binomial proportion confidence interval

File:Normal_approx_interval_and_logistic_example.png reveals problems of overshoot and zero-width intervals.]]

A commonly used formula for a binomial confidence interval relies on approximating the distribution of error about a binomially-distributed observation, $\backslash hat\; p$, with a normal distribution.

{{cite journal

|last = Wallis |first = Sean A.

|year = 2013

|title = Binomial confidence intervals and contingency tests: Mathematical fundamentals and the evaluation of alternative methods

|journal = Journal of Quantitative Linguistics

|volume = 20 |issue = 3 |pages = 178–208

|doi = 10.1080/09296174.2013.799918 |s2cid = 16741749

|url = http://www.ucl.ac.uk/english-usage/staff/sean/resources/binomialpoisson.pdf

}}

The normal approximation depends on the de Moivre–Laplace theorem (the original, binomial-only version of the central limit theorem) and becomes unreliable when it violates the theorems' premises, as the sample size becomes small or the success probability grows close to either {{math|0}} or {{math|1}} .

{{cite journal

| last1 = Brown | first1 = Lawrence D. | author1-link = Lawrence D. Brown

| last2 = Cai | first2 = T. Tony | author-link2 = T. Tony Cai

| last3 = DasGupta | first3 = Anirban

| year = 2001

| title = Interval estimation for a binomial proportion

| journal = Statistical Science

| volume = 16 | issue = 2 | pages = 101–133

| doi = 10.1214/ss/1009213286 | mr = 1861069

| zbl = 1059.62533 | citeseerx = 10.1.1.50.3025

}}

Using the normal approximation, the success probability $\backslash \; p\backslash$ is estimated by

: $\backslash \; p\; ~~\; \backslash approx\; ~~\; \backslash hat\; p\; \backslash pm\; \backslash frac\{\backslash ;\; z\_\backslash alpha\backslash \; \}\{\backslash \; \backslash sqrt\{n\backslash ;\; \}\backslash \; \}\backslash \; \backslash sqrt\{\; \backslash hat\; p\backslash \; \backslash left(1\; -\; \backslash hat\; p\; \backslash right)\backslash \; \}\backslash \; ,$

where $\backslash \; \backslash hat\; p\; \backslash equiv\; \backslash frac\{\backslash !\backslash \; n\_\backslash mathsf\{s\}\backslash !\backslash \; \}\{\; n\; \}\backslash$ is the proportion of successes in a Bernoulli trial process and an estimator for $\backslash \; p\backslash$ in the underlying Bernoulli distribution. The equivalent formula in terms of observation counts is

: $\backslash \; p\; ~~\; \backslash approx\; ~~\; \backslash frac\{\backslash !\backslash \; n\_\backslash mathsf\{s\}\backslash !\backslash \; \}\{\; n\; \}\; \backslash pm\; \backslash frac\{\backslash ;\; z\_\backslash alpha\backslash \; \}\{\backslash \; \backslash sqrt\{n\backslash ;\; \}\backslash \; \}\; \backslash sqrt\{\; \backslash frac\{\backslash !\backslash \; n\_\backslash mathsf\{s\}\backslash !\backslash \; \}\{\; n\; \}\backslash \; \backslash frac\{\backslash !\backslash \; n\_\backslash mathsf\{f\}\backslash !\backslash \; \}\{\; n\; \}\backslash \; \}\backslash \; ,$

where the data are the results of $\backslash \; n\backslash$ trials that yielded $\backslash \; n\_\backslash mathsf\{s\}\backslash$ successes and $\backslash \; n\_\backslash mathsf\{f\}\; =\; n\; -\; n\_\backslash mathsf\{s\}\backslash$ failures. The distribution function argument $\backslash \; z\_\backslash alpha\backslash$ is the $\backslash \; 1\; -\; \backslash tfrac\{\backslash !\backslash \; \backslash alpha\backslash !\backslash \; \}\{2\}\backslash$ quantile of a standard normal distribution (i.e., the probit) corresponding to the target error rate $\backslash \; \backslash alpha\; ~.$ For a 95% confidence level, the error $\backslash \; \backslash alpha\; =\; 1\; -\; 0.95\; =\; 0.05\backslash \; ,$ so $\backslash \; 1\; -\; \backslash tfrac\{\backslash !\backslash \; \backslash alpha\backslash !\backslash \; \}\{\; 2\; \}\; =\; 0.975\backslash$ and $\backslash \; z\_\{.05\}\; =\; 1.96\; ~.$

When using the Wald formula to estimate $\backslash \; p\backslash$, or just considering the possible outcomes of this calculation, two problems immediately become apparent:

• First, for $\backslash \; \backslash hat\; p\backslash$ approaching either {{math|1}} or {{math|0}}, the interval narrows to zero width (falsely implying certainty).
• Second, for values of (probability too low / too close to {{math|0}}), the interval boundaries exceed $\backslash \; [0\; ,\backslash \; 1]\backslash$ (overshoot).

(Another version of the second, overshoot problem, arises when instead $\backslash \; 1\; -\; \backslash hat\; p\backslash$ falls below the same upper bound: probability too high / too close to {{math|1}} .)

An important theoretical derivation of this confidence interval involves the inversion of a hypothesis test. Under this formulation, the confidence interval represents those values of the population parameter that would have large P-values if they were tested as a hypothesized population proportion.{{clarification needed|date=January 2024|reason=Hypothesis tests assume the population parameter value and derive $\backslash \; p\backslash$values, probabilities, for observed estimates under the assumption of that hypothesis. Confidence interval gives those values such that, assuming the center of the confidence interval as null hypothesis, estimates falling within the interval would be assigned $\backslash \; p\backslash$values too large to reject. Which is kind of what the above is saying but also not, hence why clarification is needed.}} The collection of values, $\backslash \; \backslash theta\backslash \; ,$ for which the normal approximation is valid can be represented as

: $\backslash left\backslash \{~~\; \backslash theta\; \backslash quad\; \backslash Bigg|\; \backslash quad\; y\_\{\backslash alpha/2\}\; ~~\; \backslash le\; ~~\; \backslash frac\{\backslash \; \backslash hat\; p\; -\; \backslash theta\backslash \; \}\{\backslash sqrt\{\backslash frac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; n\; \}\backslash \; \backslash hat\; p\backslash \; \backslash left(1\; -\; \backslash hat\; p\backslash right)\backslash \; \}\backslash \; \}\; ~~\; \backslash le\; ~~\; z\_\{\backslash alpha/2\}\; ~~\backslash right\backslash \}\backslash \; ,$

where $\backslash \; y\_\{\backslash alpha/2\}\backslash$ is the lower $\backslash \; \backslash tfrac\{\; \backslash alpha\; \}\{\backslash !\backslash \; 2\backslash !\backslash \; \}\backslash$ quantile of a standard normal distribution, vs. $\backslash \; z\_\{\backslash alpha/2\}\backslash \; ,$ which is the upper quantile.

Since the test in the middle of the inequality is a Wald test, the normal approximation interval is sometimes called the Wald interval or Wald method, after Abraham Wald, but it was first described by Laplace (1812).

{{cite book

|last = Laplace |first=P.S. |author-link = Pierre-Simon Laplace

|year = 1812

|title = Théorie analytique des probabilités |lang=fr

|trans-title = Analyitic Probability Theory

|publisher = Ve. Courcier

|page = 283

|url = https://archive.org/details/thorieanalytiqu00laplgoog

}}

Extending the normal approximation and Wald-Laplace interval concepts, Michael Short has shown that inequalities on the approximation error between the binomial distribution and the normal distribution can be used to accurately bracket the estimate of the confidence interval around $\backslash \; p\backslash \; :$

{{cite journal

|last = Short |first = Michael

|date = 2021-11-08

|title = On binomial quantile and proportion bounds: With applications in engineering and informatics

|journal = Communications in Statistics - Theory and Methods

|volume = 52 |issue = 12 |pages = 4183–4199

|doi = 10.1080/03610926.2021.1986540 |doi-access = free

|s2cid = 243974180 |issn = 0361-0926

}}

:$\backslash \; \backslash frac\{\backslash \; k\; +\; C\_\backslash mathsf\{L1\}\; -\; z\_\backslash alpha\backslash \; \backslash widehat\{W\}\backslash \; \}\{\backslash \; n\; +\; z\_\backslash alpha^2\backslash \; \}\; ~~\; \backslash le\; ~~\; p\; ~~\; \backslash le\; ~~\; \backslash frac\{\backslash \; k\; +\; C\_\backslash mathsf\{U1\}\; +\; z\_\backslash alpha\backslash \; \backslash widehat\{W\}\backslash \; \}\{\; n\; +\; z\_\backslash alpha^2\; \}\backslash$

with

:$\backslash \; \backslash widehat\{W\}\; \backslash equiv\; \backslash sqrt\{\; \backslash frac\{\backslash \; n\backslash \; k\; -\; k^2\; +\; C\_\backslash mathsf\{L2\}\; n\backslash \; -\; C\_\backslash mathsf\{L3\}\; k\; +\; C\_\backslash mathsf\{L4\}\backslash \; \}\{\; n\; \}\; ~\}\backslash \; ,$

and where $\backslash \; p\backslash$ is again the (unknown) proportion of successes in a Bernoulli trial process (as opposed to $\backslash \; \backslash hat\; p\; \backslash equiv\; \backslash frac\{\backslash \; n\_\backslash mathsf\{s\}\backslash \; \}\{\; n\; \}\; \backslash$ that estimates it) measured with $\backslash \; n\backslash$ trials yielding $\backslash \; k\backslash$ successes, $\backslash \; z\_\backslash alpha\backslash$ is the $\backslash \; 1\; -\; \backslash tfrac\{\backslash alpha\}\{2\}\backslash$ quantile of a standard normal distribution (i.e., the probit) corresponding to the target error rate $\backslash \; \backslash alpha\backslash \; ,$ and the constants $\backslash \; C\_\backslash mathsf\{L1\},\; C\_\backslash mathsf\{L2\},\; C\_\backslash mathsf\{L3\},\; C\_\backslash mathsf\{L4\},\; C\_\backslash mathsf\{U1\},\; C\_\backslash mathsf\{U2\},\; C\_\backslash mathsf\{U3\}\backslash$ and $\backslash \; C\_\backslash mathsf\{U4\}\backslash$ are simple algebraic functions of $\backslash \; z\_\backslash alpha\; ~.$ For a fixed $\backslash \; \backslash alpha\backslash$ (and hence $\backslash \; z\_\backslash alpha\backslash$), the above inequalities give easily computed one- or two-sided intervals which bracket the exact binomial upper and lower confidence limits corresponding to the error rate $\backslash \; \backslash alpha\; ~.$

Let there be a simple random sample $\backslash \; X\_1,\backslash \; \backslash ldots,\backslash \; X\_n\backslash$ where each $\backslash \; X\_i\backslash$ is i.i.d from a Bernoulli(p) distribution and weight $w\_i$ is the weight for each observation, with the(positive) weights $w\_i$ normalized so they sum to {{math|1}} . The weighted sample proportion is: $\backslash \; \backslash hat\; p\; =\; \backslash sum\_\{i=1\}^n\backslash \; w\_i\backslash \; X\_i\; ~.$ Since each of the $\backslash \; X\_i\backslash$ is independent from all the others, and each one has variance $\backslash \; \backslash operatorname\{var\}\backslash \{\backslash \; X\_i\backslash \; \backslash \}\backslash \; =\backslash \; p\backslash \; (1\; -\; p)\; ~~$ for every $~~\; i\backslash \; =\backslash \; 1\; ,\backslash \; \backslash ldots\backslash \; ,\; n\backslash \; ,$ the sampling variance of the proportion therefore is:{{cite web |title=How to calculate the standard error of a proportion using weighted data? |website = stats.stackexchange.com |id = 159220 / 253 |url=https://stats.stackexchange.com/a/159220/253 }}

:$\backslash \; \backslash operatorname\{var\}\backslash \{\backslash \; \backslash hat\; p\backslash \; \backslash \}\; =\; \backslash sum\_\{i=1\}^n\; \backslash operatorname\{var\}\backslash \{\backslash \; w\_i\backslash \; X\_i\backslash \; \backslash \}\; =\; p\backslash \; (\; 1\; -\; p\; )\backslash \; \backslash sum\_\{i=1\}^n\; w\_i^2\; ~.$

The standard error of $\backslash \; \backslash hat\; p\backslash$ is the square root of this quantity. Because we do not know $\backslash \; p\backslash \; (1-p)\backslash \; ,$ we have to estimate it. Although there are many possible estimators, a conventional one is to use $\backslash \; \backslash hat\; p\backslash \; ,$ the sample mean, and plug this into the formula. That gives:

:$\backslash \; \backslash operatorname\{SE\}\backslash \{\backslash \; \backslash hat\; p\backslash \; \backslash \}\; \backslash approx\; \backslash sqrt\{~\backslash hat\; p\backslash \; (1\; -\; \backslash hat\; p)\backslash \; \backslash sum\_\{i=1\}^n\; w\_i^2\; ~~\}\backslash$

For otherwise unweighted data, the effective weights are uniform $\backslash \; w\_i\; =\; \backslash tfrac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; n\; \}\backslash \; ,$ giving $\backslash \; \backslash sum\_\{i=1\}^n\; w\_i^2\; =\; \backslash tfrac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; n\; \}\; ~.$ The $\backslash \; \backslash operatorname\{SE\}\backslash$ becomes $\backslash sqrt\{\; \backslash tfrac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; n\; \}\backslash \; \backslash hat\; p\backslash \; (1-\backslash hat\; p)~\}\backslash \; ,$ leading to the familiar formulas, showing that the calculation for weighted data is a direct generalization of them.

File:Wilson_score_interval_and_logistic_example.png

The Wilson score interval was developed by E.B. Wilson (1927).

{{cite journal

| last = Wilson | first = E.B. | author-link = Edwin Bidwell Wilson

| year = 1927

| title = Probable inference, the law of succession, and statistical inference

| journal = Journal of the American Statistical Association

| volume = 22 | issue = 158 | pages = 209–212

| jstor = 2276774 | doi = 10.1080/01621459.1927.10502953

}}

It is an improvement over the normal approximation interval in multiple respects: Unlike the symmetric normal approximation interval (above), the Wilson score interval is asymmetric, and it doesn't suffer from problems of overshoot and zero-width intervals that afflict the normal interval. It can be safely employed with small samples and skewed observations. The observed coverage probability is consistently closer to the nominal value, $\backslash \; 1\; -\; \backslash alpha\; ~.$

Like the normal interval, the interval can be computed directly from a formula.

Wilson started with the normal approximation to the binomial:

:$\backslash \; z\_\backslash alpha\; \backslash approx\; \backslash frac\{~\; \backslash left(\backslash \; p\; -\; \backslash hat\{p\}\backslash \; \backslash right)\; ~\}\{\; \backslash sigma\_n\; \}\backslash$

where $\backslash \; z\_\backslash alpha\backslash$ is the standard normal interval half-width corresponding to the desired confidence $\backslash \; 1\; -\; \backslash alpha\; ~.$ The analytic formula for a binomial sample standard deviation is

$$\backslash \; \backslash sigma\_n\; =\; \backslash sqrt\{\backslash frac\{\backslash \; p\backslash \; \backslash left(\; 1\; -\; p\; \backslash right)\backslash \; \}\{n\}\; ~\}\; ~.$$

Combining the two, and squaring out the radical, gives an equation that is quadratic in $\backslash \; p\backslash \; :$

:$$

\left(\ p - \hat{p} \ \right)^2 = \frac{\; z_\alpha^2\ }{ n }\ p\ \left(\ 1 - p\ \right)\ \qquad

or $\backslash qquad$

p^2 - 2\ p\ \hat{p} + {\hat{p}}^2 = p\ \frac{\; z_\alpha^2\ }{ n } - p^2\ \frac{\; z_\alpha^2\ }{ n } ~.

Transforming the relation into a standard-form quadratic equation for $\backslash \; p\backslash \; ,$ treating $\backslash \; \backslash hat\; p\backslash$ and $\backslash \; n\backslash$ as known values from the sample (see prior section), and using the value of $\backslash \; z\_\backslash alpha\backslash$ that corresponds to the desired confidence $\backslash \; 1\; -\; \backslash alpha\backslash$ for the estimate of $\backslash \; p\backslash$ gives this:

\left(\ 1 + \frac{\; z_\alpha^2\ }{ n }\ \right)\ p^2 -

\left(\ 2\ {\hat p} + \frac{\; z_\alpha^2\ }{ n }\ \right)\ p +

\biggl(\ {\hat p}^2\ \biggr) = 0 ~,

where all of the values bracketed by parentheses are known quantities.

The solution for $\backslash \; p\backslash$ estimates the upper and lower limits of the confidence interval for $\backslash \; p\; ~.$ Hence the probability of success $\backslash \; p\backslash$ is estimated by $\backslash \; \backslash hat\; p\backslash$ and with $\backslash \; 1\; -\; \backslash alpha\backslash$ confidence bracketed in the interval

:$$

p \quad \underset{\approx}{\in}_\alpha \quad \bigl(\ w^- ,\ w^+\ \bigr) ~~ = ~~ \frac{ 1 }{~ 1 + z_\alpha^2\ / n ~} \Bigg(\ \hat p + \frac{\; z_\alpha^2\ }{ 2\ n }

~~ \pm ~~

\frac{\!\ z_\alpha\!\ }{\!\ 2\ n\!\ }\ \sqrt{\ 4\ n\ \hat p\ (1 - \hat p) + \ z_\alpha^2 ~} ~\Biggr)\

where $\backslash \; \backslash underset\{\backslash approx\}\{\backslash in\}\_\backslash alpha\backslash$ is an abbreviation for

:$\backslash \; \backslash operatorname\{\backslash mathbb\; P\}\; \backslash Bigr\backslash \{~~\; p\; \backslash in\; \backslash left(\backslash \; w^-\; ,\backslash \; w^+\backslash \; \backslash right)\; ~~\backslash Bigl\backslash \}\; =\; 1\; -\; \backslash alpha\; ~.$

An equivalent expression using the observation counts $\backslash \; n\_\backslash mathsf\{s\}\backslash$ and $\backslash \; n\_\backslash mathsf\{f\}\backslash$ is

:$$

p \quad \underset{\approx}{\in}_\alpha \quad \frac{\ n_\mathsf{s} + \tfrac{\!\ 1\!\ }{ 2 }\ z_\alpha^2\ }{ n + z_\alpha^2 }

~ \pm ~ \frac{ z_\alpha }{\ n + z_\alpha^2\ }

\sqrt{

\frac{\ n_\mathsf{s}\ n_\mathsf{f}\ }{ n } + \frac{\; z_\alpha^2\ }{ 4 } ~}\ ,

with the counts as above: $n\_\backslash mathsf\{s\}\; \backslash equiv$ the count of observed "successes", $\backslash \; n\_\backslash mathsf\{f\}\; \backslash equiv$ the count of observed "failures", and their sum is the total number of observations $\backslash \; n\; =\; n\_\backslash mathsf\{s\}\; +\; n\_\backslash mathsf\{f\}\; ~.$

In practical tests of the formula's results, users find that this interval has good properties even for a small number of trials and / or the extremes of the probability estimate, $\backslash \; \backslash hat\; p\; \backslash equiv\; \backslash frac\{\backslash !\backslash \; n\_\backslash mathsf\{s\}\backslash \; \}\{\; n\; \}~.$

Intuitively, the center value of this interval is the weighted average of $\backslash \; \backslash hat\{p\}\backslash$ and $\backslash \; \backslash tfrac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; 2\; \}\backslash \; ,$ with $\backslash \; \backslash hat\{p\}\backslash$ receiving greater weight as the sample size increases. Formally, the center value corresponds to using a pseudocount of $\backslash \; \backslash tfrac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; 2\; \}\; z\_\backslash alpha^2\backslash \; ,$ 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 $\backslash \; \backslash frac\{\backslash \; n\_\backslash mathsf\{s\}\; +\; 2\; \backslash \; \}\{\; n\; +\; 4\; \}\backslash \; ,$ 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

:$$

p_{\!\ k+1} = \hat{p} \pm z_\alpha\ \sqrt{ \tfrac{\!\ 1\!\ }{ n }\ p_k\ \left(\ 1 - p_k\ \right) ~}

with $\backslash \; p\_0\; =\; \backslash hat\{p\}\; ~.$

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,

:$$

\left\{ ~~ \theta \quad \Bigg| \quad \ y_\alpha ~~ \le ~~

\frac{\ \hat{p} - \theta\ }{\ \sqrt{ \tfrac{\!\ 1\!\ }{ n }\ \theta\ \left( 1 - \theta\right) ~}\ }

~~ \le ~~ z_\alpha ~~ \right\}\ ,

(with $\backslash \; y\_\backslash alpha\backslash$ the lower $\backslash \; \backslash alpha\backslash$ quantile)

can then be solved for $\backslash \; \backslash theta\backslash$ to produce the Wilson score interval. The test in the middle of the inequality is a score test.

File:Wilson score pdf and interval equality.png ({{sc|pdf}}) for the Wilson score interval, plus {{sc|pdf}}s at interval bounds. Tail areas are equal.]]

Since the interval is derived by solving from the normal approximation to the binomial, the Wilson score interval $~\; \backslash bigl(\backslash \; w^-\backslash \; ,\backslash \; w^+\backslash \; \backslash bigr)\; ~$ 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).

{{cite book

| last = Wallis | first = Sean A.

| year = 2021

| title = Statistics in Corpus Linguistics: A new approach

| place = New York, NY

| publisher = Routledge

| isbn = 9781138589384

| url = https://www.routledge.com/Statistics-in-Corpus-Linguistics-Research-A-New-Approach/Wallis/p/book/9781138589384

}}

{{rp|style=ama|pp= 297-313}}

After that, then also plotting a normal {{sc|pdf}} across each bound. The tail areas of the resulting Wilson and normal distributions represent 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.

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 nominal value, $\backslash \; 1\; -\; \backslash alpha\; ~.$

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 $\backslash \; \backslash left(\; w\_\backslash mathsf\{cc\}^-\; ,\; w\_\backslash mathsf\{cc\}^+\; \backslash right)\backslash$ are derived from Newcombe:

{{cite journal

| last = Newcombe | first = R.G.

| year = 1998

| title = Two-sided confidence intervals for the single proportion: Comparison of seven methods

| journal = Statistics in Medicine

| volume = 17 | issue = 8 | pages = 857–872

| pmid = 9595616

| doi = 10.1002/(SICI)1097-0258(19980430)17:8<857::AID-SIM777>3.0.CO;2-E

}}

:$$\backslash begin\{align\}$$

w_\mathsf{cc}^- &= \max \left\{ ~~ 0\ , ~~

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

\right\}\ ,\\

w_\mathsf{cc}^+ &= \min \left\{ ~~ 1\ , ~~

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

\right\} ,

\end{align}

for $\backslash \; \backslash hat\; p\; \backslash ne\; 0\backslash$ and $\backslash \; \backslash hat\; p\; \backslash ne\; 1\; ~.$

If $\backslash \; \backslash hat\; p\; =\; 0\backslash \; ,$ then $\backslash \; w\_\backslash mathsf\{cc\}^-\backslash$ must instead be set to $\backslash \; 0\backslash \; ;$ if $\backslash \; \backslash hat\; p\; =\; 1\backslash \; ,$ then $\backslash \; w\_\backslash mathsf\{cc\}^+\backslash$ must be instead set to $\backslash \; 1\; ~.$

Wallis (2021) identifies a simpler method for computing continuity-corrected Wilson intervals that employs a special function based on Wilson's lower-bound formula: In Wallis' notation, for the lower bound, let

:$~\; \backslash mathsf\{Wilson\_\{lower\}\}\backslash left(\backslash \; \backslash hat\{p\},\backslash \; n,\backslash \; \backslash tfrac\{\backslash !\backslash \; \backslash alpha\backslash !\backslash \; \}\{\; 2\; \}\backslash \; \backslash right)\backslash \; \backslash equiv\backslash \; w^-\backslash \; =\backslash \; \backslash frac\{\; 1\; \}\{~\; 1\; +\; z\_\backslash alpha^2\backslash \; /\; n\; ~\}\; \backslash Bigg(\backslash \; \backslash hat\; p\; +\; \backslash frac\{\backslash ;\; z\_\backslash alpha^2\backslash \; \}\{\backslash !\backslash \; 2\backslash \; n\backslash !\backslash \; \}$

~~ - ~~

\frac{\!\ z_\alpha\!\ }{\!\ 2\ n\!\ }\ \sqrt{\ 4\ n\ \hat p\ (1 - \hat p) + \ z_\alpha^2 ~} ~\Biggr)\ ,

where $\backslash \; \backslash alpha\backslash$ is the selected tolerable error level for $\backslash \; z\_\backslash alpha\; ~.$ Then

: $\backslash \; w\_\backslash mathsf\{cc\}^-\; =\; \backslash mathsf\{Wilson\_\{lower\}\}\backslash left(\backslash \; \backslash max\; \backslash left\backslash \{\backslash \; \backslash hat\{p\}\; -\; \backslash tfrac\{\; 1\; \}\{\backslash !\backslash \; 2\backslash \; n\backslash !\backslash \; \},\backslash \; 0\backslash \; \backslash right\backslash \},\backslash \; n,\backslash \; \backslash tfrac\{\backslash !\backslash \; \backslash alpha\backslash !\backslash \; \}\{\; 2\; \}\backslash \; \backslash right)\; ~.$

This method has the advantage of being further decomposable.

The Jeffreys interval has a Bayesian derivation, but good frequentist properties (outperforming most frequentist constructions). 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 $\backslash \; p\; =\; 0.5\; ~.$

{{cite journal

| last = Cai | first = T.T. | author-link = T. Tony Cai

| year = 2005

| title = One-sided confidence intervals in discrete distributions

| journal = Journal of Statistical Planning and Inference

| volume = 131 | issue = 1 | pages = 63–88

| doi=10.1016/j.jspi.2004.01.005

}}

The Jeffreys interval is the Bayesian credible interval obtained when using the non-informative Jeffreys prior for the binomial proportion $\backslash \; p\; ~.$ The Jeffreys prior for this problem is a Beta distribution with parameters $\backslash \; \backslash left(\; \backslash tfrac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; 2\; \},\; \backslash tfrac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; 2\; \}\; \backslash right)\backslash \; ,$ a conjugate prior. After observing $\backslash \; x\backslash$ successes in $\backslash \; n\backslash$ trials, the posterior distribution for $\backslash \; p\backslash$ is a Beta distribution with parameters $\backslash \; \backslash left(\; x\; +\; \backslash tfrac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; 2\; \},\; n\; -\; x\; +\; \backslash tfrac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; 2\; \}\; \backslash right)\; ~.$

When $\backslash \; x\; \backslash ne\; 0\backslash$ and $\backslash \; x\; \backslash ne\; n\backslash \; ,$ the Jeffreys interval is taken to be the $\backslash \; 100\backslash \; \backslash left(\; 1\; -\; \backslash alpha\; \backslash right)\backslash \; \backslash mathrm\{\%\}\backslash$ equal-tailed posterior probability interval, i.e., the $\backslash \; \backslash tfrac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; 2\; \}\backslash \; \backslash alpha\backslash$ and $\backslash \; 1\; -\; \backslash tfrac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; 2\; \}\backslash \; \backslash alpha\; \backslash$ quantiles of a Beta distribution with parameters $\backslash \; \backslash left(\backslash \; x\; +\; \backslash tfrac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; 2\; \},\backslash \; n\; -\; x\; +\; \backslash tfrac\{\backslash !\backslash \; 1\backslash !\backslash \; \}\{\; 2\; \}\backslash \; \backslash right)\; ~.$

In order to avoid the coverage probability tending to zero when $\backslash \; p\; \backslash to\; 0\backslash$ or {{math|1}} , when $\backslash \; x\; =\; 0\backslash$ the upper limit is calculated as before but the lower limit is set to {{math|0}} , and when $\backslash \; x\; =\; n\backslash$ the lower limit is calculated as before but the upper limit is set to {{math|1}} .

Jeffreys' interval can also be thought of as a frequentist interval based on inverting the p-value from the G-test after applying the Yates correction to avoid a potentially-infinite value for the test statistic.{{cn|date=March 2025}}

The Clopper–Pearson interval is an early and very common method for calculating binomial confidence intervals.

{{Cite journal

| last1 = Clopper | first1 = C.

| last2 = Pearson | first2 = E.S. | author-link2 = Egon Pearson

| year = 1934

| title = The use of confidence or fiducial limits illustrated in the case of the binomial

| journal = Biometrika

| volume = 26 | issue = 4 | pages = 404–413

| doi = 10.1093/biomet/26.4.404

}}

This is often called an 'exact' method, as it attains the nominal coverage level in an exact sense, meaning that the coverage level is never less than the nominal $\backslash \; 1\; -\; \backslash alpha\; ~.$

The Clopper–Pearson interval can be written as

:$\backslash \; S\_\{\backslash le\}\; \backslash cap\; S\_\{\backslash ge\}\backslash$

or equivalently,

:$\backslash \; \backslash left(\; \backslash inf\; S\_\{\backslash ge\}\backslash \; ,\backslash \; \backslash sup\; S\_\{\backslash le\}\; \backslash right)\backslash$

with

:$S\_\{\backslash le\}\; ~\; \backslash equiv\; ~\; \backslash left\backslash \{\; ~\; p\; ~~\; \backslash Big|\; ~~\; \backslash operatorname\{\backslash mathbb\; P\}\; \backslash left\backslash \{~\; \backslash operatorname\{Bin\}\backslash left(\; n;\; p\; \backslash right)\; \backslash le\; x\; ~\backslash right\backslash \}\; >\; \backslash tfrac\{\backslash !\backslash \; \backslash alpha\backslash !\backslash \; \}\{\; 2\; \}\; ~~\; \backslash right\backslash \}\; ~$

and

:$~\; S\_\{\backslash ge\}\; ~\; \backslash equiv\; ~\; \backslash left\backslash \{\; ~\; p\; ~~\; \backslash Big|\; ~~\; \backslash operatorname\{\backslash mathbb\; P\}\; \backslash left\backslash \{~\; \backslash operatorname\{Bin\}\backslash left(\; n;\; p\; \backslash right)\; \backslash ge\; x\; \backslash right\backslash \}\; >\; \backslash tfrac\{\backslash !\backslash \; \backslash alpha\backslash !\backslash \; \}\{\; 2\; \}\; ~~\; \backslash right\backslash \}\backslash \; ,$

where $\backslash \; 0\; \backslash le\; x\; \backslash le\; n\backslash$ is the number of successes observed in the sample and $\backslash \; \backslash mathsf\{Bin\}\backslash left(\; n;\; p\; \backslash right)\backslash$ is a binomial random variable with $\backslash \; n\backslash$ trials and probability of success $\backslash \; p\; ~.$

Equivalently we can say that the Clopper–Pearson interval is $\backslash \; \backslash left(\backslash \; \backslash frac\{\backslash !\backslash \; x\backslash !\backslash \; \}\{\; n\; \}\; -\; \backslash varepsilon\_1,\backslash \; \backslash frac\{\backslash !\backslash \; x\backslash !\backslash \; \}\{\; n\; \}\; +\; \backslash varepsilon\_2\backslash \; \backslash right)\backslash$ with confidence level $\backslash \; 1\; -\; \backslash alpha\backslash$ if $\backslash \; \backslash varepsilon\_i\backslash$ is the infimum of those such that the following tests of hypothesis succeed with significance $\backslash \; \backslash frac\{\backslash !\backslash \; \backslash alpha\backslash !\backslash \; \}\{2\}\backslash \; :$

1. H₀: $\backslash \; p\; =\; \backslash frac\{\backslash !\backslash \; x\backslash !\backslash \; \}\{\; n\; \}\; -\; \backslash varepsilon\_1\backslash$ with H_A: $\backslash \; p\; >\; \backslash frac\{\backslash !\backslash \; x\backslash !\backslash \; \}\{\; n\; \}\; -\; \backslash varepsilon\_1$
2. H₀: $\backslash \; p\; =\; \backslash frac\{\backslash !\backslash \; x\backslash !\backslash \; \}\{\; n\; \}\; +\; \backslash varepsilon\_2\backslash$ with H_A:

Because of a relationship between the binomial distribution and the beta distribution, the Clopper–Pearson interval is sometimes presented in an alternate format that uses quantiles from the beta distribution.

{{cite journal

|last = Thulin |first = Måns

|date = 2014-01-01

|title = The cost of using exact confidence intervals for a binomial proportion

|journal = Electronic Journal of Statistics

|volume = 8 |issue = 1 |pages = 817–840

|doi = 10.1214/14-EJS909 |issn = 1935-7524

|arxiv = 1303.1288 |s2cid = 88519382 |lang = en

}}

:

where $\backslash \; x\backslash$ is the number of successes, $\backslash \; n\backslash$ is the number of trials, and $\backslash \; B\backslash !\backslash left(\backslash \; p\backslash \; ;\backslash \; v\backslash \; ,\backslash \; w\backslash \; \backslash right)\backslash$ is the {{mvar|p}}th quantile from a beta distribution with shape parameters $\backslash \; v\backslash$ and $\backslash \; w\; ~.$

Thus, where:

:$$

\tfrac{\ \Gamma(n+1)\ }{\ \Gamma\!( x )\ \Gamma\!( n-x+1 )\ }\ \int_0^{ p_{\min}}\ t^{x-1}\ (1-t)^{n-x}\ \mathrm{d}\!\ t ~~ = ~~ \tfrac{\!\ \alpha\!\ }{ 2 }\ ,

:$$

\tfrac{\Gamma(n+1)}{\Gamma(x+1)\Gamma(n-x)}\ \int_0^{ p_{\max}}\ t^{x}\ (1-t)^{n-x-1}\ \mathrm{d}\!\ t ~ = ~ 1 - \tfrac{\!\ \alpha\!\ }{ 2 } ~.

The binomial proportion confidence interval is then $\backslash \; \backslash left(\backslash \; p\_\{\backslash min\}\backslash \; ,\backslash \; p\_\{\backslash max\}\; \backslash right)\backslash \; ,$ as follows from the relation between the Binomial distribution cumulative distribution function and the regularized incomplete beta function.

When $\backslash \; x\backslash$ is either {{math|0}} or $\backslash \; n\backslash \; ,$ closed-form expressions for the interval bounds are available: when $\backslash \; x\; =\; 0\backslash$ the interval is

:$\backslash \; \backslash left(\; 0\backslash \; ,\backslash \; 1\; -\; \backslash left(\backslash \; \backslash tfrac\{\backslash !\backslash \; \backslash alpha\backslash !\backslash \; \}\{\; 2\; \}\backslash \; \backslash right)^\{\; 1/n\; \}\; \backslash right)\backslash$

and when

$\backslash \; x\; =\; n\backslash$ it is

:$\backslash \; \backslash left(\backslash \; \backslash left(\; \backslash tfrac\{\backslash !\backslash \; \backslash alpha\backslash !\backslash \; \}\{\; 2\; \}\backslash \; \backslash right)^\{\; 1\; /\; n\; \}\backslash \; ,\backslash \; 1\backslash \; \backslash right)\; ~.$

The beta distribution is, in turn, related to the F-distribution so a third formulation of the Clopper–Pearson interval can be written using F quantiles:

:$$

\left(\ 1 + \frac{\ n - x + 1\ }{\ x \ F\!\left[\ \tfrac{\!\ \alpha\!\ }{ 2 }\ ;\ 2\ x\ ,\ 2\ (\ n - x + 1\ )\ \right]\ }\ \right)^{-1}

~~ < ~~ p ~~ < ~~

\left(\ 1 + \frac{\ n - x\ }{(x + 1)\ \ F\!\left[\ 1 - \tfrac{\!\ \alpha\!\ }{ 2 }\ ;\ 2\ (x + 1)\ ,\ 2\ (n - x\ )\ \right]\ }\ \right)^{-1}

where $\backslash \; x\backslash$ is the number of successes, $\backslash \; n\backslash$ is the number of trials, and $\backslash \; F\backslash !\backslash left(\backslash \; c\backslash \; ;\backslash \; d\_1\backslash \; ,\; d\_2\backslash \; \backslash right)\backslash$ is the $\backslash \; c\backslash$ quantile from an F-distribution with $\backslash \; d\_1\backslash$ and $\backslash \; d\_2\backslash$ degrees of freedom.

The Clopper–Pearson interval is an 'exact' interval, since it is based directly on the binomial distribution rather than any approximation to the binomial distribution. This interval never has less than the nominal coverage for any population proportion, but that means that it is usually conservative. For example, the true coverage rate of a 95% Clopper–Pearson interval may be well above 95%, depending on $\backslash \; n\backslash$ and $\backslash \; p\; ~.$ Thus the interval may be wider than it needs to be to achieve 95% confidence, and wider than other intervals. In contrast, it is worth noting that other confidence interval may have coverage levels that are lower than the nominal $\backslash \; 1\; -\; \backslash alpha\backslash \; ,$ i.e., the normal approximation (or "standard") interval, Wilson interval, Agresti–Coull interval,

{{cite journal

| last1 = Agresti | first1 = Alan | author-link1 = Alan Agresti

| last2 = Coull | first2 = Brent A.

| year = 1998

| title = Approximate is better than 'exact' for interval estimation of binomial proportions

| journal = The American Statistician

| volume = 52 | issue = 2 | pages = 119–126

| mr = 1628435 | jstor = 2685469

| doi=10.2307/2685469

}}

etc., with a nominal coverage of 95% may in fact cover less than 95%, even for large sample sizes.

The definition of the Clopper–Pearson interval can also be modified to obtain exact confidence intervals for different distributions. For instance, it can also be applied to the case where the samples are drawn without replacement from a population of a known size, instead of repeated draws of a binomial distribution. In this case, the underlying distribution would be the hypergeometric distribution.

The interval boundaries can be computed with numerical functions {{mono|qbeta}}

{{cite web

|title = The Beta distribution

|series = R Manual

|type = software doc

|website = stat.ethz.ch

|url=https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html

|access-date=2023-12-02

}}

in R and scipy.stats.beta.ppf

{{cite report

|section= scipy.stats.beta

|title= SciPy Manual |edition = 1.11.4

|website = docs.scipy.org

|type = software doc

|section-url = https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.beta.html

|access-date = 2023-12-02

}}

in Python.

from scipy.stats import beta

import numpy as np

k = 20

n = 400

alpha = 0.05

p_u, p_o = beta.ppf([alpha/2, 1 - alpha/2], [k, k + 1], [n - k + 1, n - k])

if np.isnan(p_o):

p_o=1

if np.isnan(p_u):

p_u=0

The Agresti–Coull interval is also another approximate binomial confidence interval.

Given $\backslash \; n\_\backslash mathsf\{s\}\backslash$ successes in $\backslash \; n\backslash$ trials, define

:$\backslash \; \backslash tilde\; n\; \backslash equiv\; n\; +\; z^2\_\backslash alpha\backslash$

and

:$\backslash \; \backslash tilde\; p\; =\; \backslash frac\{\; 1\; \}\{\backslash !\backslash \; \backslash tilde\; n\backslash !\backslash \; \}\backslash left(\backslash !\backslash \; n\_\backslash mathsf\{s\}\; +\; \backslash tfrac\{\backslash \; z^2\_\backslash alpha\; \}\{\; 2\; \}\backslash !\backslash \; \backslash right)\backslash$

Then, a confidence interval for $\backslash \; p\backslash$ is given by

:$$

\ p ~~ \approx ~~ \tilde p ~ \pm ~

z_\alpha\ \sqrt{ \frac{\!\ \tilde p\!\ }{ \tilde n }\ \left(\!\ 1 - \tilde p \!\ \right) ~}

where $\backslash \; z\_\backslash alpha\; =\; \backslash operatorname\{\backslash Phi^\{-1\}\}\backslash !\backslash !\backslash left(\backslash \; 1\; -\; \backslash tfrac\{\backslash !\backslash \; \backslash alpha\backslash !\backslash \; \}\{\; 2\; \}\backslash \; \backslash right)\backslash$ is the quantile of a standard normal distribution, as before (for example, a 95% confidence interval requires $\backslash \; \backslash alpha\; =\; 0.05\backslash \; ,$ thereby producing $\backslash \; z\_\{.05\}\; =\; 1.96\backslash$). According to Brown, Cai, & DasGupta (2001), taking $\backslash \; z\; =\; 2\backslash$ instead of 1.96 produces the "add 2 successes and 2 failures" interval previously described by Agresti & Coull.

This interval can be summarised as employing the centre-point adjustment, $\backslash \; \backslash tilde\; p\backslash \; ,$ of the Wilson score interval, and then applying the Normal approximation to this point.

:$\backslash \; \backslash tilde\; p\; =\; \backslash frac\{\backslash quad\; \backslash hat\; p\; +\; \backslash frac\{\backslash \; z^2\_\backslash alpha\backslash !\backslash \; \}\{\backslash \; 2\backslash \; n\backslash \; \}\; \backslash quad\}\{\backslash quad\; 1\; +\; \backslash frac\{\backslash \; z^2\_\backslash alpha\; \}\{\; n\; \}\; \backslash quad\}\backslash$

{{main|Arcsine transformation}}

{{further|Cohen's h}}

The arcsine transformation has the effect of pulling out the ends of the distribution.

{{cite web

|last=Holland|first=Steven

|title=Transformations of proportions and percentages

|url=http://strata.uga.edu/8370/rtips/proportions.html

|access-date=2020-09-08

|website=strata.uga.edu

}}

While it can stabilize the variance (and thus confidence intervals) of proportion data, its use has been criticized in several contexts.

{{cite journal

|last1 = Warton |first1 = David I.

|last2 = Hui |first2 = Francis K.C.

|date = January 2011

|title = The arcsine is asinine: The analysis of proportions in ecology

|journal = Ecology

|volume = 92 |issue = 1 |pages = 3–10

|doi = 10.1890/10-0340.1 |pmid = 21560670

|bibcode = 2011Ecol...92....3W |issn = 0012-9658

|hdl = 1885/152287 |hdl-access = free

|url = http://doi.wiley.com/10.1890/10-0340.1

|lang = en

}}

Let $\backslash \; X\backslash$ be the number of successes in $\backslash \; n\backslash$ trials and let $\backslash \; p\; =\; \backslash tfrac\{\; 1\; \}\{\backslash !\backslash \; n\backslash !\backslash \; \}\backslash !\backslash \; X\; ~.$ The variance of $\backslash \; p\backslash$ is

:$\backslash operatorname\{var\}\backslash \{~~\; p\; ~~\backslash \}\; =\; \backslash tfrac\{\; 1\; \}\{\backslash !\backslash \; n\backslash !\backslash \; \}\backslash \; p\backslash \; (1\; -\; p)\; ~.$

Using the arc sine transform, the variance of the arcsine of $\backslash \; \backslash sqrt\{\backslash \; p\; ~\}\backslash$ is

{{cite book

|last = Shao |first = J.

|year = 1998

|title = Mathematical Statistics

|publisher = Springer

|place = New York, NY

}}

: $\backslash \; \backslash operatorname\{var\}\; \backslash left\backslash \{~~\; \backslash arcsin\; \backslash sqrt\{\; p\; ~\}\; ~~\backslash right\backslash \}\; ~\; \backslash approx\; ~\; \backslash frac\{\backslash \; \backslash operatorname\{var\}\backslash \{~~\; p\; ~~\backslash \}\backslash \; \}\{\backslash \; 4\backslash \; p\backslash \; (1\; -\; p)\backslash \; \}\; =\; \backslash frac\{\backslash \; p\backslash \; (1\; -\; p)\backslash \; \}\{\backslash \; 4\backslash \; n\backslash \; p\backslash \; (1\; -\; p)\backslash \; \}\; =\; \backslash frac\{\backslash \; 1\backslash \; \}\{\backslash \; 4\backslash \; n\backslash \; \}\; ~.$

So, the confidence interval itself has the form

:

where $\backslash \; z\_\backslash alpha\backslash$ is the $\backslash \; 1\; \backslash \; -\backslash \; \backslash tfrac\{\backslash !\backslash \; \backslash alpha\backslash !\backslash \; \}\{2\}\backslash$ quantile of a standard normal distribution.

This method may be used to estimate the variance of $\backslash \; p\backslash$ but its use is problematic when $\backslash \; p\backslash$ is close to {{math|0}} or {{math|1}} .

{{unreferenced section|date=July 2017}}

Let $\backslash \; p\backslash$ be the proportion of successes. For $\backslash \; 0\; \backslash le\; a\; \backslash le\; 2\backslash \; ,$

: $\backslash \; t\_a\backslash \; =\backslash \; \backslash log\backslash left(\backslash \; \backslash frac\{\backslash \; p^a\backslash \; \}\{\backslash \; (1\; -\; p)^\{2-a\}\backslash \; \}\backslash \; \backslash right)\backslash \; =\backslash \; a\backslash \; \backslash log\; p\; -\; (2-a)\backslash \; \backslash log(\backslash \; 1\; -\; p\backslash \; )\backslash$

This family is a generalisation of the logit transform which is a special case with a = 1 and can be used to transform a proportional data distribution to an approximately normal distribution. The parameter a has to be estimated for the data set.

The rule of three is used to provide a simple way of stating an approximate 95% confidence interval for $\backslash \; p\backslash \; ,$ in the special case that no successes ($\backslash \; \backslash hat\; p\; =\; 0\backslash$) have been observed.

{{cite web

|first = Steve |last = Simon

|year = 2010

|title = Confidence interval with zero events

|series= Ask Professor Mean

|place = Kansas City, MO

|publisher = The Children's Mercy Hospital

|url = http://www.pmean.com/01/zeroevents.html

|archive-url = https://web.archive.org/web/20111015182854/http://www.childrensmercy.org/stats/

|archive-date=15 October 2011

}} [http://www.childrensmercy.org/stats/ Stats topics on Medical Research]

The interval is $\backslash \; \backslash left(\backslash \; 0,\backslash \; \backslash tfrac\{\backslash !\backslash \; 3\backslash !\backslash \; \}\{\; n\; \}\; \backslash right)\; ~.$

By symmetry, in the case of only successes ($\backslash \; \backslash hat\; p\; =\; 1\backslash$), the interval is $\backslash \; \backslash left(\backslash \; 1\; -\; \backslash tfrac\{\backslash !\backslash \; 3\backslash !\backslash \; \}\{\; n\; \},\backslash \; 1\backslash \; \backslash right)\; ~.$

There are several research papers that compare these and other confidence intervals for the binomial proportion.

{{cite conference

|last1 = Sauro |first1 = J.

|last2 = Lewis |first2 = J.R.

|year = 2005

|title = Comparison of Wald, Adj-Wald, exact, and Wilson intervals calculator

|conference = Human Factors and Ergonomics Society, 49th Annual Meeting (HFES 2005)

|place = Orlando, FL

|pages = 2100–2104

|url = http://www.measuringusability.com/papers/sauro-lewisHFES.pdf

|archive-url = https://web.archive.org/web/20120618053914/http://www.measuringusability.com/papers/sauro-lewisHFES.pdf

|archive-date = 2012-06-18 |df = dmy-all

}}

{{cite journal

| last = Reiczigel | first = J.

| year = 2003

| title = Confidence intervals for the binomial parameter: Some new considerations

| url = http://www.zoologia.hu/qp/Reiczigel_conf_int.pdf

| journal = Statistics in Medicine

| volume = 22 | issue = 4 | pages = 611–621

| doi=10.1002/sim.1320

| pmid = 12590417 | s2cid = 7715293

}}

Both Ross (2003)

{{Cite journal

| last = Ross | first = T.D.

| year = 2003

| title = Accurate confidence intervals for binomial proportion and Poisson rate estimation

| journal = Computers in Biology and Medicine

| volume = 33 | issue = 6 | pages = 509–531

| doi = 10.1016/S0010-4825(03)00019-2 | pmid = 12878234

| url = https://zenodo.org/record/1259565

}}

and Agresti & Coull (1998)

point out that exact methods such as the Clopper–Pearson interval may not work as well as some approximations. The normal approximation interval and its presentation in textbooks has been heavily criticised, with many statisticians advocating that it not be used.

The principal problems are overshoot (bounds exceed $\backslash \; \backslash left[\backslash \; 0,\backslash \; 1\backslash \; \backslash right]\backslash$), zero-width intervals at $\backslash \; \backslash hat\; p\backslash \; =\; 0\backslash$ or {{math|1}} (falsely implying certainty), and overall inconsistency with significance testing.

Of the approximations listed above, Wilson score interval methods (with or without continuity correction) have been shown to be the most accurate and the most robust, though some prefer Agresti & Coulls' approach for larger sample sizes. Wilson and Clopper–Pearson methods obtain consistent results with source significance tests, and this property is decisive for many researchers.

Many of these intervals can be calculated in R using packages like {{mono|binom}}.

{{cite report

|last = Dorai-Raj |first = Sundar

|date = 2022-05-02

|title = binom: Binomial confidence intervals for several parameterizations

|type = software doc.

|url = https://cran.r-project.org/web/packages/binom/index.html

|access-date = 2023-12-02 |df=dmy-all

}}

• binomial distribution#Confidence intervals
• estimation theory
• pseudocount
• CDF-based_nonparametric_confidence_interval#Pointwise_band
• Z-test#Comparing the Proportions of Two Binomials

{{reflist|25em}}

{{DEFAULTSORT:Binomial Proportion Confidence Interval}}

Category:Statistical approximations

Category:Statistical intervals