Measurement

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Measurement

Measurement is the evaluation or estimation of degree, extent, dimension, or capacity in relation to certain standards (i.e., units of measurement). As one of the most important inventions in human history, the process of measuring involves every aspect of our lives, such as time, mass, length, and space. The Greeks first developed the “foot” as their fundamental unit of length during the fourth millennium BCE. The ancient peoples of Mesopotamia, Egypt, and the Indus Valley seem to have all created systems of weight around the same period. Zero, the crucial number in the history of measurement, was first regarded as a true number by Aristotle.

The ancient Egyptians first developed a systematic method of measuring objects, which they used in the construction of pyramids. During the long period during which their civilization thrived in northeastern Africa, they also cultivated the earliest thoughts on earth measurement: geometry. Euclid of Alexandria (c. 330–c. 275 BCE), a Greek mathematician who lived in Egypt and who is regarded as the “father of geometry,” provided the proofs of geometric rules that Egyptians had devised in building their monuments. His most famous work, Elements, covers the basic definitions, postulates, propositions, and proofs of mathematical and geometric theorems. Euclid’s Elements has proven instrumental in the development of modern science and measurement.

Another great mathematician who contributed to modern measurement was Karl Friedrich Gauss. He was born on April 30, 1777, in Brunswick, Germany. At age twenty-four, Gauss published a brilliant work, Disquisitiones Arithmeticae, in which he established basic concepts and methods of number theory. In 1801, Gauss developed the method of least squares in calculating the orbital component of the motion of celestial bodies with high accuracy. Since that time the method of least squares has been the most widely used method in all of science to estimate the impact of measurement error. He was able to prove that a bell-shaped, normally distributed error curve is a basic assumption of statistical probability analysis (Gauss-Markov theorem).

Among all science and social science disciplines, there are three broad measurement theories (Michell 1986, 1990). The first and most commonly used is the classical theory of measurement. Measurement is defined by the magnitudes of the quantity and expressed as real numbers. An object’s quantitative properties are estimated in relation to one another, and ratios of quantities can be determined by the unit of measurement. The classical concept of measurement can be traced back to early theorists and mathematicians, including Isaac Newton and Euclid. The classical approach assumes that the underlying reality exists, but only quantitative attributes are measurable, and the meaningfulness of scientific theories can, and only can, be supported by the empirical relationships of various measurements.

The second theory, the representational approach, defines measurement as “the correlation of numbers and entities that are not numbers” (Nagel [1932] 1960, p. 121). For example, IQ scores can be used to measure intelligence, and the Likert scale measures personal attitudes based on a set of statements. The representational approach assumes that a reality exists and can be measured, and the goal of science is to understand this reality. However, the representational approach does not insist that only quantitative properties are measurable. Instead, measurements can be used to reflect differences at multiple levels.

Unlike the classical and representational approaches, the third approach, called the operational approach, avoids the assumption of objective reality. Instead, it emphasizes only the precisely specified operational process, such as the measurement of reliability and validity. The main concern of scientific theories is only the relationships indicated by the measurements rather than the distance between the reality and measures.

According to the different properties and relationships of the numbers, there are four different levels of measurement: nominal, ordinal, interval, and ratio. In nominal (also called categorical) measurement, names or symbols are assigned to objects, and this assignment is determined by the similarity of the to-be-measured values or attributes. The categories of assignment in most situations are defined arbitrarily, for instance, numbers assigned to individual marital status: single = 1, married = 2, separated = 3, divorced = 4, and so on, or to religious preference: Christian = 1, Jewish = 2, Muslim = 3, Buddhist = 4, and so on.

In ordinal measurement, the number assigned to the objects based on their attributes reflects an order relation among them. Examples include grades for academic performance (A, B, C …), the results of sporting events and the awarding of gold, silver, and bronze medals, and many measurements in psychology and other social science disciplines.

Interval measurements have all the features of ordinal measurements, but in addition the difference between the numbers reflects the equivalent interval of the attributes being measured. This property makes comparison among different measures of an attribute or characteristic meaningful and operations such as addition and subtraction possible. Temperature in Fahrenheit or Celsius degrees, calendar dates, and standardized intelligence tests (IQ) are a few examples of interval measurements.

In ratio measurement, objects are assigned numbers that have all the features of interval measurements, but in addition there are meaningful ratios between the numbers. In other words, the zero value is a meaningful point on the measurement scale, and operations of multiplication and division are therefore also meaningful. Examples include income in dollars, length or distance in meters or feet, age, and duration in seconds or hours.

Because measurement can be arbitrarily defined by the government, researchers, or cultural norms, it is socially constructed. The social construction of measurement is frequently encountered in social science disciplines. For instance, the U.S. Census Bureau has redefined the measure of race several times. Before the 1980 census, census forms contained questions about racial categories, but the categories included only white, black, American Indian, and specified Asian categories. The census was based on the Office of Management and Budget’s (OMB) 1977 Statistical Policy Directive Number 15, Race and Ethnic Standards for Federal Statistics and Administrative Reporting, defining four mutually exclusive single-race categories: white, black, American Indian or Alaska Native, and Asian or Pacific Islander. In addition, the standards also provided two ethnicity categories: Hispanic origin and Not of Hispanic origin. The 1980 and 1990 censuses were collected according to these standards.

By 1997, OMB modified the race/ethnicity measurement again by splitting the Asian or Pacific Islander category into two groups, creating five race categories: white, African American, American Indian or Alaska Native, Asian, and Native Hawaiian or Other Pacific Islander. In addition, the 2000 census allowed people to identify themselves as belonging to two or more races. It also created six single races and fifty-seven multiple race categories. The ethnicity measure for Hispanic doubled the total number of the race/ethnicity categories to 126. However, such an extensive number of measures causes even more problems. Many Hispanics consider their ethnic origin as a racial category and therefore choose “some other race” on the census form, leading to over 40 percent of the Texas population reported as “some other race.” The misconstruction of the categories of race and ethnicity in the U.S. census illustrates the fluid and subjective nature of measurement.

SEE ALSO Econometrics; Ethnicity; Gender; Likert Scale; Mathematics in the Social Sciences; Measurement Error; Methods, Quantitative; Racial Classification; Regression Analysis; Sampling; Scales; Survey