Measurement
Measurement begins with a fall. Drop an object one metre in the gravitational field of the Earth, and physics says it should take about 0.45 second to reach the ground. That clean little number hides a swarm of trouble. The figure for the acceleration of gravity, 9.8 metres per second squared, is precise to only two significant digits. The field itself shifts slightly with height above sea level. Extracting a square root forced a rounding-off. And then there is air resistance, the exact instant the object was released, and even the posture of the human participants timing it. Measurement is the quantification of an attribute of an object or event, a way of determining how large or small a physical quantity is against a basic reference quantity of the same kind. It is, at heart, the comparison of an unknown quantity with a known or standard one. So how did communities ever agree on what an inch or a second should be? Who decided a metre, and how is it that a unit can be tied to nothing more solid than the speed of light? And why does a quantum system seem to change the very instant you try to measure it?
Four criteria pin down any measurement of a property: type, magnitude, unit, and uncertainty. Together they let two results be compared without ambiguity. The type, or level of measurement, is a taxonomy for the character of the comparison itself. Two states of a property may be set against each other by ratio, by difference, or by ordinal preference, though this type is usually left implicit in the measurement procedure rather than spoken aloud.
Magnitude is the numerical value of the thing, usually read off a suitably chosen measuring instrument. The unit then assigns a mathematical weighting factor to that magnitude, derived as a ratio to the property of a standard artefact or a natural physical quantity. Uncertainty is the honest part. It represents the random and systemic errors of the procedure and indicates a confidence level in the result. Errors get evaluated by methodically repeating the measurement and weighing the accuracy and precision of the instrument.
The scope of all this depends on the field. In the natural sciences and engineering, measurement does not apply to nominal properties of objects, in keeping with the International Vocabulary of Metrology published by the International Bureau of Weights and Measures. Statistics and the social and behavioural sciences take a wider view, allowing nominal, ordinal, interval, and ratio scales. The discipline that studies all of this has its own name. Metrology is the science of measurement.
Charles Sanders Peirce, who lived from 1839 to 1914, made the first proposal to tie an SI base unit to an experimental standard independent of any physical artefact. He suggested defining the metre in terms of the wavelength of a spectral line. The idea directly influenced the Michelson-Morley experiment. Michelson and Morley cite Peirce and improve on his method.
Artefact-free definitions are the prize here. Fixing a unit to a physical constant or some invariable natural phenomenon means it cannot deteriorate or be destroyed, unlike a standard object kept in a vault. The only way such a unit can change is through increased accuracy in determining the value of the constant it is tied to. Every one of the seven SI base units is now defined this way, without reference to a particular physical object.
The modern system reduces all physical measurements to a mathematical combination of those seven base units: the kilogram, metre, candela, second, ampere, kelvin, and mole. Each rides on a defining constant. The second rests on the hyperfine splitting in caesium-133, the metre on the speed of light, the mass unit on the Planck constant, the ampere on the elementary charge, temperature on the Boltzmann constant, amount of substance on the Avogadro constant, and luminous intensity on the luminous efficacy of a 540-terahertz source. From these seven, derived units are built: the watt, the unit for power, comes out as metres squared times kilograms times seconds to the minus three.
Laws regulating measurement were first written to prevent fraud in commerce. Nothing in nature insists that an inch be a certain length, or that a mile beats a kilometre as a measure of distance. Standards evolved over human history first for convenience and then out of necessity, so that communities would share common benchmarks. Earlier still, many separate measurement systems existed for the varied fields of human existence, often settled by local agreements between trading partners.
The General Conference on Weights and Measures, or CGPM, sits at the top of this structure. It was established in 1875 by the Metre Convention and oversees the International System of Units. Its rulings rewrite the world's units on a scientific basis. The metre was redefined in 1983 in terms of the speed of light. The kilogram was redefined in 2019 in terms of the Planck constant. The international yard was fixed in 1960 by the governments of the United States, the United Kingdom, Australia, and South Africa as exactly 0.9144 metres.
Nations keep their own guardians of commercial measurement. In the United States, the National Institute of Standards and Technology, a division of the Department of Commerce, regulates commercial measurements. The United Kingdom relies on the National Physical Laboratory, Australia on the National Measurement Institute, South Africa on the Council for Scientific and Industrial Research, and India on its own National Physical Laboratory of India.
Draught beer and cider in Britain must still be sold by the imperial pint, and milk in returnable bottles can be too. Many Britons give their height in feet and inches and their weight in stone and pounds, even though the country officially switched to SI. Road signs there hold out as well, showing distances in miles, or in yards for short stretches, with speed limits in miles per hour.
The British systems of English units and later imperial units once ruled Britain, the Commonwealth, and the United States. In America the system became known as U.S. customary units and is still in use there and in a few Caribbean countries. These have at times been called foot-pound-second systems, after the imperial units for length, weight, and time. Yet the labels can deceive: tons, hundredweights, gallons, and nautical miles carry different values in the U.S. and imperial systems. Imperial units linger far beyond Britain too. In many Commonwealth countries counted as metricated, land is measured in acres and floor space in square feet for commercial deals, and gasoline is often sold by the gallon.
The metric system answers with decimal order. It is built on the metre for length and the kilogram for mass, and it exists in several variations with different base units that do not affect daily use. Since the 1960s the International System of Units has been the internationally recognised metric system, developed in 1960 from the metre-kilogram-second system rather than the centimetre-gram-second one. Its prefixes make conversion trivial: to go from metres to centimetres you multiply by 100, since a metre holds 100 centimetres.
A two-metre carpenter's rule folds down to a length of only 20 centimetres to fit a pocket, and a five-metre tape measure retracts into a small housing. The ruler, strictly speaking, is the instrument for ruling straight lines, while the calibrated tool for finding length is properly called a measure. Common usage blurs the two and calls both rulers, reserving the name straightedge for an unmarked rule. The older sense of measure survives only in the phrase tape measure, a tool that can measure but cannot draw a straight line.
Weighing reveals a deeper split between mass and weight. Mass is the intrinsic property of all material objects to resist changes in their momentum. Weight is the downward force produced when a mass sits in a gravitational field. In free fall, with no net gravitational forces, objects lack weight but keep their mass. A spring scale measures force but not mass, while a balance compares weight, and both need a gravitational field to work at all. Some of the most accurate instruments rest on load cells with a digital read-out, yet they too would fail in free fall.
Exactness has its own grammar. In the number 12000, all non-zero digits and any zeros between them are significant, giving two significant digits and implied limits of 11500 and 12500. Zeros added after a decimal separator signal greater exactness: 1 carries implied limits of 0.5 and 1.5, while 1.0 narrows them to 0.95 and 1.05.
Survey research turns the tools of measurement on attitudes, values, and behaviour, using questionnaires as the instrument. Like every measurement, it is vulnerable to measurement error, the departure of the recorded value from the true one. In substantive survey research, such error can produce biased conclusions and wrongly estimated effects, so results must be corrected for it when it appears.
Economics measures in three modes: physical measures, nominal price value measures, and real price measures. They differ by the variables they capture and the variables they leave out. Biology, by contrast, has generally no well established theory of measurement, though it stresses the importance of theoretical context. The context drawn from the theory of evolution leads biologists to join the theory of measurement with historicity as a fundamental notion. Among biology's most developed measurement fields are genetic diversity and species diversity.
The physicists offer competing definitions of what measurement even is. The classical definition, standard throughout the physical sciences, calls measurement the determination or estimation of ratios of quantities, an idea traced to John Wallis and Isaac Newton and foreshadowed in Euclid's Elements. Representational theory instead defines measurement as the correlation of numbers with entities that are not numbers, with its most elaborated form known as additive conjoint measurement. A third view treats every measurement as a set of observations that reduce uncertainty, expressed as a quantity, so that a range of values is assigned rather than a single one. By this account there is no neat line between estimation and measurement.
In quantum mechanics, a measurement is an action that determines a particular property of a quantum system, such as position, momentum, or energy. These measurements are always statistical samples drawn from a probability distribution, and for many quantum phenomena that distribution is discrete rather than continuous.
Measurement here does something no ruler does: it alters the very state it inspects. The act appears to work as a filter, changing the quantum state into one carrying the single measured value. Yet repeated measurements on a quantum state remain reproducible, which only deepens the puzzle. The most common interpretation holds that when a measurement is performed, the wavefunction of the system collapses to a single, definite value.
What that collapse actually means is an unresolved fundamental problem in quantum mechanics. The discipline that began by comparing an unknown quantity against a known standard ends, at its smallest scale, unable to say with certainty what happens in the instant of comparison itself.
Common questions
What is measurement and how is it defined?
Measurement is the quantification of an attribute of an object or event, used to compare it with other objects or events. It is the process of determining how large or small a physical quantity is compared to a basic reference quantity of the same kind, or the comparison of an unknown quantity with a known or standard one.
What are the seven SI base units in measurement?
The seven SI base units are the second, metre, kilogram, ampere, kelvin, mole, and candela. Each is tied to a defining constant, such as the speed of light for the metre, the Planck constant for the kilogram, and the hyperfine splitting in caesium-133 for the second.
Who first proposed defining a measurement unit without a physical artefact?
Charles Sanders Peirce, who lived from 1839 to 1914, made the first proposal to tie an SI base unit to an experimental standard independent of a physical artefact. He proposed defining the metre in terms of the wavelength of a spectral line, an idea that directly influenced the Michelson-Morley experiment.
When was the General Conference on Weights and Measures established?
The General Conference on Weights and Measures, known as the CGPM, was established in 1875 by the Metre Convention. It oversees the International System of Units, and through it the metre was redefined in 1983 in terms of the speed of light and the kilogram in 2019 in terms of the Planck constant.
What is the difference between mass and weight in measurement?
Mass is the intrinsic property of all material objects to resist changes in their momentum, while weight is the downward force produced when a mass sits in a gravitational field. In free fall, with no net gravitational forces, objects lack weight but retain their mass.
How does measurement work in quantum mechanics?
In quantum mechanics, a measurement is an action that determines a particular property such as position, momentum, or energy of a quantum system. These measurements are always statistical samples from a probability distribution, and the most common interpretation holds that the wavefunction collapses to a single, definite value when a measurement is performed.