MEASUREMENT

The process or technique of correlating numbers with things that are not patently numbered in the order of nature; also, the relation that arises from such a process. Measurement is usually affected by comparing observable phenomena with a suitable metric, although sometimes it is the result of a mathematical calculation based on data that are not directly accessible to experience. As employed in the physical sciences, the process of measurement is itself an interaction between a measuring instrument and the thing measured, and on this account is dependent for its objective validity on corrections (sometimes involving theoretical interpretations) to account for the perturbing effect of the instrument.

This article first presents a philosophical analysis of measurement in general, then considers specific problems associated with measurement in psychology, and concludes with a discussion of mathematical aspects of the measuring process.

MEASUREMENT IN GENERAL

Measurement, according to St. thomas aquinas, is the process by which the quantity of a thing is made known (In 1 sent. 8.4.2 ad 3). It is applied directly to physical bodies (1) when their discrete quantity is ascertained, e.g., by counting the number of objects in a room, or (2) when their continuous quantity is measured, e.g., by using a scale to determine individual lengths. In current practice the term measurement is sometimes applied to counting, but is more usually reserved for determinations of dimensive or continuous quantity.

Quantitative Measurement. The elements involved in direct measurement can be explained in terms of the requirements for a quantitative measurement such as the determination of length. Such measurement first presupposes a unit; the unit may be one that occurs naturally, such as the foot, or it may be one fixed by convention. The choice of a conventional unit is not completely arbitrary, but is dictated by the unit's suitability as a minimum dimension into which lengths can be divided.

Secondly, the unit used must be homogeneous with the thing measured (In 1 anal. post. 36.11). For example, if length is to be determined, the unit must be a length. Similarly, the thing measured must be uniformly structured and continuous to permit the application of the same unit to each of its parts.

A third requirement is that the unit of measurement and the object being measured must be invariant throughout the measuring process (Summa theologiae 1a2ae, 91.3 ad 3; 97.1 ad 2). This ideal is never completely realized for any physical object, since all bodies continually undergo change. Because of such variation, as well as the infinite variety of contingent circumstances that accompany any measuring process, every measurement is at best an approximation. Yet a practical invariance is not only detectable, but is more or less guaranteed by the nature of both the object measured and the standard used. For example, a person's body temperature, although varying over a small range, is held constant by natural causes. Similarly, the unit of time is determined by the rotation of the earth and the gram by the weight of one cubic centimeter of water, both of which are maintained constant through the regularity of nature's operation.

A fourth requirement is that measurement involves a judgment of comparison between the object measured and the measuring unit (Summa theologiae 1a, 79.9 ad 4). Such a judgment is an intellectual operation, although it presupposes a physical process. The program associated with operationalism to reduce every measurement to the manipulation of instruments alone thus disregards an essential feature of the measuring process. Instruments cannot measure. Ultimately they require mind, which, because of its reflexive character as a "self-reading instrument," can effect the judgment of comparison and make the measurement.

These requirements for the direct measurement of quantity or of bodily extension are applicable to spatio-temporal measurements (see time). They can be applied also to other entities, such as certain types of quality, but not without some adaptations, as will now be explained.

Qualitative Measurement. Physical qualities, because present in quantified bodies and intimately associated with the quantity of such bodies, can themselves be said to be quantified. Their quantity can be measured in two different ways, giving rise to the two measurements that are usually associated with physical quality, viz, extensive and intensive quantification from the extension of the body in which they are present; thus there is a greater amount of heat in a large body than in a small body, assuming both to be at the same temperature (cf. De virt. in comm. 11 ad 10). They receive intensive quantification, on the other hand, from the degree of intensity of a particular quality in the body (ibid.; Summa theologiae 1a, 42.1 ad 1). If two bodies are at different temperatures, for example, there is a more intense heat in the body at the higher temperature, or it is the hotter, and this regardless of the size of either.

Measurement of the extensive aspect of physical qualities, being effectively the same as the measurement of length, area, and volume, has the same requirements as that for quantitative measurement. Measurement of the intensive aspect, on the other hand, is more difficult and requires slightly different techniques.

Two possibilities suggest themselves for the measurement of a quality's intensive aspect. The simplest is to arrange objects with a given quality in the order of increasing intensity, and then number them consecutively. For example, if bodies be arranged according to increasing hotness as discernible by touch, and these bodies be numbered, the higher number indicates the greater degree of heat. This is the closest one can come to a direct intensive measurement of quality. Such a measure offers difficulties, however, because of the subjectivity of sensation and the arbitrariness of assigning numbers depending on the number of objects that happen to be compared.

The other possibility is that of determining the intensity of a quality (1) from an effect, i.e., the change the quality produces in a body other than that in which it is subjected, or (2) from a cause, i.e., the agent that produces the quality's intensity in the subject.

Effect. If the quality is an active one, i.e., if it produces alterations in other bodies, it can be measured by the effect it produces in such bodies. This is usually done through special types of bodies known as instruments. Thus heat intensity is measured by a thermometer containing a substance that expands noticeably when contacting a hot object. Similarly, the intensity of sound is measured by vibrations produced in a microphone, and light intensity by electric current generated in a photocell. In each case, the intensity of an active quality in one subject is measured by the quantity of the effect it produces in a receiving subject, which is known on this account as the measuring instrument.

Active qualities, it may be noted, can sometimes be measured independently of external alterations of the type just mentioned. If they induce pronounced quantitative changes in the subject in which they are present, they can be measured directly through measurement of the subject body. In this way the temperature of mercury in an immersion thermometer is measured simply by reading the length of its own expansion. Similarly, the wavelength of sound in a resonating chamber of variable length is measured directly, using a standing wave technique to ascertain the length of the vibrating column. Such a method of concomitant variation, however, while of theoretical interest, is of limited applicability, since it is restricted to bodies that are quantitatively sensitive to the presence of the qualities being discussed.

Cause. If a quality is not particularly active, i.e., if it does not produce pronounced effects in itself or in another body, its intensity can alternatively be measured through some type of causality required to produce it in the subject body. In this way, one measures the intensity of light on a reflecting surface by the number of footcandles emitted by the source illuminating the surface. A variation on this technique is that of using an instrumental cause to measure some modality of the principal cause that actively produces the quality. An example would be using a prism or ruled grating selectively to refract and measure the wavelength of colored light incident on an opaque surface, and in this way indirectly to measure the ability of the surface to reflect light of a particular color.

All these methods are indirect ways of measuring qualitative intensity through a cause-effect relationship. All involve techniques whereby a precise quantity is assigned to the quality being measured, and on this account are considerably more accurate and objective than direct ordinal measures of qualitative intensities. As a consequence, these constitute the type of qualitative measurement most widely used in the physical sciences.

Accuracy. As employed in physical science, a measurement cannot be made to an infinite degree of accuracy. There are two reasons why this is so. The first is that all such measurements reduce to a measurement of continuous quantity, and the only way in which number can be assigned to such quantity is in terms of a conventional unit. For infinite accuracy, this unit would have to approach zero as a limiting case. Attaining the limit would itself involve a contradiction in terms, since a number cannot be assigned to a unit of zero, or nonexistent, magnitude. The second limitation arises from specifying the conditions that attend a particular measuring process. Since these involve details that are themselves infinitely variable, they can be only approximately specified. For all practical purposes, however, it is possible to specify the range of magnitudes between which a given measurement is accurate, depending upon the unit involved and the circumstances of measurement. Some Thomistic philosophers regard such accuracy as sufficient to permit a demonstration with the certitude that is proper to physical science, although not with that proper to mathematics, while others see this as sufficient reason for questioning the strictly demonstrative character of any conclusion of modern science that is based upon a measuring process that has the above limitations.

Bibliography: w. a. wallace, "The Measurement and Definition of Sensible Qualities," The New Scholasticism 39 (1965) 1–25. m. heath, "Can Qualities Be Measured?" Thomist 18 (1955) 31–60. v. e. smith, "Toward a Philosophy of Physical Instruments," ibid. 10 (1947) 307–33. p. h. j. hoenen, Cosmologia (5th ed. Rome 1956).

[w. a. wallace]

MEASUREMENT IN PSYCHOLOGY

In psychology, the term measurement means the assigning of numbers to quantitative variations in a distinguishable attribute of behavior, or of behaviorally related objects, with the expectation that something true or predictable may be derived from their relationship with other variables. The logic of measurement is concerned primarily with the construction of a scale or measuring device, and secondly with the application of that scale to a particular behavior or object, such as occurs in psychological testing.

Requirements for Measurement. Quantitative indexes of behavior, such as number of errors, perception time, and number of words recalled are often employed in laboratories of experimental psychology, but in a fairly simple way—e.g., as a convenience for the experimenter in distinguishing and recording various performances, where the only assumption involved is that the behaviors can be properly and meaningfully ranked. The numbers themselves refer to physical units and are not commonly scaled psychologically. Psychological scaling involves the following special features:

Isolation and Identification of a Dimension. What is measured is not an object, strictly speaking, but rather some property or dimension associated with an object, either directly, at the level of sensory observation, or indirectly, through the type of indicant specified by an operational definition. Such a property must first be qualitatively distinguished from other properties and seen as capable of quantification. Not all psychological properties are measurable, for measurability depends upon whether or not the property can be conceived in a quantitative way.

Human Significance. The numbers employed in a psychological scale must represent a value indicative of the experience or performance of the human subject, as opposed to a value inherent in the physical nature of the stimulus, situation, or condition evoking the behavior. Physics, for example, using the human subject as mediating observer, employs a physical system—e.g., a thermometer, based on a law relating liquid expansion to heat—to eliminate or reduce subjective variations within the sensing organism. The psychologist, however, is directly concerned with variations in sensation and with human performance; thus the numbers he uses must reflect, or be isomorphic with, variations in psychological meaningfulness. The basic law employed must include behavior as one of its terms.

Rule for Assigning Numbers. The derivation of a psychological scale involves perceiving or performing subjects on the one hand, and stimuli (physical objects, situations, words, tasks, or problems) on the other, and a search for some functional law relating the two. Systematic variations in the responses of the subjects, when quantified in terms of the human variation itself, become the key to establishing the function relating stimuli and responses and to assigning one, and only one, number to each object.

Usable Properties of Numbers. Not all arithmetical properties of numbers are usable in psychological scaling. The number four, for example, is greater than two (order); it is also greater by a definite amount (distance or interval); and it is counted off from zero (origin—in this case, an absolute zero) and contains two by addition or division (composition or extension). Corresponding to these distinguishable properties, scales are commonly identified as ordinal, interval, or ratio scales. The difficulty of establishing an absolute zero in psychological matters restricts the use of ratio scales, and such matters offer little opportunity for using the extensive or composite properties of numbers.

Examples of Psychological Scales. Commonly employed psychological scales include psychophysical scales, attitude scales, product and mental test scales, and multidimensional and other types of scaling procedures.

Psychophysical Scales. An instance of a psychophysical scale is the measurement of the loudness of a sound based on the Weber-Fechner law, which states that the intensity of a sensation increases as the logarithm of the stimulus. A unit difference between the logarithms of two physical sound pressures is divided into ten equal steps, called decibels. The zero is set at the point of the absolute threshold, the weakest sound that can be heard, and the scale extends to cover a range of about 140 decibels.

Attitude Scales. These are commonly derived from a large number of statements of opinion, favorable and unfavorable, about some commonly known subject such as communism or about a debatable social custom or institution. Agreement among a group of judges as to how favorable or unfavorable each opinion-statement may be is transformed into a scale value, using the law of comparative judgment. This law states that the psychological difference between items is a function of the relative frequency with which the difference is perceived. The zero is placed arbitrarily low.

Product and Mental Test Scales. Product scales, used for rating specimens of handwriting, soldering joints (in trade tests), art work, or other kinds of cultural product, are similarly based on the law of comparative judgment and have an arbitrary zero point. Mental test scales, such as those used in psychological testing, are based on a statistical analysis of the performance of a group of homogeneous subjects on each item.

Other Types. Multidimensional scaling is employed to discover the number of dimensions involved in a particular phenomenon and to rate each object on the various dimensions. An example of this type of scaling is the "semantic differential," which is used to measure the connotative meaning of common words. Other scales are used for the associative value of common words or of nonsense syllables, the frequency rating of associated responses to a standard word list, the frequency rating of words in common use, and ratings of abnormal or psychotic behavior in terms of basic trait content.

Role of Measurement in Psychology. Psychological measurement is based on empiriological properties that lend themselves readily to conceptual quantification and identification in terms of observed indicants. It thus serves to supplement logical definitions and to extend these into an area of finer objective discrimination. Among its other contributions, the following may be enumerated: (1) increased precision in identifying instances of the occurrence of a property; (2) better contexts of meaningfulness, to the extent that the assignment of numbers is based on behavioral laws; (3) evaluations of the influence of empirically established relationships on tentative definitions of objects or properties; and (4) more reliable inferences of causal relationships as these are discernible through the application of the principle of concomitant variation.

Bibliography: w. s. togerson, Theory and Methods of Scaling (New York 1958). conference on the history of quantification in the sciences, Quantification of the Meaning of Measurement in the Natural and Social Sciences, ed. h. woolf (Indianapolis 1961). c. w. churchman and p. ratoosh, eds., Measurement: Definitions and Theories (New York 1959).

[w. d. commins]

MATHEMATICAL ASPECTS

From the viewpoint of the mathematician, measurement is the determination of the value of a measure function of a given attribute of an object. A measure function is a rule that correlates a set of attributes with a set of elements (usually numbers) of an algebra. For example, the measure function called "length," which one may denote by L, is a rule or set of procedures that associates the set of attributes of "extended objects" with the set of real numbers for which addition and multiplication are already defined. The length of the edge of a table may be denoted by L (t ), in which case, for example, L (t ) = 3 feet 6 inches. The rule denoted by L is the set of procedures for measuring the length of the table, which results in the correlation "length of table is 3'6"."

The element correlated with "length of table," or more generally "length of x, " is not merely a real number; it is a dimension number, i.e., a number plus a dimension, as, for example, "3 feet." The term "feet" refers to a unit of measurement previously established. "To measure" thus signifies that one knows a set of procedures and a unit of measurement (a dimension) such that, by applying the set of procedures, he can associate a unique number of units of measurement with a given measurable.

Requirements for Measure Functions. A minimal set of conditions imposed on a measure function are: (1) if m (x ₁) and m (x ₂) are measurements, then one and only one of m (x ₁) is equal to m (x ₂) or m (x ₁) is less than m (x ₂) or m (x ₁) is greater than m (x ₂) holds; (2) x ₁ is equal to x ₂ if and only if m (x ₁) is equal to m (x ₂), where x ₁ and x ₂ denote measurables, e.g., the length of this table and the length of that table; and (3) if m (x ₁) is less than m (x ₂) and m (x ₂) is less than m (x ₃), then m (x ₁) is less than m (x ₃). For a measure function using the set (or any subset of the set) of real numbers as dimension numbers, these conditions are easily satisfied.

In defining the measure function, one must also define the rule, i.e., the method of assigning a dimension number to each instance of the attribute. An illustration of this method can be seen in fixing the age of some person P. Evidently "age of P " is measured from the point "birth of P. " Moreover, to measure age one must define a dimension, i.e., a unit of measurement. In this case "calendar year," in its ordinary meaning, can serve as the unit of measurement, which is also called the metric. One correlates the birth of P with a point on a given calendar year. The age of P at birth is defined as zero years, whereas the age of P at present is the number of calendar years from zero to the present date. This process defines a mapping of two points in the set of calendar years to two events in the life of P, and consequently defines the age of P as the "distance" between the two points in the set of calendar years.

Mapping. The problems associated with mapping may be seen in the following example. Suppose that one wishes to measure food preferences of a set of adults aged 20 to 30. He then must define a measure function that maps a set of foods into the set of elements of some algebra. The most obvious algebraic set to select is the set of nonnegative integers. The unit of measurement is the preferential attitude of an adult toward a food. The measure function m (_p) is definable as follows: m (_p) (F ) = order of F in the ranking by person. So m (_p) (cheese) = 3 means that a given individual ranks cheese third on his list.

There are difficulties involved in this procedure, however. For example, it is not clear how to define m (_p) (F ) = zero. The metric is neither precise nor unambiguous. The measure function gives different results at different times and for different persons. The fact that, for two individuals, the measure function for the same food gives different integers does not enable one to make any significant comparison between the two measures. Finally, the fact that the measure function for a given individual yields m (_p) (F ₁) = a and m (_p) (F ₂) = b does not provide any significant conclusion concerning the relation of the two preferences.

Whether or not a measure function can be defined for a particular attribute is an empirical problem. But when a measure function can be found that correlates the measurables with some algebra, particularly that of the real number system, the whole apparatus of mathematics becomes available for inferences. Assuming translation from mathematical equations to attributes, the mechanical derivation of mathematical consequences then suggests phenomena that may be related to the measurables.

In the empirical sciences, concrete models of the metric are often constructed, e.g., clocks, meters, and scales. Some concrete entity or objective phenomenon is then used to determine the number of units to be mapped to a particular appearance of the attribute being measured. Such an instrument can be considered to define the method of determining the measure.

Error in Measurement. There are two major sources of error in measurement: (1) the definition of the measure function, and (2) the construction and use of the measuring instrument. Since the use of the measuring instrument involves the recognition of the coincidence of "points," the degree of accuracy with which the points can be seen to coincide influences the accuracy of the measurement. The size of the unit used also affects the accuracy. For example, a length measured by a measuring instrument graduated in inches is accurate only to within ½"; thus a length of 12" is really a length between 11½" and 12½". If the length is measured with an instrument graduated to ½", a length of 12" means a length between 11¾" and 12¼". Further errors arise if, unknown to the observer, the conditions under which the measurements are taken cause the measuring instrument to be altered, or the attribute measured to be affected.

Bibliography: c. g. hempel, Fundamentals of Concept Formation in Empirical Science (Chicago 1952). h. margenau, The Nature of Physical Reality: A Philosophy of Modern Physics (New York 1959). j. g. kemeny, A Philosopher Looks at Science (Princeton 1959). s. s. stevens, "On the Theory of Scales of Measurement," Science 103 (1946) 677–80.

[l. o. kattsoff]