On the Theory of Measurement

CIS343

 

The Numerological Fallacy

In a book on statistical analysis in the social sciences, the following definition of measurement was given: 

    “Measurement is the process of assigning numbers according to a rule.”

 

Clearly this is not sufficient for measurement.  For example, suppose I assign numbers in the range of 0 to 9 according to this rule:

          Assign to each person the last digit of their social security number.

Absolutely no measurement has occurred.  Measurement does not begin with numbers, but with a concept.  We will refer to the idea that measurement is simply the assignment of numbers according to a rule The Numerological Fallacy.

 

The Priority of the Concept

What is missing in the numerological fallacy is an understanding of this: In order to have measurement, one must have a prior concept of what is to be measured.  Therefore, the conceptualization precedes the design of a method of measurement.  As we proceed to develop measurement techniques, we use the underlying concept to test the applicability of the proposed method.  If, for example, our proposed method for measuring weight tells us that the jockey who rode the winning horse in the Kentucky Derby weighs more than most lineman for the Philadelphia Eagles, we become highly suspicious.

 

Difficulties

As we seek a measurement method for a concept we encounter two problems:

     w It may be easy to find a measure in some contexts, but difficult in others.

     w We may be fuzzy in our understanding of the concept we wish to measure.

We look at each of these in turn.

 

Differing Contexts

Distance is a concept about which there is no controversy.  We begin with the basic method of employing a measuring object (ruler, tape measure, etc.) that has been calibrated with an agreed upon standard.  We will refer to this as our base method.  For short distances we can lay down this object and read the measurement.  For greater distances we use mathematics to devise methods of measurement that can be calibrated to this base method.

 

Problems arise, however, when we begin to talk about distance in space.  Some distances can be calibrated back to our base method, but others cannot.  We can, for example, send radio signals to the moon and use the time that elapses before the signal returns to calculate distance.  Since this same method can be used on earth, it can be calibrated back to our base method.  But, what about measuring distances to astronomical objects that are too far away for this to work?  At some point astronomers use differences in what is called the red shift to measure distance.  For more distance objects they employ a diagram called the Hertzsprung-Russell diagram and use positions on this diagram as a measure of distance.

 

A legitimate question, at this point, is this: are we still measuring the same concept?  Unraveling this question requires philosophical analysis.  The point is clear, however, that as the context of application changes, a concept that was initially well understood may become murky.

 

Fuzzy Concepts

Other concepts are fuzzy from the very outset.  Sports offers many examples.  In the ranking of NCAA football teams a coaches poll will differ from a sportswriters poll which will differ from “computerized” rankings.  Another example.  Who is the better quarterback - the one who has the best quarterback rating or the one who leads his team to Super Bowl victory?

 

Even as we accept that some concepts are simply not amenable to clear cut measurement, there are principles we can use in our search for a good measure.  In most cases we seek a scalar measure, one whereby any quantity we obtain is either less than, equal to, or greater than some other quantity.  Underlying scalar quantities is the notion of difference, which is slightly more general.  Here we can apply the rubric of philosopher Gilbert Ryle: “A difference in order to be a difference must make a difference.”  In other words, if we want to say that quantity A is greater than quantity B, there should be a practical implication.  I.e., it may be that something else stands in direct (or inverse) proportion to the original measurement.  As a simple example, in the grocery store we will expect to pay more for 2 gallons of milk than for one gallon.

 

The idea, then, if the original concept is fuzzy, to determine whether it can be measured in a practical sense, we need to find a practical implication to relate it to.  We will refer to this as the Difference Principle.

 

The Application Context

This discussion of measurement arises from our desire to make more precise the concept of locality.  In applying the Difference Principle to this, we will need to find some measure of locality which can be related to an operating system performance difference.  In other words, if process A has a greater degree of locality than process B we want to be able to infer that in some specific context process A will perform better than process B.  Process A may execute more quickly than process B, or it may have a better hit ratio in cache memory, or it may generate less page faults.  We will see that finding that ideal measure of locality is not a simple task.

 

The problem of finding a measure of locality and its appropriate application context is explored further in “Measuring Locality” note and in the “Process Characteristics” project.  We may find that different contexts require different methods of measurement, to be applicable.