Rule of three (statistics)

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 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 = 0.05 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 \frac{-\ln(\alpha)}{n}, where 1-\alpha is the desired confidence level. ==Extension==

The Vysochanskij–Petunin inequality shows that the rule of three holds for unimodal distributions with finite variance beyond just the binomial distribution, and gives a way to change the factor 3 if a different confidence is desired. Chebyshev's inequality removes the assumption of unimodality at the price of a higher multiplier (about 4.5 for 95% confidence). Cantelli's inequality is the one-tailed version of Chebyshev's inequality. ==See also==