*) is the modern form of the metric system. It is the only system of measurement with an official status in nearly every country in the world. It comprises a coherent system of units of measurement starting with seven base units, which are the second (the unit of time with the symbol s), metre (length, m), kilogram (mass, kg), ampere (electric current, A), kelvin (thermodynamic temperature, K), mole (amount of substance, mol), and candela (luminous intensity, cd). The system allows for an unlimited number of additional units, called derived units, which can always be represented as products of powers of the base units.^{[Note 1]} Twenty-two derived units have been provided with special names and symbols.^{[Note 2]} The seven base units and the 22 derived units with special names and symbols may be used in combination to express other derived units,^{[Note 3]} which are adopted to facilitate measurement of diverse quantities. The SI system also provides twenty prefixes to the unit names and unit symbols that may be used when specifying power-of-ten (i.e. decimal) multiples and sub-multiples of SI units. The SI is intended to be an evolving system; units and prefixes are created and unit definitions are modified through international agreement as the technology of measurement progresses and the precision of measurements improves.”

(End from Wikipedia)

Considering there are only seven base units, the concept of derived units is pretty important. A derived unit is just a combination of base units such as miles per gallon or pounds per square inch. Derived units can be really complex. For example, the unit of electrical conductance, the siemens (S), is defined as s^{3}A^{2}/kgm^{2}. But you can literally combine any or all of the base units, raise each of them to any power (positive or negative) and have yourself a lovely time trying to figure out what it measures. (As it says above, this allows for an infinite number of possible units.)

An interesting type of derived unit is the “unitless” measure. A commonly known example is Mach number, which is the ratio of the speed of an object to the speed of sound. Technically the unit of Mach is (meters per second) / (meters per second). But, in the sense that units can “cancel” the way numerators and denominators can, the resulting unit appears to be nonexistent – thus, “unitless.” Something that is not universally understood is that Mach number changes with atmospheric pressure (or, more generally, with the density of the material the sound is traveling through). That is, Mach number is not the equivalent of speed or velocity. Many of us have been told the heuristic of counting seconds between a flash of lightning and the resulting thunder to figure out how far away the lightning was. (The five second per mile heuristic translates to roughly Mach 1-million.) But this is based on the speed of sound at the atmospheric pressure found at sea level. Trick question: what’s the Mach number of a vehicle traveling in space? Sound doesn’t propagate in a vacuum, so the denominator in the calculation of Mach is 0. So the answer is Mach infinity. (A fun fact is that the space shuttles used to enter the atmosphere at about Mach 26; infinite to 26 in the blink of an eye!)

Another important aspect of units is conversion, like between meters and feet. (Some people may remember the Mars landing disaster that happened because this conversion wasn’t done.) Such conversions are merely mathematical so that they are not any different than a rose by any other name. Which unit you use is mere preference. In my experience, for some reason, metrologists tend to prefer furlongs per fortnight instead of miles per hour.

It’s also important to understand that the same unit name can be defined differently. For example there is the imperial ton as well as the metric ton. An interesting historic example is the cubit which has been defined to be many different lengths. And, of course, units are not always well defined, as when the foot was (supposedly) based on the length of the king’s foot. But that’s why we have standards.

**Units in Abstract and the Generalized (Mathematical) Definition of Measure**

Some metrologists would probably end the discussion of measure with the SI units and their derivatives. But it doesn’t take much thought about “units” to realize that there are plenty of things we use as units that don’t seem to be based on the SI units. The phrase, “you can’t compare apples and oranges” is effectively saying that apples and oranges are two different *units*. A way of discussing this is in terms of a generalized definition of measure. This definition is pretty straight forward, so I’ll start with it and then elucidate. From Wikipedia:

“In mathematical analysis, a *measure* on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume.”

Focus on the relation of “set” and “unit.” We might have a bunch of apples in front of us. We have a *set* of apples. We can pull out a subset of 5 apples. In this context, the unit of measure is “apple”. The size, or measure, of the subset is 5. You can substitute “unicorn” or any other object for “apple” and have it act as a unit. As such, the SI units are a subset of the more general concept of measure. You can stand on a plot (set) of land and measure its area in square meters.

The International System of Quantities and ISO/IEC 80000 are a generalized standard of quantities. The SI units are derivable from this standard, but the quantities defined by this standard go beyond the SI units. For example, there are information technology units such as bit and byte.

The reason I bring these quantities up is to emphasize that the definition of measure allows for a very broad range of measurements. All you have to do is define a set of things (such as apples) and establish a way of assigning sizes to subsets. (Of course, there are mathematical conditions required on how you assign the sizes as described in the Wikipedia article.) For future discussion, I want to point out that the SI units, including their derivatives, are the only physical units. I will call all other units “conceptual.” For example, bits and bytes are electromagnetically instantiated on disks, but you can’t really hold a bit or byte in your hand – they are conceptual. There are plenty of units other than those in the ISO standard, such as apples and unicorns. These also fall into the categories of physical and conceptual. In future blogs on metrology, I will discuss the relationship between conceptual and physical units in order to distinguish what types of units represent measurements of physical objects.

Before moving on, I want to address one of the technical requirements for a measure. A measure is a function onto the real numbers. That is, to formally state the size of a subset is to provide a number. In Part I of this metrology series I point out that the eyeball is the most common measurement device. But the eyeball rarely constructs a number during its measurement. This illustrates the difference between “in practice” and “in theory”. In practice we don’t need to know the numeric size of something to know whether it will go through a door. But in theory we could quantify it if we need to.

**Analyzing Sets of Measurements (aka Statistics)**

I’m not going to go into statistics very much, but it is an important aspect of analyzing measurements. A single measurement can be useful, but it’s by grouping sets of measurements that give us deeper understanding. Roughly speaking, data is grouped across space or time or both. How many gorillas are there at a moment in time and where are they? How has their population changed over time? What are the demographics of a particular issue and how has that changed?

One of the main ways of analyzing sets of data is to determine their distribution curve (more properly called their probability distribution). The standard bell curve is probably the most well-known. But there are lots of other distributions. After determining the distribution, there are lots of analyses that can be done on the data: determining mean, median, standard deviation, and many more complicated characteristics. It is also possible to take many sets of data, each with their own distribution, and combine them. My first point is that the level of analysis can get very sophisticated and such techniques provide deeper insights than individual measurements. My second point is that these analyses are ultimately based on individual physical measurements. My third point is that the complexity of analysis is what sometimes leads to the famous expression about types of lies.

**Summary and Looking Ahead**

I started with the SI base units and then provided the formal generalized definition of measure and its relation to unit. This illustrated the broad nature of “measure”. Along the way, I mentioned that some units measure physical objects while others are conceptual. In future metrology blogs I will discuss how to tell when a unit relates to a physical object. This will aid in discussing what science is and, in particular, in demonstrating that the so called “soft” social sciences are not only sciences, but that they are *physical* sciences.

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