Fundamental considerations in quantitative research

Whether numbers obtained through an experimental procedure are considered measurements is, on the one hand, largely a matter of how measurement is defined. On the other hand, the nature of the measurement process has important implications for scientific research. Firstly, many arithmeitic operations are only justified for measurements either in the classical sense described above, or in the sense of interval and ratio-level measurements as defined by Stevens (which arguably describe the same thing). Secondly, quantitative relationships between different properties which feature in most natural theories and laws imply that the properties have a specific type of quantitative structure; namely, the structure of a continuous quantity. The reason for this is that such theories and laws display a multiplicative structure (for example Newton's second law).

Continuous quantities are those for which magnitudes can be represented as real numbers and for which, therefore, measurements can be expressed on a continuum. Continuous quantities may be scalar or vector quantities. For example, SI units are physical units of continuous quantitative properties, phenomena, and relations such as distance, mass, heat, force and angular separation. The classical concept of quantity described above necessarily implies the concept of continuous quantity.

Recording observations with numbers does not, in itself, imply that an attribute is quantitative. For example, judges routinely assign numbers to properties such as the perceived beauty of an exercise (e.g. 1-10) without necessarily establishing quantitative structure in any sort of rigorous fashion. A researcher might also use the number 1 to mean "Susan", 2 to mean "Michael", and so on. This, however, is not a meaningful use of numbers: the researcher can arbitrarily reassign the numbers (so that 1 means "Michael" and 2 means "Susan") without losing any information. Put another way, facts about numbers (for example, that 2 is greater than 1, that 5 is two more than 3, and that 8 is twice 4) don't mean anything about the names corresponding to those numbers. A person's name is not, therefore, a quantitative property.

Whether counts of objects or observations are considered measurements is also largely a matter of how measurement is defined. Again, though, an important consideration is the manner in which resulting numbers are used. Counts are not measurements of continuous quantities. If, for example, a researcher were to count the number of grains of sand in a specified volume of space on a beach, the result denumerates how many separate grains there are; i.e. the number of separate distinguishable entities of a specific type. Arithmetic operations, such as addition, have meaning only in this specific sense. For instance, combining 5 and 4 grains of sand gives 9 grains of sand. The numbers used in this case are therefore the natural numbers.