Monthly Archives: May 2019

Essence of Measurement

The usual definition of measurement is the assignment of numbers to something according to specified rules, where empirical evidence can be used to validate the results (Magnusson, 1966, see p.1).

But at least 40 different ways of using the term “measure” have been identified (Lorge, 1951). The same term can mean the act of weighing, the balance by which weighing is done, the grams that are used to balance an object, or the numeral that expresses the result of the weighing process. It can be used to refer to any instrument used as a basis for comparison, even when that comparison involves processes of estimation or judgment.

“Measure” is used more frequently to refer to acts of subjective estimates than to precise objective determination.

It is not likely that consensus will emerge regarding one meaning for “measure” or associated terms. Most people, however, expect some quantity will be used to express the outcome of some measurement process. In general, measurement involves assigning a class of numerals to a class of objects by applying a specific set of rules or procedures. (See Numbers and Numerals All measurement involves a person’s use of perceptual faculties, either unaided or extended by instrumentation of some sort.

Thus the classes of objects, the conceptual organization of discourse about those objects, the nature of instruments used, and the training of the observer all influence the results of a measurement process. Some observations are made directly — for example, the observer lays the ruler on the face of an object and compares the ends of the face with the markings on the ruler. The property of interest is observed directly.

Other observations are indirect — for example, the length of a column of mercury in a thermometer has been correlated with variations in temperature. In this case the effects of temperature on the column of mercury are observed, and then an inference is made about temperature by observing the column of mercury.

In scientific measurement, regardless of the field, the conditions for observation are carefully specified in terms of time, place, and circumstance. Observations are independently verified under the same conditions, with the results reported in terms of probable error.

Lorge (1951) makes the important observation that what is observed depends upon man’s conceptual equipment to translate sensory experiences into the notion of a property. Frequently the notion of a property will change as attempts are made to measure it. Detailed description of the property is essential.

Statements about a property are empirical in that they depend on what is experienced. Science demands that observations be reproducible — measurements must be reproducible. Assuming the property remains constant, measurement of the property under the same conditions by different observers should yield the same result.

Lorge, I. (1951). The fundamental nature of measurement. In E. F. Lindquist (ed.), Educational measurement. Washington, D.C.: American Council on Education. (pp. 533-559).

Magnusson, D. (1966). Test theory. Reading, MA: Addison-Wesley.

Numbers and Numerals

Number is a mathematical concept that is used to develop various formal systems of mathematical reasoning. Generally, the concept rests at the tacit level of understanding in the sense that we assume that we know what it means, and that others have the same meaning in mind. But the concept becomes more complex as we play with it.

Consider the formal number systems.

  • There are natural numbers, which is the set [1, 2, 3, …].
  • There are whole numbers, which is the set of natural numbers with the added element 0.
  • Within the set of whole numbers there is the subset : [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]. Each of the ten elements in this subset is called a digit. Digits are combined in various ways with other symbols to represent numbers from a variety of number sets.
  • There are integers, which is based on the property of symmetry around 0: the set is […, -3, -2, -1, 0, 1, 2, 3, …]. We say that integers can be negative or positive, depending on their location with respect to 0.
  • There are rational numbers, which is the set of integers plus all of the ratios of two integers, p/q, where q is not 0. (Thus, ½ is a rational number, as is 312/311.)
  • Then there are numbers that cannot be expressed as the ratio of integers, such as the ratio of the circumference of a circle to its diameter, called pi. These are called irrational numbers.
  • The set of rational numbers combined with the set of irrational numbers is called the set of real numbers.

There are larger sets of numbers, but the point is made that number is a mathematical concept that is a building block for formal systems of mathematical reasoning.

For most purposes, especially in the social sciences, in measurement discourse numerical references are to elements in the set of real numbers. What does that mean?

Usually we use the term “number” to refer to the result of counting specific units, or the result of performing some group of mathematical operation on that result. For example, six (6) cars went through the red traffic light at this intersection on Monday between 7:00 a.m. and 9:00 a.m. The unit is a car that has certain characteristics (passed through the intersection in the two-hour period when the traffic light was red). The number “6” represents the result of someone counting the defined units. (The same principle applies if the counting was done by a set of instruments set up at the intersection.)

Suppose the results of five days of counting were 6, 7, 8, 10, and 5. What does the statement mean: In the five days an average of 7 1/5 cars went through the red light. The number “7 1/5” is the result of the mathematical operations of adding the five numbers and dividing the total by the number 5 (which is the result of counting the number of days that cars were counted). This is a rational number (it can be expressed as the ratio of integers 36/5). The process for deriving the number 7 1/5 is clear, but what does it mean?

Mathematically speaking, it means that 7 1/5 cars going through the red light every day for five days is the same total number of cars that were actually observed going through a red light in that period. Without knowing the actual counts for each day, we usually assume that about 7 cars were observed each day, maybe a few more on a particular day and a few less on another day. Presumably the observer did not see a part of a car go through the intersection.


A numeral is a symbol that represents a number. Numerals have been devised to facilitate mathematical reasoning. The result of counting six cars can be represented by the Arabic numeral “6” or the Roman numeral “vi”. Arabic numerals are much easier to work with.

In measurement discourse, it is essential that we clearly define the unit that is being counted, the counting process and the mathematical operations performed. That is, it is essential that we define the numbers that are represented by the numerals in a particular situation. Otherwise, confusion lurks in the background.

For fascinating accounts of some relationships between numerals and numbers see Isaac Asimov,  (1977). Asimov on numbers. Garden City, NY: Doubleday.

Managing Evaluator Anxiety

6Do not be anxious about anything, but in every situation, by prayer and petition, with thanksgiving, present your requests to God. 7And the peace of God, which transcends all understanding, will guard your hearts and your minds in Christ Jesus. (Philippians 4:6-7)

Do not be anxious… Do not be apprehensive or fearful. Do not worry about something that might harm you in some way at some time.

I cannot prevent feeling anxious, but I can manage anxiety with the actions listed here by Paul. I can talk about the situation with God, and ask for help. That helps me realize that alone I cannot do what matters most.

Anxiety reminds me that I am not self sufficient in being what I was created to be. I am dependent on my loving Creator for knowing what that is, and then acting on that knowledge. In this sense anxiety is a friend.

In every situation… This includes everything that happens, or does not happen, in an evaluation. Stakeholders quarrel as the evaluation is being planned, torrential rain makes it impossible to travel to villages to interview and observe per the data collection plan, evaluation team members become ill, etc. In these and innumerable other circumstances instead of fretting about how an evaluation might be flawed I can, and should, pray for guidance to move forward.

With thanksgiving… in every situation… I commit to giving my anxieties to God.