How has measurement broadened its scope?

Concept and definition of measurement beyond the physical and engineering perspective

Figure 1. A triple point water cell where ice, water and vapour coexist (by the author)

Prologue

Last week, I wrote about uncertainty with a message for the metrologist who tends to stick to the definition of a term from his point of view. Measurement might be another topic that the metrologist might claim as his own. In the early 20th century, measurement was not popular only in physics and engineering but also in experimental psychology and behavioural Science. In fact, according to the Google Books Ngram viewer, the use of the word ‘measurement’ increased steadily from the 1800s to 1940 and then more than doubled by the 1980s, as shown in Figure 2. This increase is peculiar compared to the fluctuating use of the word ‘science’. It may be related to the trend of using the word ‘uncertainty’ since measurement is an active action to reduce uncertainty.

In this article, I will explore how the definition of measurement has broadened its scope beyond the physical and engineering perspective. This theme would underpin the theory of measurement not only for metrology but also for the social sciences. With such a foundation for the theory, I want to write an article about the theory soon.

Figure 2. Google Books Ngram Viewer of words including measurement

Campbell’s view

By the time Norman Campbell published a book in 1920, measurement was accepted as representing properties of objects or events by assigning numbers to them in such a way as to reproduce empirical relationships in the numerical domain. This idea was primarily Helmholtz’s intellectual legacy. In his book, Campbell asked, “Why can and do we measure some properties of bodies while not measuring others?” and asked, “What is the difference between the properties that determine the possibility or impossibility of measuring them?” [1]

He divided quantities into two main types: fundamental (e.g. mass and length) and derived (e.g. temperature, density). Both require an empirical property of order in the bodies to be measured. Fundamental quantities, however, require a physical addition operation. The physical connection of two rods with different length units follows the additivity of the numerical operation. On the other hand, the physical addition of two bottles of water of the same temperature gives the same temperature, whereas the numerical addition gives twice the temperature value. The addition of temperature as a derived quantity in a physical domain cannot be represented by arithmetic addition in a numerical domain. However, the volume of water obeys additivity in both domains: we can consider volume as a fundamental quantity. Note that measurement is about assigning a number to a property of an object, not to the object itself.

Figure 3. A homomorphism is a mapping between two algebraic structures of the same type, preserving the operations in the structures. Within a set A of a physical domain, how to make a physical operation ⊕ should be defined, and what condition to satisfy ⇔. For example, we can define the operator ⊕ as putting mass blocks together on a pan of a balance. The balanced condition of both arms can be defined as ⇔.

More mathematically, given a mapping from a physical to a numerical domain, the mapping preserves an empirical structure (e.g. physical addition in a physical domain) in an algebraic structure (e.g. arithmetic addition in a numerical domain). The physical structure is homomorphic to the arithmetic structure (See Figure). Homomorphism answers Campbell’s question: The fundamental quantities are homomorphic, but the derived ones are not. The intuition of a physical addition convinces us that commutativity and associativity are also valid. We can have more mathematical discussions with commutativity and associativity. I hope to try it in the future, not too much further.

Homomorphism is so important that it allows us to construct a measurement scale for measuring a fundamental quantity.[2] The scale is a set of standards with correctly assigned numerical values. We can measure a given object by comparing it to the scale. On the other hand, derived quantities do not require the construction of a specific scale because they can be measured thanks to a physical law that relates them to other fundamental quantities. For example, temperature can be defined by measuring the pressure (P), volume (V) and number of moles (n) of the gas through the ideal gas law: PV = nRT, and density (ρ) can be defined as the ratio of mass (m) to volume (V): ρ = m/V.

Nowadays, the physical or chemical properties of materials and systems are often classified as either intensive or extensive, depending on how the properties change when their sizes (or extent) change. Roughly speaking, the fundamental and derived quantities have extensive and intensive properties, respectively. Campbell’s work contributed to a deeper understanding of the nature of measurement, seeing that the classification into intensive or extensive properties is still in effect.[3]

In 1932, the British Association for the Advancement of Science appointed a committee of physicists and psychologists to consider and report on the possibility of quantitative estimates of sensory events in psychology and behavioural Science. However, the two sides of the Committee couldn’t reach a common understanding on measurement. The physicists vehemently opposed the possibility of actually making measurements in behavioural Science. Although the psychologists claimed it was possible to quantify sensations, the physicists denied it, mainly because direct estimation of sensations was not feasible, and additivity was inconceivable to them.

Campbell was deeply involved in the 1930s controversy and influenced the Committee’s report. The report had an enormous impact in the years that followed and continues to this day, leading to an essentially parallel development of measurement science in both scientific disciplines.

Figure 4. Stevens’ monumental treatise that was published in Science [4]

Stevens’ view

While this debate was raging in the Committee, Stevens, who was not involved in the discussion, was at the same time working on problems similar to those being considered by the Committee. Since Campbell’s view allowed for only one type of scale and an empirical operation of addition, he felt that it was necessary to develop a more general theory of measurement that could increase the number of possible scale types of measurement. In 1946, he published the monumental treatise in Science. (see Figure 3.)

Figure 5. Fourfold classification of scales of measurement (from Stevens’ paper)

If he accepted measurement as assigning numbers to objects or events according to rules, then there could be different rules for assigning numbers, leading to other kinds of scales and various types of measurement. To increase the number of possible scales of measurement, he realised that the problem should be to make explicit “(a) the various rules for the assignment of numerals, (b) the mathematical properties (or group structure) of the resulting scales, and (C) the statistical operations applicable to measurements made with each type of scale.”

Finally, he proposed four types of scales, which are so famous that they are still used today. His classification is based on transformations that leave the form of the scale invariant. Although we can now see a similar table on Wikipedia because their compilation of the original table omits a critical column containing his central idea, I would like to show you the original version in Figure 4. The table shows that he shifted his focus from empirical operations (e.g. order, additivity...) to transformations (column 3) and the invariant statistic (column 4) through the transformations. The invariance of a statistic means that the functional relationship is satisfied with the statistic in both domains, connected by a mapping x’ = f(x). For example, in interval scale, we can easily show that if f(x) = ax + b, then [mean of domain X’] = [a * (mean of domain X) + b].

Nominal scales are involved in classification operations and numbers to distinguish one class of objects from another. Any bi-univocal transformation is allowed because identification is still possible. Bi-univocal relation is a mathematical term meaning one-to-one. A bi-univocal relation between sets A and B means that an element a from set A is related to one and only one element b from set B, which itself is related only to a. Examples of such scales are the sex of a human being, the colour of a paint sample, the ISO two-letter country code, and the sequence of amino acids in a polypeptide, expressed also in words, by alphanumeric codes, or by other means.

Ordinal scales result from the operation of ranking objects or individuals according to some criterion, but the intervals between ranks may not be equal. In other words, ordinal scales provide a relative ordering of objects without necessarily implying equal differences between them. However, since any ‘order-preserving (monotonically increasing)’ transformation will leave the scale form invariant, this scale has the structure of what might be called an order-preserving group. An example of this scale is the scale of hardness of minerals, earthquakes, or wind intensity. A Likert scale is also an ordinal scale commonly used in surveys and questionnaires to assess attitudes, opinions, or perceptions. It typically consists of several items or statements to which respondents indicate their level of agreement or disagreement on a scale. We should be more cautious when interpreting the mean and standard deviation of the Likert scale after a survey, as these statistics are not invariant in an ordinal scale.

Interval scales maintain the rank order of objects or individuals but also have equal intervals between adjacent points on the scales. However, interval scales lack a true zero, i.e. zero does not indicate the absence of the measured attribute. They remain invariant to linear positive transformations. Temperatures in Fahrenheit or Celsius are good examples.

Ratio scales have all the characteristics of interval scales, with the added feature of an actual zero point. This zero point indicates the absence of the measured attribute, and ratios between measurements are meaningful. They are invariant under any simple multiplicative transformation or similarity. Their measurements are the estimations of the ratios between the magnitude of a continuous quantity and a unit of measurement of the same kind. Therefore, most measurements in the physical sciences and engineering are made on ratio scales. Absolute temperature is a good example, as are mass and length.

In summary, Stevens’ efforts to introduce these types of scales met with resistance from physicists and traditionalists in the scientific community. Physicists, in particular, were used for precise and objective measurements of physical quantities, often expressed in standard units with well-defined properties. The introduction of scale types for psychological attributes was met with scepticism because these attributes were perceived as inherently subjective and less amenable to precise measurement.

VIM’s view

The International Vocabulary of Metrology (VIM) is a guide that aims to disseminate scientific and technological knowledge about metrology by harmonising its basic terminology worldwide. Its 4th edition is currently under revision. It can be accepted as the official position of metrologists on measurement. VIM defines measurement as the process of experimentally obtaining one or more quantity values that can reasonably be attributed to a quantity, where quantity is a property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference.[5]

The quantity value is defined as “number and reference together expressing the magnitude of a quantity”. There are several examples: length of a given rod ( 5.34 m or 534 cm), electric impedance of a given circuit element at a given frequency, where j is the imaginary unit [(7 + 3j) Ω], refractive index of a given sample of glass (1.32), Rockwell C hardness of a given sample (43.5 HRC) and mass fraction of cadmium in a given sample of copper (3 µg/kg or 3 ·10^−9). The refractive index as a quantity seems to have no reference. However, if we consider the refractive index as the ratio of the speed of light in a medium to that in a vacuum, its reference is implicitly (m/s)/(m/s), as if that of the mass fraction were µg/kg.

VIM clearly states in NOTE 1 that “measurement does not apply to nominal properties”, which is the same as Stevens’ nominal scale and in NOTE 2, that “measurement implies comparison of quantities or counting of entities”. Since comparison yields quality or ranking, NOTE 2 allow Stevens’ ordinal scales to be implicitly accepted as measurement. In addition, although counting does not necessarily require a reference, we can consider counting as having a reference of 1 to meet the first definition of measurement.

VIM notes another: “Measurement presupposes a description of the quantity commensurate with the intended use of a measurement result, a measurement procedure, and a calibrated measuring system operating according to the specified measurement procedure, including the measurement conditions”. Even if this statement seems too comprehensive, I expect that the Committee would like to see measurement as a well-planned and well-prepared action before measurement. However, they may not want to be too strict in interpreting the document by adding an ANNOTATION: “experimentally obtaining” in the definition is to be construed in a broad sense of not including only direct physical comparison but also of using models and calculations that are based on theoretical considerations.

Epilogue

We have explored how the concept of measurement has expanded its scope from physics and engineering to the social sciences by introducing the views of Campbell and Stevens. By reviewing the VIM view, we can see how Stevens’ idea contributes to its expansion and the development of measurement theory. Considering that the VIM explicitly excludes the assignment of a nominal scale from measurement and implicitly includes the comparison of quantities or the counting of entities in measurement, I think that the levels of measurement except the nominal scale can be accepted as a measurement from a metrological point of view. In the same context, we can consider any Likert scale survey result as a measurement. However, we should be more careful when interpreting statistics such as mean and standard deviation, which are not invariant. Stevens’ ratio scale has the same property as the quantities of the basic units. The definition of their measurements in the VIM is inherited from Stevens’ idea of the ratio scale.

I was so impressed by Stevens’ ingenuity and efforts to overcome the objections of physicists and sceptics. I end this article with a list of reasons for his success, which I will use in future lessons.

First, he emphasised the importance of systematically quantifying psychological attributes, even if they were not directly observable or easily comparable to physical quantities. He argued that applying rigorous measurement techniques could enable meaningful and useful assessments of psychological phenomena. Second, he underscored the importance of reliability and validity in psychological measurement. He advocated for the development of standardised measurement instruments and robust psychometric methods to ensure the consistency and accuracy of measurements across different contexts and populations.

Third, he adopted a pragmatic approach to measurement, recognising that while psychological attributes might not be directly comparable to physical quantities, they could still be quantified meaningfully. He emphasised the utility of measurement in advancing scientific understanding and practical applications in fields such as psychology, sociology, and education. Finally, he encouraged interdisciplinary collaboration between psychologists, statisticians, and other experts to refine measurement techniques and address concerns raised by critics. By fostering dialogue and collaboration across disciplines, he sought to bridge the gap between traditional physical measurements and the emerging field of psychological measurement. His work remains influential in shaping contemporary approaches to measurement theory and practice in diverse fields of study.

References

1. Data Modeling for Metrology and Testing in Measurement Science, edited by Franco Pavese and Alistair B. Forbes (ISBN 978–0–8176–4592–2) pp. 31
2. Wikipedia, Homomorphism
3. Wikipedia, Intensive and extensive properties
4. S. S. Stevens, Science, New Series, Vol. 103, №2684 (1946), pp. 677–680
5. The International Vocabulary of Metrology (VIM)