Rule of three (statistics)
In
The rule is useful in the interpretation of
Derivation
A 95% confidence interval is sought for the probability p of an event occurring for any randomly selected single individual in a population, given that it has not been observed to occur in n Bernoulli trials. Denoting the number of events by X, we therefore wish to find the values of the parameter p of a binomial distribution that give Pr(X = 0) ≤ 0.05. The rule can then be derived[2] either from the Poisson approximation to the binomial distribution, or from the formula (1−p)n for the probability of zero events in the binomial distribution. In the latter case, the edge of the confidence interval is given by Pr(X = 0) = 0.05 and hence (1−p)n = .05 so n ln(1–p) = ln .05 ≈ −2.996. Rounding the latter to −3 and using the approximation, for p close to 0, that ln(1−p) ≈ −p (Taylor's formula), we obtain the interval's boundary 3/n.
By a similar argument, the numerator values of 3.51, 4.61, and 5.3 may be used for the 97%, 99%, and 99.5% confidence intervals, respectively, and in general the upper end of the confidence interval can be given as , where is the desired confidence level.
Extension
The
See also
Notes
- three standard deviations] as definitely significant" – and claimed it for his new journal of significance testing, Biometrika. Even Darwin late in life seems to have fallen into the confusion. (Ziliak and McCloskey, 2008, p. 26; parenthetic gloss in original)
- ^ "Professor Mean" (2010) "Confidence interval with zero events", The Children's Mercy Hospital. Retrieved 2013-01-01.
References
- Eypasch, Ernst; Rolf Lefering; C. K. Kum; Hans Troidl (1995). "Probability of adverse events that have not yet occurred: A statistical reminder". BMJ. 311 (7005): 619–620. PMID 7663258.
- Hanley, J. A.; A. Lippman-Hand (1983). "If nothing goes wrong, is everything alright?". JAMA. 249 (13): 1743–5. S2CID 44723518.
- Ziliak, S. T.; D. N. McCloskey (2008). The cult of statistical significance: How the standard error costs us jobs, justice, and lives. University of Michigan Press. ISBN 0472050079