Binomial Proportion Confidence Interval - Wilson Score Interval

Wilson Score Interval

The Wilson interval is an improvement (the actual coverage probability is closer to the nominal value) over the normal approximation interval and was first developed by Edwin Bidwell Wilson (1927).

\frac{{ {\hat p + \frac{{1}}{{2n}} z_{1- \alpha / 2}^2 \pm z_{1- \alpha / 2} \sqrt {\frac{{\hat p\left( {1 - \hat p} \right)}}{n} + \frac{{z_{1- \alpha / 2}^2}} {{4n^2}} }} }} {{ {1 + \frac{{1}}{n}} z_{1- \alpha / 2}^2 }}

This interval has good properties even for a small number of trials and/or an extreme probability. The center of the Wilson interval

\frac{{ {\hat p + \frac{{1}}{{2n}} z_{1- \alpha / 2}^2 } }} {{ {1 + \frac{{1}}{n}} z_{1- \alpha / 2}^2 }}

can be shown to be a weighted average of and, with receiving greater weight as the sample size increases. For the 95% interval, the Wilson interval is nearly identical to the normal approximation interval using instead of .

The Wilson interval can be derived from Pearson's chi-squared test with two categories. The resulting interval

can then be solved for to produce the Wilson interval.

The test in the middle of the inequality is a score test, so the Wilson interval is sometimes called the Wilson score interval.

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