Editing Binomial distribution (section)

==== Wilson (score) method ====
{{Main|Binomial proportion confidence interval#Wilson score interval}}

The notation in the formula below differs from the previous formulas in two respects:<ref name="Wilson1927">{{Citation |last = Wilson |first=Edwin B. |date = June 1927 |title = Probable inference, the law of succession, and statistical inference |url = http://psych.stanford.edu/~jlm/pdfs/Wison27SingleProportion.pdf |journal = Journal of the American Statistical Association |volume=22 |issue=158 |pages=209–212 |access-date= 2015-01-05 |doi = 10.2307/2276774 |url-status=dead |archive-url = https://web.archive.org/web/20150113082307/http://psych.stanford.edu/~jlm/pdfs/Wison27SingleProportion.pdf |archive-date = 2015-01-13 |jstor = 2276774 }}</ref>
* Firstly, {{math|''z''<sub>''x''</sub>}} has a slightly different interpretation in the formula below: it has its ordinary meaning of 'the {{math|''x''}}th quantile of the standard normal distribution', rather than being a shorthand for 'the {{math|(1 − ''x'')}}th quantile'.
* Secondly, this formula does not use a plus-minus to define the two bounds. Instead, one may use <math>z = z_{\alpha / 2}</math> to get the lower bound, or use <math>z = z_{1 - \alpha/2}</math> to get the upper bound. For example: for a 95% confidence level the error <math>\alpha</math>&nbsp;=&nbsp;0.05, so one gets the lower bound by using <math>z = z_{\alpha/2} = z_{0.025} = - 1.96</math>, and one gets the upper bound by using <math>z = z_{1 - \alpha/2} = z_{0.975} = 1.96</math>.
: <math>\frac{
    \widehat{p\,} + \frac{z^2}{2n} + z
    \sqrt{
        \frac{\widehat{p\,}(1 - \widehat{p\,})}{n} +
        \frac{z^2}{4 n^2}
    }
}{
    1 + \frac{z^2}{n}
}</math><ref>
{{cite book
 | chapter = Confidence intervals
 | chapter-url = http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm
 | title = Engineering Statistics Handbook
 | publisher = NIST/Sematech
 | year = 2012
 | access-date = 2017-07-23
}}</ref>