(This is the third chapter of the first volume of The Scientific Experience, by Herbert Goldstein, Jonathan L. Gross, Robert E. Pollack and Roger B. Blumberg. The Scientific Experience is a textbook originally written for the Columbia course "Theory and Practice of Science". The primary author of "Measurement" is Jonathan L. Gross, and it has been edited and prepared for the Web by Blumberg. It appears at MendelWeb, for non-commercial educational use only. Although you are welcome to download this text, please do not reproduce it without the permission of the authors. An ascii version (47K) is also available.)
Deciding by observation the amount of a given property a subject possesses is called measurement. In a more liberal sense, the term "measurement" often refers to any instance of systematic observation in a scientific context. Measurement is always an empirical procedure, such as reckoning the mass of an object by weighing it, or evaluating the amount a student has learned by giving an examination.
By way of contrast, quantification is a kind of theorizing, such as refining the concept of mass, to explain observed resistance to force. Therefore, it might seem that quantification precedes measurement, or perhaps that after some preliminary observations to develop a method of quantification, one thereafter makes measurements according to that quantification. Such a picture omits much of the hard work typically involved in the creative process.
In original scientific investigations, the relationship between quantification and measurement is a "feedback loop". That is, the first set of measurements might suggest that the initial quantification was overly simplistic or even partially wrong, in which case the quantification is appropriately modified. Then some more measurements are made. Perhaps they indicate that not all the problems have been resolved, and that the quantification should be further refined. Then there are more measurements. And so on.
The pot of gold at the end of the quantification rainbow is called a "mathematical model". In general, a model is a construction that represents an observed subject or imitates a system. Museums of science often display physical models of atoms and molecules, for example, or of the solar system. Here is the sense in which a mathematical abstraction can be a model: the relationships among the values of the mathematical variables in the abstraction can imitate the relationships among their respective counterparts in the system being modeled.
For instance, there are models of the solar system that enable us to predict solar eclipses, far before their occurrence. Such models might be considered valid only to the extent that they predict accurately, independent of the attention given to detail. Analogously, a`valid model of some aspect of the economy would be one that accurately predicts future economic behavior.
This chapter describes the steps along the way from empirical observations to the formulation of models. Developing a model is what wins a Nobel Prize. Perfecting the methods of observation is a step along the way, but "scientific truth" is an empirical demonstration that a model is valid.
3.1 Nominal, ordinal, and interval scales
We have defined measurement in a sufficiently broad sense that it applies to any procedure that assigns a classification or a value to observed phenomena. Informally, the data that result from a measurement procedure are also called measurements.
Collecting data is not often a matter of writing down whatever you see, an infinite task whose outcome would be a few critical observations, completely hidden in an undifferentiated mess of irrelevant details. The collection of data requires structure, including an experimental design and a method of observation.
Part of an experimental design is the creation of a "scale" for the measurements, based on the quantification of the phenomena to be observed. There are three major classes of scales, called "nominal" scales, "ordinal" scales, and "interval" scales.
A nominal scale is a qualitative categorization according to unordered distinctions. Consider, for example, an attempt to assign a value "male" and "female" while measuring gender. We might say that female is "category 0" and male "category 1", but we might equally well reverse those numeric labels, because they have nothing to do with femaleness or maleness.
Another example of a nominal scale is the department an undergraduate chooses for "major" emphasis. Some academic institutions, such as the Massachusetts Institute of Technology, have assigned a number to every department. When asked to identify his or her major, an M.I.T. student is likely to respond something like "Course 8" (which means physics) or"Course 21" (which means humanities).  However, the numeric designations are entirely arbitrary. Thus, even at M.I.T., classification of students according to major department is nominal!
An ordinal scale makes ranked distinctions. For instance, lexicographic ("dictionary") order is an example of an ordinal scale. It depends on only one property, the sequence of letters in the word. The lexicographic order of words, such as
need not be consistent with any notion of order that is derived from the meanings of the words. Military rank is another example of an ordinal scale.
An interval scale is based on the real numbers, so that each unit on the scale expresses the same degree of difference, no matter where on the scale it is located. For instance, weight, length, and duration of time are interval scales.
The atomic number of the elements of chemistry is regarded as an interval scale, even though the realizable values are all integers. The difference it expresses between elements one apart is always one proton in the nucleus. The point is that the type of a scale -- nominal, ordinal, or interval -- depends on the underlying nature of the quantification, rather than on the observed existence of particular values, or even on the physical possibility of existence.
If the value of zero on an interval scale represents a total absence of the property being measured, then the scale is sometimes called a ratio scale. For instance, whereas Celsius temperature and Kelvin temperature are both interval scales, Kelvin temperature is a ratio scale but Celsius temperature is not. The difference, in classical physics, is that zero on the Kelvin scale means absolute zero, the case in which all motion stops. Here are some multiple-choice questions designed to illustrate what is missing when an interval scale fails to be a ratio scale and that the concepts involved are somewhat subtle.
Question 1: Suppose it is 10° Celsius on Sunday and 20° on Monday. Does that make it twice as warm on Monday? Choose only one of the following answers.