Measurement Theory
Measurement theory describes the methodology associated with the relation of numbers and mathematical operations to physical objects and processes. In a very practical sense, measurement theory deals with mathematical measurement and the analysis of mathematical measurement. A key component of mathematical analysis is error analysis that allows the estimations of measurement accuracy and precision. The development of measurement theory also involves justification of measurement and estimation of uniqueness of the measurement of physical phenomena. Measurements are considered unique if there is only one possible way to mathematically characterize the phenomena.
Measurement theory is as ancient as formalized mathematics. Aspects of measurement theory were included in Euclid's elements and since that time measurement theory has evolved critical axioms that form the basis of modern data analysis. Measurement theory itself is broadly described as the set of rules related to the quantitative measurement of properties of physical entities (e.g., length) or the measurement of the duration between phenomena (e.g., time). Geometric measurements represent the spatial and dimensional properties of objects.
Well-ordered measurements are those in which the ultimate numerical assignments of a measurement closely correspond to the real or original order obtained in the actual observation or measurement process.
Measurement theory extends to a number of types of measurement. Extended measurements are those measurements that are created by the combination of measured attributes or that are linked together in a series or chain of measurements to describe multiple objects. Comparative measurement relates the properties of a body or system to a known standard (e.g., weight and other standardized unit measures). Differential measurements usually deal with measurement of differences between systems or in the measurement of the intervals between phenomena. Conjointed measurements describe those attributes of a system or body that cannot be measured directly but can only be assigned a numerical quantity when related to changes in measurable phenomena (e.g., the measurement of intelligence by test scores).
Historically, the estimation of error in measurement evolved from direct estimation of numerical accuracy to a complex array of statistical techniques. Powerful computer-based data-analysis techniques referred to by mathematicians and statisticians as bootstrap statistics now allow the accurate determination of the reliability of data based upon only a small and representative sample of measurements. The techniques, invented in 1977 by Stanford University mathematician Bradley Efron, have found wide use in almost all fields of scholarship, including subjects as diverse as politics, economics, biology, and astrophysics.
Most scientific measurements are specified to contain some range of accuracy or degree of error. Errors are usually described as directed or random, and encompass deviations in mathematical measurements from the actual or true state. Errors can result from a variety of sources ranging from random human error to systematic errors that propagate through a series of mathematical equations. Modern error analysis still relies on the fundamental techniques advanced by French-Italian astronomer Joseph-Louis Lagrange (born Giuseppe Luigi Lagrandgia, 1736-1813) and French mathematician and astronomer Pierre-Simon Laplace.
Prior to the formulation of quantum theory, measurement theory treated measurement error as a problem resolvable by refinement of technology and statistical techniques. The advent of quantum theory, however, provided limitations on the resolvability of error and made the actual measurement of phenomena an integral factor in scientific models. Electrons, for example, are be found to be in one of two possible spin states. Prior to any determination of this spin state the electron is described by a state vector that is a linear superposition of those two states. Accordingly, electron spin is not an intrinsic property of the electron unless, and until, it is measured. According to the Copenhagen interpretation of quantum theory, the act of measurement collapses the state vector and results in an electron with a defined spin.
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