nominal value, . Just as the Wilson interval mirrors Pearson's chi-squared test, the Wilson interval with continuity correction mirrors the equivalent Yates' chi-squared test. The following formulae for the lower and upper bounds of the Wilson score interval with continuity correction are derived from Newcombe (1998). However, if p = 0, must be taken as 0; if p = 1, is then 1. Wallis (2021) identifies a simpler method for computing continuity-corrected Wilson intervals that employs functions. For the lower bound, let , where is the selected error level for . Then . This method has the advantage of being further decomposable. Jeffreys interval The Jeffreys interval has a Bayesian derivation, but it has good frequentist properties. In particular, it has coverage properties that are similar to those of the Wilson interval, but it is one of the few intervals with the advantage of being equal-tailed (e.g., for a 95% confidence interval, the probabilities of the interval lying above or below the true value are both close to 2.5%). In contrast, the Wilson interval has a systematic bias such that it is centred too close to p = 0.5. The Jeffreys interval is the Bayesian credible interval obtained when using the non-informative Jeffreys prior for the binomial proportion . The Jeffreys prior for this problem is a Beta distribution with parameters , it is a conjugate prior. After observing successes in trials, the posterior distribution for is a Beta distribution with parameters . When and , the Jeffreys interval is taken to be the equal-tailed posterior probability interval, i.e., the and quantiles of a Beta distribution with parameters . These quantiles need to be computed numerically, although this is reasonably simple with modern statistical software. In order to avoid the coverage probability tending to zero when or , when the upper limit is calculated as before but the lower limit is set to 0, and when the lower limit is calculated as before but the upper limit is set to 1. Clopper–Pearson interval The Clopper–Pearson interval is an early and very common method for calculating binomial confidence intervals. This is often called an 'exact' method, as is attains the nominal coverage level in an exact sense, meaning that the coverage level never is less than the nominal . The Clopper–Pearson interval can be written as or equivalently, with where 0 ≤ x ≤ n is the number of successes observed