In books of research about education, the definition of *measurement *offered by F.N. Kerlinger (1910-1991) is often quoted and widely followed:

“The assignment of numerals to objects or events according to rules” (*Foundations of Behaviorial Research*, 3rd edition, 1986. p. 391).

Unfortunately, this is a woefully inadequate definition of *measurement* because it is far too inclusive. It allows the inclusion of nominal numerals and ordinal numerals as concepts which can be used in measurement. Nominal and ordinal numerals do not denote cardinal value and thus can not ever represent units of measurement in either their interval or ratio form. There are always rules required in the application of symbols, otherwise the script or figures (alpha, numeric, ideographic, iconic, etc.) can not be symbols. The crucial question is what rule applies in what case. In the case of the Kerlinger definition of *measurement*, the allowance of any rule to be used with numerical symbols is mistaken, and it fails to delineate the essential characteristics of measurement.

The mathematical and scientific concept of measurement is the appropriate one to use in conceptualizing measurement as a procedure used in conducting research about eduation. Measurement, as a mathematical concept, is the application of a system in the task of assigning a cardinal number to indicate the size, amount or intensity of each subset of a set. Measurement is a comparison of the size, amount or intensity of something with a standard unit of size, amount or intensity of the something. The something being measured is a subset of a set of phenomena. A cardinal number is assigned to the size, amount or intensity of the subset, and the number is a multiple of the standard unit of size, amount or intensity of something. Unequivocal examples of measurement include

(a) using the centimeter (a standard unit of measurement of size) to determine the length (some cardinal number of centimeters) of a stick (a physical thing, i.e. a subset of the set of all physical things denoted by the term *stick*),

(b) using the gram (a standard unit of measurement of amount) to determine the mass (some cardinal number of grams) of a stick (a physical thing, i.e. a subset of the set of all physical things denoted by the term *stick*),

(c) using the centimeter and the second (a standard unit of measurement of size and amount) to determine the speed (some cardinal number of centimeters per second) at which a stick is moving in a stream of water (a physical thing, i.e. a subset of the set of all physical things denoted by the term *stick*).

The essential elements of measurement are

(1) a standard unit of size, amount or intensity of a subset of a set of phenomena,

(2) a size, amount or intensity of a subset of a set of phenomena,

(3) a comparison of the standard unit with the subset of the set of phenomena,

(4) an assignment of a cardinal number to the size, amount or intensity of the subset as a multiple of the standard unit of size, amount or intensity of a subset of the set of phenomena.

In applying a system of measurement, one of the subsets of a set is selected and nominated as the basic unit of the system by which all other subsets of the set of phenomena are to be compared. The number assigned to the basic unit is distributive, associative and commutative, i.e. the unit can be added to itself to form regular intervals of units, and the regular intervals constitute a scale.

A scale is a system of ordered standard units used for comparing the size, amount or intensity of a subset of a set of phenomena. If the scale has an absolute value of zero (0), the multiple units on the scale form ratios. If the scale does not have an absolute zero (0), it is not possible to find ratios on the scale. Such a scale shows intervals, but no ratios.

For example, a degree Centigrade is a standard unit for measuring temperature. Water freezes at 0 degrees and boils at 100 degrees Centigrade at one atmosphere of pressure. But 40 degrees Centigrade is not twice the temperature of 20 degrees Centigrade. It is not possible to establish ratios of intervals on the Centigrade scale because there is no absolute value of zero (0).

On the other hand, the meter is a standard unit for measuring length. The definition of the meter has undergone several historical revisions, and its current definition is the distance traveled by light rays in a specified time (a very small fraction of a second, 1/299,792,459^{th} of a second). On a scale consisting of multiples of meters, there is an absolute value of zero (0). Thus it is possible to calculate ratios of measurements in meters. Two meters is half the length of four meters. The ratio is one to two (1:2).

Thus, in scales of measurement, there can be interval scales and ratio scales. Interval scales assign only values of units to a subset of a set of phenomena (e.g. 20, 40, 80 degrees Centigrade). Ratio scales assign both values of units and ratios of comparison to a subset of a set of phenomena, e.g. 20, 40, 80 meters with ratios of 20:40 (1:2), 40:80 (1:2) and 20:80 (1:4) respectively.

In the natural sciences, standard units of measurement have been established as part of the conceptual theory of natural sciences, e.g. in addition to meters and degrees Centigrade, there are kilograms, degrees Kelvin, volts, amperes, watts, ohms, kilojoules, etc.

In the social sciences, including scientific educology, no such standard units of measurement have been established in conceptual theory. For example, in educological theory, there are no units of intelligence, no units of studying, no units of teaching, no units of knowing, no units of appropriate practice, etc.

While social scientists in general, and scientific educologists in particular, make claims to measuring phenomena, they are false claims. There are no appropriate units of measurement as yet established to conduct measurement of educational phenomena, and without units of measurement, the task of measuring is impossible.

Of course, in discourse about education, it is common to find the expressions *measurement *and *tests and measurements.* But the use of the term *measurement *in relation to scientific educological research is either a mistake, a conceit or a deception.

It is a mistake when educological researchers do not recognize and do not understand that they are operating with an ill conceived definition of the term *measurement.*

It is a conceit when social scientific researchers know that they are operating with an ill conceived definition of the term *measurement*, but want recognition as legitimate scientific researchers, and thus claim that they do conduct measurement, but the measurement is of a different nature (i.e. measurement without units) from that used in the natural sciences.

And it is a deception when social scientific researchers know and understand that they are operating with an ill conceived definition of the term *measurement* and wilfully assert (falsely) that the definition of the term *measurement* is adequately conceived i.e. conceived as “the assignment of numerals to objects or events according to rules” (F. Kerlinger, 1986, p. 391).

Putting a wine label on a bottle of vinegar does not turn the vinegar into wine. And putting the label of *measurement* on research instruments which do not provide measurement units does not turn those instruments into measurement instruments.

Without measurement (i.e. without a system of units of measurement to assign cardinal values to amount, size or intensity of subsets of a set of phenomena), there are still many ways and means of expressing data about educational phenomena in cardinal numerical form from observations and tests. The cardinal numerical data can readily be used to test hypotheses about educational phenomena for the purpose of evaluating educological theory. Data expressed in cardinal numerical form from tests and observations facilitate enumeration (counting) of subsets of sets of phenomena, and statistical analysis can be performed on the cardinal numbers generated from the enumeration.

Intelligence tests are a model case of expressing data in numerical form as cardinal numbers (without measurement units) and using that data to enumerate, analyze and establish position of intellectual performance. The process begins with the development of the intelligence test. A set of reasoning processes is conceived as part of exemplifying intelligence. Test items are generated which are judged by test makers to require the use of those reasoning processes. The test is scored as some number correctly answered out of the total items on the test.

A random sample is selected from a population. The individuals in the randomly selected sample are tested. Their scores are analyzed statistically to establish the distribution and mean of the scores. A standard deviation is calculated to establish the position of the individual scores within all of the scores. Different versions of the test are administered and analyzed to establish consistency of results among the tests so that reliability is established for the tests. Any individual test score indicates the position of the individual’s score in relation to all other scores.

The position is a rank of a subset (i.e. one person’s score) of a set (i.e. all person’s scores) of correctly answered test items. But there is no indication of the value between positions. Thus a score of 120 has a higher rank than a score of 60, but the interval between 120 and 60 does not indicate an intelligence which is twice that of the score of 60. Nor does the interval between 120 and 60 indicate a difference of 60 units of intelligence (because there are no units of intelligence as yet established). It is an indication of position, but not an interval measure nor a ratio measure of intelligence.

Thus, with intelligence tests, there is position, but there is no measured unit of intelligence. There is no interval of intelligence. There is no scale of intelligence. Hence there is no measurement. There is no measurement in the sense of making a comparison of the size, amount or intensity of a subset of phenomena with a standard unit of the phenomena so that the subset can be described numerically as some multiple of the standard unit of the phenomena.

And so it is with all other standardized tests (and indeed for all tests, standardized or otherwise) which are used in social scientific (including educological scientific) research. They establish positions for scores within a range of scores on the test. But they do not use any units of measurement to assign cardinal numbers which give value to amount, size or intensity of subsets of a set of phenomena, expressed as some number of standard units of an attribute, property or characteristic. Therefore, they are not instruments for measuring, but they are instruments for enumerating (i.e. counting) and ranking.

After a century of development of observational research instruments for educological scientific research (and for all social scientific research), no scientific educological instrument has been developed which expresses data as measures of educational phenomena (e.g. no measurement units of intelligence, learning or knowing have been developed). No scientific educological instrument has been developed which expresses data as intervals or ratios of units of measurement of educational phenomena. The current state of affairs in instruments for scientific educological research is that there are many observation schedules, behavioral inventories, questionnaires, instruments for expressions of preference, rating guidelines, scoring instructions, aptitude tests, attitude tests, achievement tests, etc., but no instruments measuring educational phenomena and expressing those measures as some interval or ratio unit of measurement.

Scientific educological research instruments over the past century have been developed to express data about educational phenomena as numerals to denote

(1) categories (with nominal numerals),

(2) ratings (with nominal numerals),

(3) rankings (with ordinal numerals) and

(4) enumerations (with cardinal numbers).

Thus in scientific educological research, there is no measurement as yet developed (it may never be developed!) of educational phenomena, but there is plenty of enumeration, ranking and rating of educational phenomena.