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In statistical `F33f`_`[estimation theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Estimation_theory]`_`f, the `!coverage probability`!, or `!coverage`! for short, is the `F33f`_`[probability`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Probability]`_`f that a `F33f`_`[confidence interval`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Confidence_interval]`_`f or `F33f`_`[confidence region`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Confidence_region]`_`f will include the `F33f`_`[true value`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Statistical_parameter]`_`f (parameter) of interest. It can be defined as the `F33f`_`[proportion of instances`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Empirical_probability]`_`f where the interval surrounds the true value as assessed by `F33f`_`[long-run frequency`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Frequentist_probability]`_`f.`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f]
In statistical prediction, the `!coverage probability`! is the `F33f`_`[probability`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Probability]`_`f that a `F33f`_`[prediction interval`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prediction_interval]`_`f will include an out-of-sample value of the `F33f`_`[random variable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Random_variable]`_`f. The `!coverage probability`! can be defined as the `F33f`_`[proportion of instances`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Empirical_probability]`_`f where the interval surrounds an out-of-sample value as assessed by `F33f`_`[long-run frequency`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Frequentist_probability]`_`f. `:cite-ref-severinietal2000-2-0[`F5bf`_`[2`#cite-note-severinietal2000-2]`_`f]
>>Contents
• `F0af`_`[Concept`#concept]`_`f
• `F0af`_`[Probability Matching`#probability-matching]`_`f
• `F0af`_`[Formula`#formula]`_`f
• `F0af`_`[See also`#see-also]`_`f
• `F0af`_`[Notes`#notes]`_`f
• `F0af`_`[References`#references]`_`f
-─
>>Concept
The fixed `F33f`_`[degree of certainty`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Certainty]`_`f pre-specified by the analyst, referred to as the `*confidence level`* or `*confidence coefficient`* of the constructed interval, is effectively the `!nominal coverage probability`! of the procedure for constructing confidence intervals. Hence, referring to a "nominal confidence level" or "nominal confidence coefficient" (e.g., as a synonym for `*nominal coverage probability`*) generally has to be considered `F33f`_`[tautological`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Tautology_(language)]`_`f and misleading, as the notion of `*confidence level`* itself inherently implies `F33f`_`[nominality`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_versus_nominal_value]`_`f already.`:cite-ref-3[`F5bf`_`[a`#cite-note-3]`_`f] The nominal coverage probability is often set at 0.95. By contrast, the (true) coverage probability is the `*actual`* probability that the interval contains the parameter.
If all assumptions used in deriving a confidence interval are met, the nominal coverage probability will equal the coverage probability (termed "true" or "actual" coverage probability for emphasis). If any assumptions are not met, the actual coverage probability could either be less than or greater than the nominal coverage probability. When the actual coverage probability is greater than the nominal coverage probability, the interval is termed a `!conservative (confidence) interval`!; if it is less than the nominal coverage probability, the interval is termed `!anti-conservative`!, or `!permissive`!. For example, suppose the interest is in the `F33f`_`[mean`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Expected_value]`_`f number of months that people with a particular type of `F33f`_`[cancer`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cancer]`_`f remain in `F33f`_`[remission`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Remission_(medicine)]`_`f following successful treatment with `F33f`_`[chemotherapy`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Chemotherapy]`_`f. The confidence interval aims to contain the unknown mean remission duration with a given probability. In this example, the coverage probability would be the real probability that the interval actually contains the true mean remission duration.
A discrepancy between the coverage probability and the nominal coverage probability frequently occurs when approximating a `F33f`_`[discrete distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Probability_distribution]`_`f with a `F33f`_`[continuous one`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Continuous_probability_distribution]`_`f. The construction of `F33f`_`[binomial confidence intervals`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Binomial_proportion_confidence_interval]`_`f is a classic example where coverage probabilities rarely equal nominal levels.`:cite-ref-4[`F5bf`_`[3`#cite-note-4]`_`f]`:cite-ref-5[`F5bf`_`[4`#cite-note-5]`_`f]`:cite-ref-6[`F5bf`_`[5`#cite-note-6]`_`f] For the binomial case, several techniques for constructing intervals have been created. The Wilson score interval is one well-known construction based on the `F33f`_`[normal distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Normal_distribution]`_`f. Other constructions include the Wald, exact, Agresti-Coull, and likelihood intervals. While the Wilson score interval may not be the most conservative estimate, it produces average coverage probabilities that are equal to nominal levels while still producing a comparatively narrow confidence interval.
The "probability" in `*coverage probability`* is interpreted with respect to a set of hypothetical repetitions of the entire data collection and analysis procedure. In these hypothetical repetitions, `F33f`_`[independent`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Independence_(probability_theory)]`_`f data sets following the same `F33f`_`[probability distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Probability_distribution]`_`f as the actual data are considered, and a confidence interval is computed from each of these data sets; see `F33f`_`[Neyman construction`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Neyman_construction]`_`f. The coverage probability is the fraction of these computed confidence intervals that include the desired but unobservable parameter value.
>>Probability Matching
In estimation, when the coverage probability is equal to the nominal coverage probability, that is known as probability matching. `:cite-ref-ghoshmukerjee1998-7-0[`F5bf`_`[6`#cite-note-ghoshmukerjee1998-7]`_`f]
In prediction, when the coverage probability is equal to the nominal coverage probability, that is known as predictive probability matching.`:cite-ref-severinietal2000-2-1[`F5bf`_`[2`#cite-note-severinietal2000-2]`_`f]
>>Formula
The construction of the confidence interval ensures that the probability of finding the true parameter ϑ ϑ {\\displaystyle \\vartheta } in the `F33f`_`[sample`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sampling_(statistics)]`_`f-dependent interval ( T u , T v ) {\\displaystyle (T_{u},T_{v})} is (at least) γ γ {\\displaystyle \\gamma } :
P ( T u ≤ ≤ ϑ ϑ ≤ ≤ T v ) ≥ ≥ γ γ ( for any allowed parameter ϑ ϑ ) {\\displaystyle P\\left(T_{u}\\leq \\vartheta \\leq T_{v}\\right)\\geq \\gamma \\quad ({\\text{for any allowed parameter }}\\vartheta )}
>>See also
• `F33f`_`[Binomial proportion confidence interval`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Binomial_proportion_confidence_interval]`_`f
• `F33f`_`[Confidence distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Confidence_distribution]`_`f
• `F33f`_`[False coverage rate`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=False_coverage_rate]`_`f
• `F33f`_`[Interval estimation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Interval_estimation]`_`f
>>Notes
`:cite-note-3`!a.`! `F0af`_`[↑`#cite-ref-3]`_`f However, some textbooks use the terms `*nominal confidence level`* or `*nominal confidence coefficient`*, and `*actual confidence level`* or `*actual confidence coefficient`* in the sense of "nominal" and "actual coverage probability"; cf., for instance, `:citerefwackerlymendenhallschaeffer2008`aWackerly, Dennis; Mendenhall, William; Schaeffer, Richard L. (2008), `*Mathematical Statistics with Applications`* (7th ed.), Cengage Learning, p. 437, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-1-111-79878-9.
>>References
`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f Dodge, Y. (2003). `*The Oxford Dictionary of Statistical Terms.`* OUP, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-19-920613-9, p. 93.
`:cite-note-severinietal2000-2`!2.`! `F0af`_`[↑`#cite-ref-severinietal2000-2-0]`_`f `:citerefseverinimukerjee-rghosh-m2002`aSeverini, T; Mukerjee, R; Ghosh, M (2002). "On an exact probability matching property of right-invariant priors". `*Biometrika`*. `!89`! (4): 952–957. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1093/biomet/89.4.952. `F33f`_`[JSTOR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=JSTOR_(identifier)]`_`f 4140551.
`:cite-note-4`!3.`! `F0af`_`[↑`#cite-ref-4]`_`f `:citerefagresticoull-brent1998`aAgresti, Alan; Coull, Brent (1998). "Approximate Is Better than "Exact" for Interval Estimation of Binomial Proportions". `*The American Statistician`*. `!52`! (2): 119–126. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:1998AmSta..52..119A. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.2307/2685469. `F33f`_`[JSTOR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=JSTOR_(identifier)]`_`f 2685469.
`:cite-note-5`!4.`! `F0af`_`[↑`#cite-ref-5]`_`f `:citerefbrowncai-t-tonydasgupta-anirban2001`aBrown, Lawrence; Cai, T. Tony; DasGupta, Anirban (2001). "Interval Estimation for a binomial proportion" (PDF). `*Statistical Science`*. `!16`! (2): 101–117. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1214/ss/1009213286. Archived (PDF) from the original on 23 June 2010. Retrieved 17 July 2009.
`:cite-note-6`!5.`! `F0af`_`[↑`#cite-ref-6]`_`f `:citerefnewcombe1998`aNewcombe, Robert (1998). "Two-sided confidence intervals for the single proportion: Comparison of seven methods". `*Statistics in Medicine`*. `!17`! (2, issue 8): 857–872. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1002/(SICI)1097-0258(19980430)17:8<857::AID-SIM777>3.0.CO;2-E. `F33f`_`[PMID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=PMID_(identifier)]`_`f 9595616. Archived from the original on 5 January 2013.
`:cite-note-ghoshmukerjee1998-7`!6.`! `F0af`_`[↑`#cite-ref-ghoshmukerjee1998-7-0]`_`f `:citerefghosh-mmukerjee-r1998`aGhosh, M; Mukerjee, R (1998). `*Recent developments on probability matching priors`*. New York Science Publishers. pp. 227–252.
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