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Toward Clarification of Confusion in the Concept of Percentile

Precentile is a fundamental concept in educational statistics (Kendall & Buckland, 1971). Nevertheless, no agreement has been reached among textbooks regarding definition and computation of percentile. In this paper, four percentile definitions and two computation formula have been identified from thirty educational statistics textbooks, and two definitions are recommended towards establishment of convention for the concept of percentile. Percentile Definition Different definitions of percentile can be found form many introductory textbooks in educational statistics. To facilitate an empirical study of these conflicting concepts, a random sample of thirty testbooks were selected and four assorted definitions were identified. Definition 1 Percentiles are points in a distribution below or at which a given percent, p, of the cases lie and about or at which (100-p) percent of the cases lie (Anderson, Sweeney & Williams, 1991; Ott, 1993). Definition 2 Percentiles are points in a distribution below which a given percent, p, of the cases lie and about which (100-p) percent of the cases lie (Agresti & Agresti, 1979; Ferguson, 1996; Henderson, 1964, Moore, 1985; Sincich, 1982; Zehna, 1970). Definition 3 Percentiles are points in a distribution at or below which a given percent, p, of the cases lie (Chao, 1969; Dayton & Stunkard, 1971; Devore & Peck, 1990; Friedman, 1972; Gravetter & Wallnau, 1992; Hays, 1967; Heiman, 1992; Hill & Kerber, 1967; Lumsden, 1969; Moore & McCable, 1989; Wolf, 1962). Definition, 4 Percentiles are points in a distribution below which a given percent, p, of the cases lie (Cornell, 1956; Dowine Health, 1974; Fischer, 1973; Freund, 1967; Guilford & Fruchter, 1978; Harshbarger, 1997; Hopkins & Glass, 1978; Jaeger, 1990; Naiman, Rosenfeld & Zirkel, 1972; Runyon & Haber, 1971; Sprinthall, 1994). In statistics, the probability of obtaining a point of score in a discrete distribution is called a probability mass function (Casella & Berger, 1990). A probability mass function (pmf) does not always equal zero at the point where a percentile is calculated. Thus, inclusion or exclusion of the pmf in the cumulative probability, p, of the percentile definitions may result in different values of percentile, and confusion could arise from the inconsistency among these percentile definitions. Percentile Computation The necessity of establishing a convention for the definition of percentile is further related to percentile computations. Two percentile formulas, one using uniform interpolation and the other using a cumulative distribution function (cdf), have been identified from the textbook examination. The method based on a cumulative distribution function is illustrated in many books through the use of a z table for a standard normal distribution (e.g., Heiman, 1992; Ott 1993; Runyon & Haber, 1972; and sprinthall, 1994). Nonetheless, cumulative distribution functions are generally unknown in most empirical studies. Thus, the method of uniform interpolation is introduced in most statistics books to estimate percentiles from an empirical data base (e.g., Anderson, Sweeney & Williams, 1991; Harshbarger, 1977; Heiman, 1992; Ott, 1993). In an intermediate order-statistics book, David (1981) elaborated the computation of percentile for continuous and discrete distributions. He wrote: Suppose first that X is a continuous variate with strictly increasing cdf P(x). Then the equation P(x) = p 0 [is less than] p [is less than] 1, (2.5.1) has a unique solution, say x = [[Xi].sub.p], which we call the (population) quantile of order p. Thus, [[Xi].sub.1/2] is the median of the distribution. If P(x) is not strictly increasing, P(x) = p may hold in some interval, interval which case any point in the interval would serve as a quantile of order p. …