confidence for the estimate of gives this: where all of the values in parentheses are known quantities. The solution for estimates the upper and lower limits of the confidence interval for . Hence the probability of success is estimated by or the equivalent The practical observation from using this interval is that it has good properties even for a small number of trials and / or an extreme probability. Intuitively, the center value of this interval is the weighted average of and , with receiving greater weight as the sample size increases. Formally, the center value corresponds to using a pseudocount of , the number of standard deviations of the confidence interval: add this number to both the count of successes and of failures to yield the estimate of the ratio. For the common two standard deviations in each direction interval (approximately 95% coverage, which itself is approximately 1.96 standard deviations), this yields the estimate , which is known as the "plus four rule". Although the quadratic can be solved explicitly, in most cases Wilson's equations can also be solved numerically using the fixed-point iteration with . The Wilson interval can also be derived from the single sample z-test or Pearson's chi-squared test with two categories. The resulting interval, can then be solved for to produce the Wilson score interval. The test in the middle of the inequality is a score test. The interval equality principle Since the interval is derived by solving from the normal approximation to the binomial, the Wilson score interval has the property of being guaranteed to obtain the same result as the equivalent z-test or chi-squared test. This property can be visualised by plotting the probability density function for the Wilson score interval (see Wallis 2021: 297-313) and then plotting a normal pdf at each bound. The tail areas of the resulting Wilson and normal distributions, representing the chance of a significant result in that direction, must be equal. The continuity-corrected Wilson score interval and the Clopper-Pearson interval are also compliant with this property. The practical import is that these intervals may be employed as significance tests, with identical results to the source test, and new tests may be derived by geometry. Wilson score interval with continuity correction The Wilson interval may be modified by employing a continuity correction, in order to align the minimum coverage probability, rather than the average coverage probability, with the