**Excerpted from an article on the Web by:**

*Warren S. Sarle, SAS Institute Inc.
SAS Campus Drive
Cary, NC 27513, USA
saswss@unx.sas.com*

URL: ftp://ftp.sas.com/pub/neural/measurement.html

Originally published in the Disseminations of the International Statistical Applications Institute, 4th edition, 1995, Wichita: ACG Press, pp. 61-66. Revised March 18, 1996.

Copyright (C) 1996 by Warren S. Sarle, Cary, NC, USA.

Permission is granted to reproduce this article for educational purposes only, retaining the author’s name and copyright notice. ,

**Contents**

* What is measurement?

* Why should I care about measurement theory?

* What are permissible transformations?

* What are levels of measurement?

* What about binary (0/1) variables?

* Is measurement level a fixed, immutable property of the data?

* Isn’t an ordinal scale just an interval scale with error?

* What does measurement level have to do with discrete vs. continuous?

* Don’t the theorems in a statistics textbook prove the validity of statistical methods without reference to measurement theory?

* Does measurement level detemine what statistics are valid?

* But measurement level has been shown empirically to be irrelevant to statistical results, hasn’t it?

* What are some more examples of how measurement level relates to statistical methodology?

* What’s the bottom line?

Measurement theory is a branch of applied mathematics that is useful in measurement and data analysis. The fundamental idea of measurement theory is that measurements are not the same as the attribute being measured. Hence, if you want to draw conclusions about the attribute you must take into account the nature of the correspondence between the attribute and the measurements.

The mathematical theory of measurement is elaborated in:

Krantz, D. H., Luce, R. D., Suppes, P., and Tversky, A. (1971). Foundations of measurement. (Vol. I: Additive and polynomial representations.). New York: Academic Press.

Suppes, P., Krantz, D. H., Luce, R. D., and Tversky, A. (1989). Foundations of measurement. (Vol. II: Geometrical, threshold, and probabilistic respresentations). New York: Academic Press.

Luce, R. D., Krantz, D. H., Suppes, P., and Tversky, A. (1990). Foundations of measurement. (Vol. III: Representation, axiomatization, and invariance). New York: Academic Press.

Measurement theory was popularized in psychology by S. S. Stevens, who originated the idea of levels of measurement. His relevant articles include:

Stevens, S. S. (1946), On the theory of scales of measurement. Science, 103, 677-680.

Stevens, S. S. (1951), Mathematics, measurement, and psychophysics. In S. S. Stevens (ed.), Handbook of experimental psychology, pp 1-49). New York: Wiley.

Stevens, S. S. (1959), Measurement. In C. W. Churchman, ed., Measurement: Definitions and Theories, pp. 18-36. New York: Wiley. Reprinted in G. M. Maranell, ed., (1974) Scaling: A Sourcebook for Behavioral Scientists, pp. 22-41. Chicago: Aldine.

Stevens, S. S. (1968), Measurement, statistics, and the schemapiric view. Science, 161, 849-856.

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