CS610 Research Methods - Measurement

Copyright 1995 Richard Halstead-Nussloch, Ph.D.
All Rights Reserved

Definition

To measure something is to assign numbers to represent the object or a property of the object. Measurement is at the heart of science, and plays a central role in research. To understand measurement requires an understanding of : Measurement is extremely important in computer science research, because it is so widely applied, yet so often misunderstood.

Numbers

Since measurement is the assignment of numbers, one can learn much about measurement by examining the structure of numbers. Three structural facts about numbers are important:
  1. Order: Numbers are ordered.
  2. Distance: Differences between numbers are ordered, and can represent a distance between two objects on a property.
  3. Origin: The series of numbers has a unique origin, represented by zero.

Objects and Properties

Without delving too deeply into philosophy, one "knows" an object by its properties. An unpeeled orange is known by its spherical shape, orange color, dimpled skin texture, etc. In measurement, one assigns a number to a property of an object. Objects can then be distinguished and sometimes compared by those numbers, which represent the measure of that property. Of course, multiple properties can be measured and used to distinguish and compare objects in a multidimensional fashion. In research, one often measures multiple instances of an object for statistical purposes.

The Assignment of Numbers to Objects and Their Properties

Torgerson [TORG58] discusses the process of measurement. In his discussion, I see a practical value of measurement in computer science research: for some, but not necessarily all properties, it is possible to give empirical meaning to none, one, two, or sometimes all three of the characteristics of numbers listed above. The assignment of numbers to objects and their properties, the actual process of measurement, thus determines whether the numbers actually have the empirical meaning or not.

Torgerson goes on to discuss that the three characteristics of numbers give rise to two different ways to distinguish among different kinds of measurement: As one takes measurements, i.e., makes assignment of numbers to objects and their properties, both the type of scales and kinds of measurements must be taken into account.

Types of Scales

Scientists, philosophers, mathematicians, etc. have provided a wide diversity of options for the types of scales to be used in measurement. A scale is a mapping between numbers and a property of an object or a set of objects possessing that property. According to Torgerson, numbers are assigned to the objects so that the relations between the numbers reflect the relations between the objects themselves with respect to the property. Types of scales are most often distinguished by the relations between the numbers that are a reasonably true reflection or representation of the relations between the objects on the property measured. According to Stevens [STEV46], the major types most often include:

Nominal Scale. Hays [HAYS63] describes a nominal or categorical scale as a rule for arranging observations into equivalence classes, so that observations falling into the same set are thought of as qualitatively the same and those in different classes as qualitatively different in some respect. In general, each observation is placed in one and only one class, making the classes mutually exclusive and exhaustive. The nominal or categorical scale gives empirical meaning to none of the three characteristics of numbers described above as important for measurement. Because it does not provide any empirical meaning to the numbers, measurement purists, such as Torgerson, do not consider the nominal scale a bona fide scale. Stevens, however, discusses a nominal scale with a basic empirical operation of determining equality, and permissable statistics of number of cases, mode, and contingency correlation. An example nominal scale is the assignment of numbers to ball players.

Ordinal Scale. An ordinal scale is a rule for arranging observations into an order, that is, ordering objects based on a common property. An example measurement is hardness. The ordinal scale gives empirical meaning to one property of numbers--order. Although numbers assigned in ordering the objects can be and are often manipulated by arithmetic, the answer is not reliably interpretable as a meaningful statement about the objects or their properties. Stevens includes the basic empirical operation of determination of greater or less; he includes the median and percentiles as permissible statistics.

Interval Scale. An interval scale is a rule for taking observations and measurements that orders the objects on a common property and provides distances between two objects with respect to their common property. For example temperature, which can be 40 degrees F warmer at 2:00 PM than it was at 2:00 AM the night before. The interval scale gives empirical meaning to two properties of numbers--order and distance. Numbers assigned to objects through an interval scale can be manipulated by linear functions and still preserve magnitudes and empirical interpretability. For example, one easily can move between the Fahrenheit and Celsius scales of temperature. Stevens includes determination of equality of intervals or differences as the basic empirical operations for interval scales. His list of permissible statistics includes the mean, standard deviation, and both the rank-order and product-moment correlation.

Ratio Scale. A ratio scale is a rule for taking observations and measurements that orders the objects on a common property, provides distances between two objects with respect to their common property, and has a true zero point or origin. Length is an example of a ratio scale. The ratio scale gives empirical meaning to all three of the characteristics of numbers--order, distance, and origin. Any of the ordinary arithmetic operations can be applied to differences between objects observed on a ratio scale, and the scale remains meaningfully interpretable. For example, if Sam buys 1 foot of bubble gum tape and Pat buys 2 feet of bubble gum tape, the statement that Pat has twice as much gum (2 feet / 1 foot) is meaningful. Stevens includes the determination of equality of ratios as the basic empirical operation for a ratio scale; he lists the coefficient of variation as the permissible statistic.

Kinds of Measurement

Torgerson distinguishes among three kinds of measurement by the kinds of information the numbers represent. Any particular scale might involve any mixture of the following three kinds of measurement: Derived measurements. Derived measurements obtain meaning through laws relating the property to other properties. Density is an example in that it derives its meaning through its relationship to volume and mass.

Fiat measurements. Fiat measurements obtain meaning simply by arbitrary definition. The meaning depends on presumed, intuitive, or common-sense relationships between the measured property and the target concept. Fiat measures are often called indicators. Some examples include economic indicators, preference indicators, and complexity indicators.

Fundamental measurements. Fundamental measurements assign numbers to objects according to natural laws to represent the property. There is no presupposition that other variables need to be measured. Examples include length, resistance, number, and volume. Fundamental measurements are almost always directly interpretable with respect to empirical meaning.

References

[HAYS63]
Hays, W.L. Statistics. Holt, Rinehart, and Winston: New York, 1963.
[STEV46]
Stevens, S.S. On the Theory of Scales of Measurement. Science. Vol 103, Number 2684, pages 677-680, 1946.
[TORG58]
Torgerson, W.L. Theory and methods of scaling. John Wiley and Sons: New York, 1958.

Bibliography

[DOMI80]
Dominowski, R.L. Research methods. Prentice Hall: Englewood Cliffs, NJ, 1980.