A quantitative attribute is one that exists in a range of magnitudes, and can therefore be measured. Measurements of any particular quantitative property are expressed as a specific quantity, referred to as a unit, multiplied by a number. Examples of physical quantities are distance, mass, and time. Many attributes in the social sciences, including abilities and personality traits, are also studied as quantitative properties and principles.
Numbers obtained through an experimental procedure are considered measurements depends upon how measurement is defined. The nature of measurement depends upon 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. 110) 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.
Any object is characterized by many attributes, such as colour and mass, only some of which constitute continuous quantities. For example, the mass of a specific grain of sand is a continuous quantity whereas the grain, as an object, is not. Thus, the mass of a grain of sand can be used as a unit of mass because it is possible to estimate the ratio of the mass of another object to the mass of a grain of sand, given an appropriate instrument.
In the social sciences, it is also common to count frequencies of observations; i.e. frequencies of observable outcomes in an experiment. Examples include the number of correct scores on an assessment of an ability, and the number of statements on a questionnaire endorsed by respondents. Provided each observable outcome is the manifestation of an underlying quantitative attribute, such frequencies will generally indicate relative magnitudes of that attribute. Strictly speaking, however, counts and frequencies do not constitute measurement in terms of a unit of continuous quantity.
