Concise Table of Three-Place Probabilities of the Symmetric Binomial Cumulative Distribution for Sample Sizes to 100

Abstract

This paper contains a table suitable for a number of statistical tasks. The symmetric binomial cumulative distribution, tabled here, applies to statistical sign tests (Dixon & Mood, 1946; Dixon & Massey, 1957) and to tests of the significance of the difference between correlated proportions (NcNemar, 1947; 1955). It also applies to confidence limits of the median and to significance tests of the index of order association. The latter two topics have been appropriately discussed in a recent textbook of statistics (Senders, 1955, pp. 426429, 455-457). The present table supplements a companion table of critical values at 12 probability levels for the same distribution for sample sizes to 1,000 ( MacKinnon, 1959). The three-place entries presented here permit the determination of twotail and central probabilities with error less than .0005. Consequently, they allow the determination of one-tail probabilities or major cumulative probabilities extending from one extreme to or beyond the midpoint of the distribution with error less than .00025. Thus any continuous area under the symmetric binomial histogram may be obtained by subtraction with error less than .0005, including, of course, the area corresponding to the probability of obtaining a particular number of occurrences of a selected class. The range of sample size to which the table applies is 100. A table of five-place entries divided into several sections has been published in Europe (Van Wijngaarden, 1950). This table, like the present one, uses for row captions or designations a variable other than the sample size; the substitution reduces the number of necessary rows, in the present case, from 100 to 43. The unbroken layout of the present table, moreover, expedites its use for several statistical purposes, especially for the uses involving a constantsample-size series of entries, later to be described. Further facilitation results from italicization in the entries and from a chart and instructions, both in the footnote to the table. These devices render more or less automatic the task of relating the entry to a desired probability, e.g., one-tail or central. They also permit specifying the probability exactly if it can be so expressed in three digits, or in four when the fourth digit is 5. In all other cases they allow locating the probability within a narrow range, the scope of the range having been detailed above. One may, of course, find the two-tail probability via the normal approxi