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The level of measurement of a variable in mathematics and statistics is a classification that was proposed in order to describe the nature of information contained within numbers assigned to objects and, therefore, within the variable. The levels were proposed by Stanley Smith Stevens in his 1946 article On the theory of scales of measurement. Different mathematical operations on variables are possible, depending on the level at which a variable is measured.
Classification levels According to the classification scheme, in statistics the kinds of descriptive statistics and significance tests that are appropriate depend on the level of measurement of the variables concerned. Stevens proposed four levels of measurement: Nominal measurement In this type of measurement, names are assigned to objects as labels. This assignment is performed by evaluating, by some procedure, the similarity of the to-be-measured instance to each of a set of named exemplars or category definitions. The name of the most similar named exemplar or definition in the set is the "value" assigned by nominal measurement to the given instance. If two instances have the same name associated with them, they belong to the same category, and that is the only significance that nominal measurements have. For practical data processing the names may be numerals, but in that case the numerical value of these numerals is irrelevant. The only comparisons that can be made between variable values are equality and inequality. There are no "less than" or "greater than" relations among the classifying names, nor operations such as addition or subtraction. "Nominal measurement" was first identified by psychologist Stanley Smith Stevens in the context of a child learning to categorize colors (red, blue and so on) by comparing the similarity of a perceived color to each of a set of named colors previously learned by ostensive definition. Other examples include: geographical location in a country represented by that country's international telephone access code, the marital status of a person, or the make or model of a car. The only kind of measure of central tendency is the mode. Statistical dispersion may be measured with a variation ratio, index of qualitative variation, or via information entropy, but no notion of standard deviation exists. Variables that are measured only nominally are also called categorical variables. In social research, variables measured at a nominal level include gender, race, religious affiliation, political party affiliation, college major, and birthplace. Ordinal measurement In this classification, the numbers assigned to objects represent the rank order (1st, 2nd, 3rd etc.) of the entities measured. The numbers are called ordinals. The variables are called ordinal variables or rank variables. Comparisons of greater and less can be made, in addition to equality and inequality. However operations such as conventional addition and subtraction are still meaningless. Examples include the Mohs scale of mineral hardness; the results of a horse race, which say only which horses arrived first, second, third, etc. but no time intervals; and most measurements in psychology and other social sciences, for example attitudes like preference, conservatism or prejudice and social class. The central tendency of an ordinally measured variable can be represented by its mode or its median; the latter gives more information. Interval measurement The numbers assigned to objects have all the features of ordinal measurements, and in addition equal differences between measurements represent equivalent intervals. That is, differences between arbitrary pairs of measurements can be meaningfully compared. Operations such as addition and subtraction are therefore meaningful. The zero point on the scale is arbitrary; negative values can be used. Ratios between numbers on the scale are not meaningful, so operations such as multiplication and division cannot be carried out directly. But ratios of differences can be expressed; for example, one difference can be twice another. The central tendency of a variable measured at the interval level can be represented by its mode, its median or its arithmetic mean; the mean gives the most information. Variables measured at the interval level are called interval variables, or sometimes scaled variables, though the latter usage is not obvious and is not recommended. Examples of interval measures are the year date in many calendars, and temperature in Celsius scale or Fahrenheit scale. About the only interval measures commonly used in social scientific research are constructed measures such as standardized intelligence tests (IQ). Ratio measurement The numbers assigned to objects have all the features of interval measurement and also have meaningful ratios between arbitrary pairs of numbers. Operations such as multiplication and division are therefore meaningful. The zero value on a ratio scale is non-arbitrary. Variables measured at the ratio level are called ratio variables. Most physical quantities, such as mass, length or energy are measured on ratio scales; so is temperature measured in kelvins, that is, relative to absolute zero. The central tendency of a variable measured at the ratio level can be represented by its mode, its median, its arithmetic mean, or its geometric mean; as with an interval scale, however, the arithmetic mean gives the most useful information. Social variables of ratio measure include age, length of residence in a given place, number of organisations belonged to or number of church attendances in a particular time. The interval and ratio measurement levels are sometimes collectively called "true measurement", although it has been argued that this usage reflects a lack of understanding of the uses of ordinal measurement. Only ratio or interval scales can correctly be said to have units of measurement. Debate on classification scheme There has been, and continues to be, debate about the merits of the classifications, particularly in the cases of the nominal and ordinal classifications (Michell, 1986). Thus, while Stevens' classification is widely adopted, it is not universally accepted (for example, Velleman & Wilkinson, 1993). * Among those who accept the classification scheme, there is also some controversy in behavioural sciences over whether the mean is meaningful for ordinal measurement. Mathematically it is not, but many behavioural scientists use it anyway. This is often justified on the basis that ordinal scales in behavioural science are really somewhere between true ordinal and interval scales; although the interval difference between two ordinal ranks is not constant, it is often of the same order of magnitude. Thus, some argue, that so long as the unknown interval difference between ordinal scale ranks is not too variable, interval scale statistics such as means can meaningfully be used on ordinal scale variables. L. L. Thurstone made progress toward developing a justification for obtaining interval-level measurements based on the law of comparative judgment. Further progress was made by Georg Rasch, who developed the probabilistic Rasch model which provides a theoretical basis and justification for obtaining interval-level measurements from counts of observations such as total scores on assessments. | ||||||||
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