# Variation

COMPONENTS AND MEASURES OF VARIATION

USE OF VARIATION IN SOCIAL SCIENCES

BIBLIOGRAPHY

Variation refers to the degree of dispersion, diversity, or inequality in a distribution: the extent to which observations of an attribute are similar to or different from one another. Applications of variation in the social sciences include income inequality, the degree to which incomes are concentrated among a few households or spread broadly across a population, and religious diversity, which describes whether a single religion dominates the cultural landscape or multiple religions exist side by side.

## COMPONENTS AND MEASURES OF VARIATION

Popular quantitative measures of variation include the following:

Range : The difference between extreme values.

Variance : The sum of the squared distances from each observation to the mean divided by the number of observations (for a population) or divided by one less than the number of observations (for a sample).

Standard deviation : The square root of the variance. The standard deviation is the average distance between a set of observations and the mean.

Coefficient of variation : The standard deviation divided by the mean. Graphically the coefficient of variation describes the peakedness of a unimodal frequency distribution. Because the coefficient of variation scales to the mean, this measure often is used in comparative analysis.

Gini coefficient : Bounded by zero and one, where lower values correspond to greater equality, the Gini coefficient is defined as twice the area between the Lorenz curve and an equality diagonal. (To construct a Lorenz curve, one ranks the observations from lowest to highest on the variable of interest and then plots the cumulative proportion of the population against the cumulative proportion of the variable of interest.)

Theils T statistic : Particularly appropriate for hierarchical, nested, or aggregate data, Theils T statistic is the product of the population share of each observation, the quotient of the observation and the population average, and the natural logarithm of the quotient of the observation and the population average, summed over all observations. Unlike the Gini coefficient and the coefficient of variation, Theils T statistic is sensitive to the number of underlying observations.

In addition to these widely used metrics, researchers sometimes create original measures to isolate a particular type of variation. For example, in American Apartheid (1993), Douglas Massey and Nancy Denton develop five dimensions of variation to identify racial hypersegregation in U.S. metropolitan areas.

Whereas measures of central tendency, such as the mean, median, and mode, describe the center of gravity of a distribution, measures of variation express how concentrated the observations are around the average. Both types of measures are necessary to describe most social science data sets. For instance, with regard to the test scores in a school, the median expresses the score of a typical student. Schools desire high median scores. However, a high level of variation in test scores indicates that some students excel, while others lag behind their peers. This could suggest differences among students in family resources, native intelligence, or other characteristics over which a school has little control, but high variation also can result from inconsistent teacher quality or administrative choices to overallocate resources to high-achieving students and/or underallocate resources to low-achieving students.

The ease of computing variation can hide the difficulties in its interpretation. Although variation reflects differences among individuals, it is a property of a group, not an individual. Correlation between variation and another group-level variable does not imply a corresponding relationship across individuals; improperly asserting that it does is an instance of the ecological fallacy. Similarly variation is relative. A measure of variation can only be described as high or low in comparison to the variation of another group, the same group in a different time period, or a predefined standard value.

## USE OF VARIATION IN SOCIAL SCIENCES

Whether variation in a particular attribute is preferred is an issue of context and perspective. A high degree of variation may be preferable, as in the case of racial diversity within communities, in which variation may imply integration rather than segregation. A low degree of variation may be preferable; for instance, a good medical intervention consistently will improve the health of those who receive treatment and have few side effects. Alternatively, preference for high or low variation may depend on ones theoretical or ethical perspective. For instance, one view of income inequality is that higher levels of variation reflect a more efficient economic system that rewards hard work (or talent) and punishes laziness (or ineptitude). An alternative view is that income inequality suggests nondemocratic use of monopoly power or outright oppression of the poor.

One way to reduce subjective judgment is to view variation as a predictor variable rather than an outcome variable. An example of such an application is research that relates economic inequality to population health. In a 1975 article Samuel Preston popularized this issue by pointing out an inherent nonlinearity in the relationship of health to income. For individuals with low incomes, an increase in economic resources will be accompanied by significant health gains, but as an individuals income rises to the top of the distribution, subsequent increases in income will yield diminishing returns in terms of health. Thus at least among nations with high average incomes, lower variation of incomes should be associated with better health. Although subsequent scholarship seemed to confirm this correlation, there is debate over causation. Some scholars assert that inequality leads to stress and stress leads to poor health. Other researchers point to mediating factors, such as race and access to care, to explain the correlation between inequality and health.

Individuals with low incomes are more likely to have poor health, and individuals with high incomes are more likely to have good health, on average. Thus the suggestion that an increase in inequality that results from the poor getting poorer would decrease population health seems reasonable. However, advocates of the inequality hypothesis also must explain why an inequality increase that occurs when the rich get richer also would reduce health. Until they address this seeming paradox, the burden of proof lies with those who claim that income variation is an operative factor for health. This exemplifies the central challenge in using variation in social science analysis; to be a useful tool, variation must be both measurable and meaningful.

## BIBLIOGRAPHY

Kachigan, Sam Kash. 1991. Multivariate Statistical Analysis: A Conceptual Introduction. 2nd ed. New York: Radius.

Massey, Douglas S., and Nancy A. Denton. 1993. American Apartheid: Segregation and the Making of the Underclass. Cambridge, MA: Harvard University Press.

Mullahy, John, Stephanie Robert, and Barbara Wolfe. 2004. Health, Income, and Inequality. In Social Inequality, ed. Kathryn Neckerman, 523544. New York: Russell Sage Foundation.

Preston, Samuel H. 1975. The Changing Relation between Mortality and Level of Economic Development. Population Studies 29: 231248.

J. Travis Hale

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## variation (statistical)

variation (statistical) The greater part of empirical research is concerned with the characteristics of groups, or aggregate social entities, rather than individual cases; that is, with men or women in general, rather than any particular man or woman. A range of statistical measures of association are employed to describe the features of groups, or types of case in the aggregate. Most of them are based on the normal distribution, the Binomial distribution, or the Poisson distribution, from which a number of statistical summary measures are derived. The mean, mode, and median provide measures of central tendency, the most common or typical value in the distribution, which coincide when the distribution is normal. Measures of dispersion attempt to concentrate information about the general pattern in single summary statistics. These include the range, mean deviation, quartile deviation, decile range, and the standard deviation, which is by far the most important. One of the properties of the normal distribution is that about 68 per cent of all cases are contained within one standard deviation on either side of the mean, about 95 per cent lie within two standard deviations, and some 99.73 per cent lie within three standard deviations on either side of the mean. The standard deviation of a distribution thus summarizes a great deal of information about the overall dispersion, the degree of clustering or concentration around the mean. The variance is the square of the standard deviation, and has the property of being additive. Analysis of variance measures variance within sub-groups in the data-set, and between the averages of these sub-groups. See also MEASUREMENT; SAMPLING.

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## variation

var·i·a·tion / ˌve(ə)rēˈāshən/ • n. 1. a change or difference in condition, amount, or level, typically with certain limits: regional variations in house prices| the figures showed marked variation from year to year. ∎  Astron. a deviation of a celestial body from its mean orbit or motion. ∎  Math. a change in the value of a function due to small changes in the values of its argument or arguments. ∎  (also mag·net·ic var·i·a·tion) the angular difference between true north and magnetic north at a particular place. ∎  Biol. the occurrence of an organism in more than one distinct color or form. 2. a different or distinct form or version of something: hurling is an Irish variation of field hockey. ∎  Mus. a version of a theme, modified in melody, rhythm, harmony, or ornamentation, so as to present it in a new but still recognizable form: there is an eleven-bar theme followed by seven variations and a coda | fig. variations on the perennial theme of marital discord. ∎  Ballet a solo dance as part of a performance. DERIVATIVES: var·i·a·tion·al / -shənl/ adj.

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## variation

variation The differences between individuals of a plant or animal species. Variation may be the result of environmental conditions; for example, water supply and light intensity affect the height and leaf size of a plant. Differences of this kind, acquired during the lifetime of an individual, are not transmitted to succeeding generations since the genes are not affected. Genetic variation, due to differences in genetic constitution, is inherited (see continuous variation; discontinuous variation). The most important sources of genetic variation are mutation and recombination (see also crossing over). It is also increased by outbreeding. Wide genetic variation improves the ability of a species to survive in a changing environment, since the chances that some individuals will tolerate a particular change are increased. Such individuals will survive and transmit the advantageous genes to their offspring.

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## variation (in music)

variation, in music, a compositional device in which certain features of a musical unit, e.g., phase, are altered while others are retained in a subsequent statement of the unit. Modifications include melodic, harmonic, and rhythmic. Variation is fundamental in Western music, serving to identify the unique features of a composition by partitioning those features. Gregorian chant exhibits much melodic variation, and all music from the Middle Ages through the 20th cent. employs the technique in some form. Specifically the term refers to a musical form, also called "theme and variations," in which the varied item is an entire brief movement. The form originated in baroque dance suites, in which all movements have the same theme, and was popular during the 18th and 19th cent. Bach's Goldberg Variations and Beethoven's Diabelli Variations are famous examples of the genre.

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## variation

variation. Piece of mus. which is a varied version of a well-known tune or of an orig. theme specially comp. as basis for variations. Some variations follow the orig. tune closely, others make the briefest reference to it, sometimes harmonically rather than thematically. Popular form with composers from 16th cent., as in ‘divisions on a ground’. In some comps., one movement takes the form of a theme with variations, e.g. the finale of Beethoven's Eroica sym., and movements of several Haydn syms. Among famous sets of variations are Beethoven's on a waltz by Diabelli, Brahms's on a theme by Haydn, Schumann's Études symphoniques, Strauss's Don Quixote, Elgar's Enigma, Rachmaninov's Rhapsody on a theme of Paganini, and Britten's Variations on a Theme of Frank Bridge.

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## variation

variation Differences displayed by individuals within a species, and which may be favoured or eliminated by natural selection. In sexual reproduction, reshuffling of genes in each generation ensures the maintenance of variation. The ultimate source of the variation is mutation, which produces fresh genetic material.

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## variation

variation Differences displayed by individuals within a species, and which may be favoured or eliminated by natural selection. In sexual reproduction, reshuffling of genes in each generation ensures the maintenance of variation. The ultimate source of the variation is mutation, which produces fresh genetic material.

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## variation

variation Differences displayed by individuals within a species, and which may be favoured or eliminated by natural selection. In sexual reproduction, reshuffling of genes in each generation ensures the maintenance of variation. The ultimate source of the variation is mutation, which produces fresh genetic material.

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## variation

variation
1. See measures of variation.

2. (contractual variation) The process of changing a contract to supply goods or services, and the results of any change to the contract as represented in a contract amendment.

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## variation

variation In biology, differences between members of the same species. Variation occurs naturally due to heredity and to differences in the environment during development. See also adaptation; evolution

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## variation

variation In music, a variety of treatments upon a single theme. Successive statements of the theme are altered by such means as simple elaboration, change of key, or change of time signature.

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## variation (in biology)

variation, in biology: see Darwinism.