Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-07T21:03:38.953Z Has data issue: false hasContentIssue false
  • English
  • Français

Bias and Runs in Dice Throwing and Recording: A Few Million Throws

Published online by Cambridge University Press:  01 January 2025

Gudmund R. Iversen ,
Willard H. Longcor ,
Frederick Mosteller ,
John P. Gilbert  and
Cleo Youtz
Get access Rights & Permissions [Opens in a new window]

Abstract

An experimenter threw individually 219 different dice of four different brands and recorded even and odd outcomes for one block of 20,000 trials for each die—4,380,000 throws in all. The resulting data on runs offer a basis for comparing the observed properties of such a physical randomizing process with theory and with simulations based on pseudo-random numbers and RAND Corporation random numbers. Although generally the results are close to those forecast by theory, some notable exceptions raise questions about the surprise value that should be associated with occurrences two standard deviations from the mean. These data suggest that the usual significance level may well actually be running from 7 to 15 percent instead of the theoretical 5 percent.

The data base is the largest of its kind. A set generated by one brand of dice contains 2,000,000 bits and is the first handmade empirical data of such size to fail to show a significant departure from ideal theory in either location or scale.

Keywords

Standard DeviationPublic PolicyRandom NumberData BaseEmpirical Data
Type
Original Paper
Information
Psychometrika , Volume 36 , Issue 1 , March 1971 , pp. 1 - 19
Copyright
Copyright © 1971 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

The analysis was facilitated by a National Science Foundation grant GS-341 and and its continuation GS-2044X. It forms part of a larger study of data analysis.

†

Mr. Longcor is from Waukegan, Illinois; the other authors are from Harvard University. Dr. Iversen has moved to the University of Michigan.

References

Anderson, T. W. An introduction to multivariate statistical analysis, 1958, New York: WileyGoogle Scholar
Bartlett, M. S. Tests of significance in factor analysis. British Journal of Psychology, Stat. Sec., 1950, 3, 77–85CrossRefGoogle Scholar
Feller, W. An introduction to probability theory and its applications, Vol. I, Third edition, New York: Wiley, 1968Google Scholar
Fry, T. C. Probability and its engineering uses, Second edition, (pp. 312–316). Princeton, New Jersey: Van Nostrand, 1965Google Scholar
Kerrich, J. E. An experimental introduction to the theory of probability, 1946, Copenhagen: Belgisk Import Co.Google Scholar
Kullback, S. Information theory and statistics. New York: Wiley. 1959, 302–302Google Scholar
Mood, A. M. The distribution theory of runs. Annals of Mathematical Statistics, 1940, 11, 367–392CrossRefGoogle Scholar
Mosteller, F., Youtz, C., & Zahn, D. The distribution of sums of rounded percentages. Demography, 1967, 4, 850–858CrossRefGoogle ScholarPubMed
Owen, D. B. Handbook of statistical tables, 1962, Reading, Mass.: Addison-WesleyGoogle Scholar
The RAND Corporation. A million random digits with 100,000 normal deviates, 1955, Glencoe, Illinois: Free PressGoogle Scholar