Distribution

Classification of Distributions

Distributions may be classified along several dimensions—continuous versus discrete, univariate versus multivariate, modeling versus sampling—or according to mathematical characteristics of their associated functions. For introduction and comprehensive exposition, see the volumes in Johnson and Kotz's series, Distributions in Statistics, and their revisions (e.g., Johnson, Kotz, & Balakrishnan, 1994), as well as the little handbook by Evans, Hastings, and Peacock (2000).

Associated Functions

Mathematically specified distributions have associated with them a variety of functions. The most basic is the distribution function (also known as the cumulative distribution function), which is defined as the probability α that the variate X assumes a value less than or equal to x and is usually denoted Fx(x), or simply F(x). Probably the best known of the associated functions is the probability density function, denoted f(x), which, in continuous distributions, is the first derivative of the distribution function with respect to x (and which, in discrete distributions, is sometimes called the probability mass function). For example, in the normal family, the bell-shaped curve depicting the probability density function is a more common representation than the ogive depicting the distribution function. Two of the most useful for social science are the quantile function, which, among other things, provides the foundation for whole-distribution measures of inequality, such as Pen's Parade, and the hazard function.

Of course, all the associated functions are related to each other in specified ways. For example, as already noted, among continuous distributions, the probability density function is the first derivative of the distribution function with respect to x. The quantile function, variously denoted G(α) or Q(α) or F−1(α), is the inverse of the distribution function, providing a mapping from the probability α to the quantile x. Important relations between the two include the following (Eubank, 1988; Evans, Hastings, & Peacock, 2000):

Distributional Parameters

Distributions also have associated with them a number of parameters. Of these, three are regarded as basic—the location, scale, and shape parameters. The location parameter is a point in the variate's domain, and the scale and shape parameters govern the scale of measurement and the shape, respectively. It is sometimes convenient to adopt a uniform notation across distributional families, such as that in Evans, Hastings, and Peacock (2000), which uses lower-case, italicized English letters a, b, and c to denote the location, scale, and shape parameters, respectively. Variates differ in the number and kind of basic parameters. For example, the normal distribution has [Page 277]two parameters, a location parameter (the mean) and a scale parameter (the standard deviation).

Subdistribution Structure

Distributions are useful for representing many phenomena of special interest in social science. These include, besides inequality, the subgroup structure of a population. In general, two kinds of subgroup structure are of interest, and they may be represented by versions of the distributional operations of censoring and truncation. In the spirit of Kotz, Johnson, and Read (1982, p. 396) and Gibbons (1988, p. 355), let censoring refer to selection of units by their ranks or percentage (or probability) points; and let truncation refer to selection of units by values of the variate. Thus, the truncation point is the value x separating the subdistributions; the censoring point is the percentage point α separating the subdistributions. For example, the subgroups with incomes less than $20,000 or greater than $80,000 each form a truncated subdistribution; the top 5% and the bottom 10% of the population each form a censored subdistribution.

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