An Essay on The Ruler: What is Measurement?

Consider the ubiquitous 12 inch ruler (or its multiples, the yardstick, tape measure,or any metric equivalent) and how useful it is in everyday life. What properties produce such wide usage?

1. A ruler is independent of its construction.

Rulers come in many forms; wood, plastic, fabric and metal. Their sizes and fractional divisions are made for a variety of applications. But these are only variations on the general form. They all implement the idea of a straight line.

2. A ruler implements the idea of a unit.

The fundamental idea contained in the structure of a ruler is that of a unit - the inch. The inch is a universally agreed upon quantity of length:


It is treated as a fact, yet is really a fiction because any specific representation of this length is conditioned by some degree of error.

3. The unit is the same everywhere along a ruler.

The abstract unit-inch is of equal length across any part of the ruler. These "inches"are exchangeable. A reordering of the twelve inch-units produces the same ruler. This has great utility because it is not always possible to start measuring from the end of the ruler.

4. The inch is the concretization of the unit.

Concrete representation solidifies the idea of the inch ­ all "inches" are the same. Hence we can connect concrete inches end-to-end and by doing so "construct" the total measures of inches by concatenation. We go beyond merely asserting the exchangeability of units, we demonstrate it through their concrete realization.

5. Concrete units operationalize additivity.

Obtaining total length by concatenating inches demonstrates that additivity holds. Since combining inches by physical concatenation gives the result predicted by additivity, rulers implement what Physicist Norman Campbell called fundamental measurement.

6. Rulers employ the natural number system

Numerals are used to designate sequence and order as in 1, 2, 3, ... We count by means of the ordered numerals to obtain the total.

7. Rulers implement a correspondence between units and numbers.

The markings on a ruler implement a one-to-one correspondence between theordered natural numbers and the unit-inches. The left-most edge or marker of the ruler is the beginning and indicates the absence of any units, i.e., none. The natural numbers mark the unit-inches as successive amounts concatenated to the right. Though the natural numbers themselves are ordinal, their combination with inch-units makes them interval/linear on a ruler.

The ruler is a sophisticated device. It contains a history of ideas and concepts that are fundamental to measurement. The crucial concept is that the abstract mathematical properties of a ruler transcend the concrete instrument. So long as social science lacks rulers, our mathematical manipulations can never transcend their context.

Mark H. Stone
Adler School of Professional Psychology
Chicago, Illinois

Stone M.H. (1996) An essay on the ruler: what is measurement? Rasch Measurement Transactions 10:2 p. 502.

Stone M.H. (1996) An essay on the ruler: what is measurement? Rasch Measurement Transactions 10:2 p. 502.

An essay on the ruler: what is measurement? Stone M.H. … Rasch Measurement Transactions, 1996, 10:2 p. 502

Please help with Standard Dataset 4: Andrich Rating Scale Model

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

To be emailed about new material on
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from welcomes your comments:

Your email address (if you want us to reply):


ForumRasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website,

Coming Rasch-related Events
July 31 - Aug. 3, 2017, Mon.-Thurs. Joint IMEKO TC1-TC7-TC13 Symposium 2017: Measurement Science challenges in Natural and Social Sciences, Rio de Janeiro, Brazil,
Aug. 7-9, 2017, Mon-Wed. In-person workshop and research coloquium: Effect size of family and school indexes in writing competence using TERCE data (C. Pardo, A. Atorressi, Winsteps), Bariloche Argentina. Carlos Pardo, Universidad Catòlica de Colombia
Aug. 7-9, 2017, Mon-Wed. PROMS 2017: Pacific Rim Objective Measurement Symposium, Sabah, Borneo, Malaysia,
Aug. 10, 2017, Thurs. In-person Winsteps Training Workshop (M. Linacre, Winsteps), Sydney, Australia.
Aug. 11 - Sept. 8, 2017, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets),
Aug. 18-21, 2017, Fri.-Mon. IACAT 2017: International Association for Computerized Adaptive Testing, Niigata, Japan,
Sept. 15-16, 2017, Fri.-Sat. IOMC 2017: International Outcome Measurement Conference, Chicago,
Oct. 13 - Nov. 10, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps),
Oct. 25-27, 2017, Wed.-Fri. In-person workshop: Applying the Rasch Model hands-on introductory workshop, Melbourne, Australia (T. Bond, B&FSteps), Announcement
Jan. 5 - Feb. 2, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps),
Jan. 10-16, 2018, Wed.-Tues. In-person workshop: Advanced Course in Rasch Measurement Theory and the application of RUMM2030, Perth, Australia (D. Andrich), Announcement
Jan. 17-19, 2018, Wed.-Fri. Rasch Conference: Seventh International Conference on Probabilistic Models for Measurement, Matilda Bay Club, Perth, Australia, Website
April 13-17, 2018, Fri.-Tues. AERA, New York, NY,
May 25 - June 22, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps),
June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps),
Aug. 10 - Sept. 7, 2018, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets),
Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps),
The HTML to add "Coming Rasch-related Events" to your webpage is:
<script type="text/javascript" src=""></script>


The URL of this page is