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. According to Stevens' theory of scales, different mathematical operations on variables are possible, depending on the level at which a variable is measured. Various meters Measurement is the estimation of a physical quantity such as length, temperature, or time. ... In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... Stanley Smith Stevens (1906-1973) was an American psychologist best known as the founder of Harvards Psycho-Acoustical Laboratory and credited with the introduction of Stevens power law. ... Year 1946 (MCMXLVI) was a common year starting on Tuesday (link will display full 1946 calendar) of the Gregorian calendar. ...

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. Nominal data is sometimes referred to as discrete data, because you only have discrete values Descriptive statistics are used to describe the basic features of the data in a study. ... In statistics, a result is significant if it is unlikely to have occurred by chance, given that a presumed null hypothesis is true, but is not improbable if the null hypothesis is false. ...

Stevens proposed four levels of measurement:

• nominal (or categorical)
• ordinal
• interval
• ratio

Interval and ratio variables are also grouped together as continuous variables.

In the paper in which Stevens introduced the classification Scheme, he also proposed the definition that is widely cited in texts in some version: "Measurement is the assignment of numbers to objects or events according to a rule". Many remain unaware this definition has received criticism on a number of grounds (e.g. Duncan, 1984; Michell, 1986, 1999). However, the scheme is widely used.

Nominal measurement

In this type of measurment, 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. A name is a label for a thing, person, place, product (as in a brand name), and even an idea or concept, normally used to distinguish one from another. ... Look up similarity in Wiktionary, the free dictionary. ... Look up category in Wiktionary, the free dictionary. ... For alternative meanings see definition (disambiguation) A definition may be a statement of the essential properties of a certain thing, or a statement of equivalence between a term and that terms meaning. ... A numeral is a symbol or group of symbols that represents a number. ... This article discusses the use of the word Number in Mathematics. ... Stanley Smith Stevens (1906-1973) was an American psychologist best known as the founder of Harvards Psycho-Acoustical Laboratory and credited with the introduction of Stevens power law. ... Red is any of a number of similar colors evoked by light consisting predominantly of the longest wavelengths of light discernible by the human eye, in the wavelength range of roughly 625–750 nm. ... YOU SUCK!!!!! ... An ostensive definition conveys the meaning of a term by pointing out examples of what is defined by it. ... A persons marital status describes their relationship with a significant other. ... In statistics, central tendency is an average of a set of measurements, the word average being variously construed as mean, median, or other measure of location, depending on the context. ... In descriptive statistics, statistical dispersion (also called statistical variability) is quantifiable variation of measurements of differing members of a population within the scale on which they are measured. ... The variation ratio is the percent of cases which are not the mode. ... Qualitative variation (QV) allows us to assess the degree of statistical dispersion in nominal distributions. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... Social research refers to research conducted by social scientists (primarily within sociology, but also within other disciplines such as social policy, human geography, social anthropology and education). ... Gender in common usage refers to the sexual distinction between male and female. ... For other uses, see Race (disambiguation). ... The following is a list of religions. ... An academic major is a general scholarly pursuit in a specific area of study which often yields, at the end of a tenure usually of four years, a bachelors degree. ... The place of birth is the place where a person was born. ...

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 many 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. This article needs to be cleaned up to conform to a higher standard of quality. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... This article or section does not cite any references or sources. ... Psychology (from Greek: ψυχή, psukhē, spirit, soul; and λόγος, logos, knowledge) is an academic / applied discipline involving the scientific study of mental processes and behavior of humans and animals. ... The social sciences are groups of academic disciplines that study the human aspects of the world. ... Attitude is a hypothetical construct that represents an individuals like or dislike for an item. ... This article deals with conservatism as a political philosophy. ... For with(out) prejudice in law, see Prejudice (law). ... Social class refers to the hierarchical distinctions between individuals or groups in societies or cultures. ... In statistics, central tendency is an average of a set of measurements, the word average being variously construed as mean, median, or other measure of location, depending on the context. ... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ...

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. A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. ... In, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... Note that this article includes some hyperlinked dates whose format is configurable in Special pages | Preferences. What you see may not be what the author intended. ... A page from the Hindu calendar 1871-72. ... Fig. ... The degree Celsius (symbol: °C) is an SI derived unit of temperature. ... Fahrenheit is a temperature scale named after the German physicist Gabriel Fahrenheit (1686–1736), who proposed it in 1724. ...

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 organizations belonged to or number of church attendances in a particular time. Unsolved problems in physics: What causes anything to have mass? The U.S. National Prototype Kilogram, which currently serves as the primary standard for measuring mass in the U.S. Mass is the property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... For other uses of this word, see Length (disambiguation). ... The kelvin (symbol: K) is a unit increment of temperature and is one of the seven SI base units. ... Absolute zero is the lowest possible temperature where nothing could be colder, and no heat energy remains in a substance. ... In, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ... The effects of ageing on a human face Elderly woman Ageing or aging is the process of systems deterioration with 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. The former Weights and Measures office in Middlesex, England. ...

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 by no means universally accepted (for example, Velleman & Wilkinson, 1993). [1]

Duncan (1986) observed that Stevens' classification nominal measurement is contrary to his own definition of measurement. Stevens (1975) said on his own definition of measurement that "the assignment can be any consistent rule. The only rule not allowed would be random assignment, for randomness amounts in effect to a nonrule". However, so-called nonimal measurement involves arbitrary assignment, and the "permissible transformation" is any number for any other. This is one of the points made in Lord's (1953) satirical paper On the Statistical Treatment of Football Numbers.

Among those who accept the classification scheme, there is also some controversy in behavioural sciences over whether the mean is meaningful for ordinal measurement. In terms of measurement theory, it is not, because the arithmetic operations are not made on numbers that are measurements in units, and so the results of computations do not give numbers in units. However, many behavioural scientists use means for ordinal data 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. For example, applications of measurement models in educational contexts often indicate that total scores have a fairly linear relationship with measurements across a range of an assessment. 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. Louis Leon Thurstone (29 May 1887–29 September 1955) was a psychometrician most notable for his contributions to factor analysis with regard to psychological tests. ... Conceived by L. L. Thurstone, the law of comparative judgment (LCJ) is a general mathematical representation of a discriminal process, which is any process in which a comparison is made between pairs of a collection of entities with respect to magnitudes of an attribute, trait, attitude, and so on. ... Rasch models are probabilistic measurement models which currently find their application primarily in psychological and attainment assessment, and are being increasingly used in other areas, including the health profession and market research. ...

References

• Babbie, E., The Practice of Social Research, 10th edition, Wadsworth, Thomson Learning Inc., ISBN 0-534-62029-9
• Duncan, O. D. (1984). Notes on social measurement: historical and critical. New York: Russell Sage Foundation.
• Lord, F.M. (1953). On the Statistical Treatment of Football Numbers. Reprint in Readings in Statistics, Ch. 3, (Haber, A., Runyon, R.P., and Badia, P.) Readingm Mass: Addison-Wesley, 1970.
• Michell, J. (1986). Measurement scales and statistics: a clash of paradigms. Psychological Bulletin, 3, 398-407.
• Stevens, S.S. (1946). On the theory of scales of measurement. Science, 103, 677-680.
• Stevens, S.S. (1951). Mathematics, measurement and psychophysics. In S.S. Stevens (Ed.), Handbook of experimental psychology (pp. 1-49). New York: Wiley.
• Stevens, S.S. (1975). Psychophysics. New York: Wiley.
• Velleman, P. F. & Wilkinson, L. (1993). Nominal, ordinal, interval, and ratio typologies are misleading. The American Statistician, 47(1), 65-72. [On line] http://www.spss.com/research/wilkinson/Publications/Stevens.pdf

Earl Robert Babbie (January 8, 1938 - present) is a professor of sociology and behavioral sciences at Chapman University, United States. ...

External links

• Hyperstat — Measurement Scales
• Measurement theory: Frequently asked questions