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numerical analysis

Branch of applied mathematics that studies methods for solving complicated equations using arithmetic operations, often so complex that they require a computer, to approximate the processes of analysis (i.e., calculus). The arithmetic model for such an approximation is called an algorithm, the set of procedures the computer executes is called a program, and the commands that carry out the procedures are called code. An example is an algorithm for deriving π by calculating the perimeter of a regular polygon as its number of sides becomes very large. Numerical analysis is concerned not just with the numerical result of such a process but with determining whether the error at any stage is within acceptable bounds.

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.

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.

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.

References

• Michell, J. (1993). The origins of the representational theory of measurement: Helmholtz, Hölder, and Russell. Studies in History and Philosophy of Science, Vol. 24 No. 2, 185-206.
• Nagel, E. (1932). Measurement. Erkenntnis, 2, 313-33, reprinted in A. Danato and S. Morgenbesser (Eds.), Philosophy of Sciences (pp. 121-140). New York: New American Library.