Toward clarification of confusion in the concept of percentile

Percentile 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 formulas 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, 1966; 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 (Co, 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; Freuna 1967; Guilford & Fruchter, 1978; Harshbarger, 1997; Hopkins & Glass, 1978; Jaega; 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 centile 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

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, in which case any point in the interval would serve as a quantile of order p. When X is discrete xi sub p can still be defined by a generation of (2.5.1.), namely, Pr{X

Hence, a percentile estimate can be a point or an interval, depending on the distribution of the data. The point estimate of percentile can be treated as a special case of interval estimates in which the length of interval is zero. Because few educational statistics textbooks contain the interval estimation of percentile, a convention has yet to be established regarding the method of identifying the percentile position in a given distribution.

Discussion

Based on the brief review of percentile concept from thirty current textbooks in educational statistics, a convention is needed for standardizing the definition and computation of percentile. Logically, the criteria of mutual-exclusivity and comprehensivity are pertinent to the examination of percentile definitions. Among the four percentile definitions, Definition 1 does not comply to the criterion of mutual-exclusivity because the point at which a percentile is calculated has been accounted for in both percents, p and (100-p), of the definition. On the other hand, the criterion of comprehensivity is not observed in Definition 2 because neither category, p or (100-p), has been specified in the definition to include the point at which a percentile is located. In Definitions 3 and 4, only the data in the category of percent p are specified, and the remaining data of the distribution, by default, are included in the (100)-p) category. Thus, Definitions 3 and 4 conform to the criteria of mutual-exclusivity and comprehensivity, and should be considered as possible definitions towards the establishment of convention for percentile.

Nonetheless, a convention of percentile definition does not necessarily guarantee a unique value of percentile at every point of discrete or continuous distributions. In case the result of a percentile estimate is an interval, any points in the interval would serve as the estimate of the percentile (David, 1981). Hence, the convention is desired to further specify which point of the interval is the estimate of the percentile.

A well-established convention should be based on implicit consent of the majority (Fowler & Fowler, 1984). However, to the author's best knowledge, no one has yet addressed the necessity of establishing a convention f the concept of percentile among the authors of these textbooks. In general, development of implicit consent needs a thorough discussion about percentile definitions and computations. Thus, this report is a modest effort to generate the indispensable discussion through a review of educational statistic textbooks.

References

Agresti, A. & Agresti, B. E (1979). Statistical Methods for the Social Sciences. San Francisco, CA: Dellen Publishing Company.

Anderson, D.R., Sweeney, D.J. & Williams, T.A. (1991). Introduction to Statistics, (2nd ed.). NY: West Publishing Company.

Casella, G. & Berger, R.L. (1990). Statistical Inference. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software.

Chao, L.L. (1969). Statistics: Methods and Analyses. NY: McGraw-Hill Book Company.

Cornell, F.G. (1956). The Essentials of Educational Statistics. NY: John Wiley & Sons.

David, H.A. (1981). Order Statistics, (2nd ed.). NY: John Wiley & Sons.

Dayton, C.M. & Stunkard, C.L. (1971). Statistics for Problem Solving. NY: McGraw-Hill Book Company.

Devore, J., & Peck, R. (1990). Introductory Statistics. NY: West Publishing Company.

Dowine, N.M. & Health, R.W. (1974). Basic Statistical Methods, (4th ed.). NY: Harper & Row Publishers.

Fischers, F.E. (1973). Fundamental Statistical Concepts. San Francisco, CA: Canfield Press.

Fowler, F.G. & Fowler, H.W. (1984). The Pocket Oxford Dictionary of Current English. Oxford: Clarendon Press.

Freund, J.E. (1967). Modern Elementary Statistics (3rd ed.). Englewood Cliffs, NJ: Prentice-Hall.

Friedman, H. (1972). Introduction to Statistics. NY: Random House.

Ferguson, G.A. (1966). Statistical Analysis in Psychology and Education. NY: McGraw-Hill Book Company.

Guilford, J.P. & Fruchter, B. (1978). Fundamental Statistics in Psychology and Education, (6th ed.). NY: McGraw-Hill Book Company.

Gravetter, F.J. & Wallnau, L.B. (1992). Statistics for the Behavioral Sciences, (3rd rd.). NY: West Publishing Company.

Harshbarger, T.R. (1977). Introductory Statistics: A Decision Map, (2nd ed.). NY: Macmillan Publishing Co.

Hays, W.L. (1967). Basic Statistics. Belmont, CA: Brooks/Cole Publishing Company.

Heiman, G.W. (1992). Basic Statistic for the Behavioral Sciences. Boston: Houghton MacMillan Company.

Henderson, N.K. (1964). Statistical Research Methods in Education and Psychology. Hong Kong: Hong Kong University Press.

Hill, J.E. & Kerber, A. (1967). Models, Methods and Analytical Procedures in Education Research. Detroit: Wayne State University Press.

Hopkins, K.D. & Glass, G.V. (1978). Basic Statistics for Behavioral Sciences. Englewood Cliffs, NJ: Prentice-Hall.

Jaeger, R.M. (1990). Statistics, A Spectator Sport. (2nd ed.). Newbury Park: Sage Publications.

Kendall, M.G. & Buckland, W.R. (1971). A Dictionary of Statistical Terms. Hafuer Publishing Company.

Lumsden, J. (1969). Elementary Statistical Method. Nedlands, Australia: University of Western Australia Press.

Moore, D.S. (1985). Statistics Concepts and Controversies. (2nd ed.). NY: W.H. Freeman and Company.

Moore, D.S. & McCable, G.P. (1989). Introduction to the Practice of Statistics. NY: W.H. Freeman and company.

Naiman, A., Rosenfeld, R., & Zirkel, G. (1972). Understanding Statistics, (2nd rd.). NY: McGraw-Hill Book Company.

Ott, R.L. (1993). An Introduction to Statistical Method and Data Analysis, (4th ed.). belmont. CA: Duxbury Press.

Runyon, R.P. & Haber, A. (1971). Fundamentals of Behavioral Statistics, (2nd ed.). Menlo Park, CA: Addison-Wesley Publishing Company.

Sincich, T. (1982). Statistics by Analysis. San Francisco, CA: Dellen Publishing Company.

Sprinthall, R.C. (1994). Statistical Analysis; (4th rd.). Boston: Allyn and Bacon.

Wolf, EL. (1962). Elements of Probability and Statistics. NY: McGraw-Hill Book Company.

Zehna, P.W. (1970). Probability Distributions and Statistics. Boston: Allyn and Bacon.

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