View source for Confidence interval

{{Short description|Range to estimate an unknown parameter}}
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[[File:Normal distribution 50% CI illustration.svg|thumb|upright=1.3|Each row of points is a sample from the same normal distribution. The colored lines are 50% confidence intervals for the mean, ''μ''. At the center of each interval is the sample mean, marked with a diamond. The blue intervals contain the population mean, and the red ones do not.]]
[[File:Confidence intervals visual.jpg|thumb|This probability distribution highlights some different confidence intervals. ]]
Informally, in frequentist statistics, a '''confidence interval''' ('''CI''') is an interval which is expected to typically contain the [[Parameter|parameter]] being estimated. More specifically, given a confidence level <math>\gamma</math> (95% and 99% are typical values), a CI is a ''random'' interval which contains the parameter being estimated <math>\gamma</math>% of the time.<ref>{{Cite book|last=Zar|first=Jerrold H.|title=Biostatistical Analysis|publisher=Prentice Hall|year=199|isbn=978-0130815422|edition=4th|location=Upper Saddle River, N.J.|pages=43–45|oclc=39498633}}</ref><ref name="Dekking">{{Cite journal|last1=Dekking|first1=Frederik Michel|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|date=2005|title=A Modern Introduction to Probability and Statistics|url=https://link.springer.com/book/10.1007/1-84628-168-7|journal=Springer Texts in Statistics|language=en-gb|doi=10.1007/1-84628-168-7|isbn=978-1-85233-896-1|issn=1431-875X}}</ref> The '''confidence level''', '''degree of confidence''' or '''confidence coefficient''' represents the long-run [[Empirical probability|proportion]] of CIs (at the given confidence level) that theoretically contain the true value of the parameter; this is tantamount to the [[Coverage probability|nominal coverage probability]]. For example, out of all intervals computed at the 95% level, 95% of them should contain the parameter's true value.<ref name=":3">{{Cite book|title=Introductory statistics|author=Illowsky, Barbara|others=Dean, Susan L., 1945-, Illowsky, Barbara., OpenStax College|isbn=978-1-947172-05-0|location=Houston, Texas|oclc=899241574}}</ref>

Factors affecting the width of the CI include the [[Sample size determination|sample size]], the [[Statistical dispersion|variability]] in the sample, and the confidence level.<ref>{{Cite journal|last=Hazra|first=Avijit|date=October 2017|title=Using the confidence interval confidently|journal=Journal of Thoracic Disease|volume=9|issue=10|pages=4125–4130|doi=10.21037/jtd.2017.09.14|issn=2072-1439|pmc=5723800|pmid=29268424 |doi-access=free }}</ref> All else being the same, a larger sample produces a narrower confidence interval, greater variability in the sample produces a wider confidence interval, and a higher confidence level produces a wider confidence interval.<ref>{{Cite book|last1=Khare|first1=Vikas|url=http://www.vlebooks.com/vleweb/product/openreader?id=none&isbn=9780128189054|title=Ocean Energy Modeling and Simulation with Big Data Computational Intelligence for System Optimization and Grid Integration|last2=Nema|first2=Savita|last3=Baredar|first3=Prashant|date=2020|publisher=Butterworth-Heinemann |isbn=978-0-12-818905-4|language=English|oclc=1153294021}}</ref>

== Definition ==
Let <math>X</math> be a random sample from a [[Probability distribution|probability distribution]] with [[Statistical parameter|statistical parameter]] <math>\theta</math>, which is a quantity to be estimated, and <math>\varphi</math>, representing quantities that are not of immediate interest. A confidence interval for the parameter <math>\theta</math>, with confidence level or coefficient <math>\gamma</math>, is an interval <math>(u(X), v(X))</math> determined by [[Random variable|random variable]]s <math>u(X)</math> and <math>v(X)</math> with the property:
: <math>P(u(X) < \theta < v(X)) = \gamma \quad \text{ for every } (\theta, \varphi).</math>

The number <math>\gamma</math>, whose typical value is close to but not greater than 1, is sometimes given in the form <math>1 - \alpha</math> (or as a percentage <math>100%\cdot(1 - \alpha)</math>), where <math>\alpha</math> is a small positive number, often 0.05.

It is important for the bounds <math>u(X)</math> and <math>v(X)</math> to be specified in such a way that as long as <math>X</math> is collected randomly, every time we compute a confidence interval, there is probability <math>\gamma</math> that it would contain <math>\theta</math>, the true value of the parameter being estimated. This should hold true for any actual <math>\theta</math> and <math>\varphi</math>.<ref name=Dekking/>

=== Approximate confidence intervals ===
In many applications, confidence intervals that have exactly the required confidence level are hard to construct, but approximate intervals can be computed. The rule for constructing the interval may be accepted as providing a confidence interval at level <math>\gamma</math> if

: <math>P(u(X) < \theta<v(X)) \approx\ \gamma \quad \text{ for every }(\theta,\varphi)</math>

to an acceptable level of approximation. Alternatively, some authors<ref>{{cite book |first=George G. |last=Roussas |year=1997 |title=A Course in Mathematical Statistics |edition=2nd |publisher=Academic Press |page=397}}</ref> simply require that

: <math>P(u(X) < \theta < v(X)) \ge\ \gamma \quad \text{ for every }(\theta,\varphi),</math>

which is useful if the probabilities are only partially identified or [[Imprecise probability|imprecise]], and also when dealing with [[Probability distribution#Discrete probability distribution|discrete distributions]]. Confidence limits of the form

: <math>P(u(X) < \theta) \ge\ \gamma</math> &nbsp;&nbsp;and&nbsp;&nbsp; <math>P(\theta < v(X)) \ge \gamma</math>

are called ''conservative'';<ref name=Cox-Hinkley-1974>{{cite book |author1=Cox, D.R. |author2=Hinkley, D.V. |year=1974 |title=Theoretical Statistics |publisher=Chapman & Hall}}</ref>{{rp|style=ama|page= 210}} accordingly, one speaks of conservative confidence intervals and, in general, regions.

=== Desired properties ===
When applying standard statistical procedures, there will often be standard ways of constructing confidence intervals. These will have been devised so as to meet certain desirable properties, which will hold given that the assumptions on which the procedure relies are true. These desirable properties may be described as: validity, optimality, and invariance.

Of the three, "validity" is most important, followed closely by "optimality". "Invariance" may be considered as a property of the method of derivation of a confidence interval, rather than of the rule for constructing the interval. In non-standard applications, these same desirable properties would be sought:

==== Validity ====
This means that the nominal [[Coverage probability|coverage probability]] (confidence level) of the confidence interval should hold, either exactly or to a good approximation.

==== Optimality ====
This means that the rule for constructing the confidence interval should make as much use of the information in the data-set as possible.

Recall that one could throw away half of a dataset and still be able to derive a valid confidence interval. One way of assessing optimality is by the length of the interval so that a rule for constructing a confidence interval is judged better than another if it leads to intervals whose lengths are typically shorter.

==== Invariance ====
In many applications, the quantity being estimated might not be tightly defined as such.

For example, a survey might result in an estimate of the median income in a population, but it might equally be considered as providing an estimate of the logarithm of the median income, given that this is a common scale for presenting graphical results. It would be desirable that the method used for constructing a confidence interval for the median income would give equivalent results when applied to constructing a confidence interval for the logarithm of the median income: Specifically the values at the ends of the latter interval would be the logarithms of the values at the ends of former interval.

=== Methods of derivation ===
For non-standard applications, there are several routes that might be taken to derive a rule for the construction of confidence intervals. Established rules for standard procedures might be justified or explained via several of these routes. Typically a rule for constructing confidence intervals is closely tied to a particular way of finding a [[Point estimation|point estimate]] of the quantity being considered.

==== Summary statistics ====
{{further|Summary statistics}}
This is closely related to the [[Method of moments (statistics)|method of moments]] for estimation. A simple example arises where the quantity to be estimated is the population mean, in which case a natural estimate is the sample mean. Similarly, the sample [[Variance|variance]] can be used to estimate the population variance. A confidence interval for the true mean can be constructed centered on the sample mean with a width which is a multiple of the square root of the sample variance.

==== Likelihood theory ====
Estimates can be constructed using the maximum likelihood principle, the likelihood theory for this provides two ways of constructing confidence intervals or confidence regions for the estimates.

==== Estimating equations ====
The estimation approach here can be considered as both a generalization of the [[Method of moments (statistics)|method of moments]] and a generalization of the maximum likelihood approach. There are corresponding generalizations of the results of maximum likelihood theory that allow confidence intervals to be constructed based on estimates derived from [[Estimating equations|estimating equations]].{{cn|date=December 2021}}

==== Hypothesis testing ====
If hypothesis tests are available for general values of a parameter, then confidence intervals/regions can be constructed by including in the {{nobr|100 {{mvar|p}} %}} confidence region all those points for which the hypothesis test of the [[Null hypothesis|null hypothesis]] that the true value is the given value is not rejected at a significance level of {{nobr|(1 − {{mvar|p}}).}}<ref name=Cox-Hinkley-1974/>{{rp|style=ama|at=§ 7.2 (iii)}}

==== Bootstrapping ====
{{further|Bootstrapping (statistics)}}
In situations where the distributional assumptions for the above methods are uncertain or violated, resampling methods allow construction of confidence intervals or prediction intervals. The observed data distribution and the internal correlations are used as the surrogate for the correlations in the wider population.

==== Central limit theorem ====
{{further|Central limit theorem}}
The central limit theorem is a refinement of the [[Law of large numbers|law of large numbers]]. For a large number of [[Independent and identically distributed random variables|independent identically distributed]] random variables <math>X_1, ..., X_n,</math> with finite variance, the average <math>\overline{X}_n</math> approximately has a normal distribution, no matter what the distribution of the <math>X_i</math> is, with the approximation roughly improving in proportion to <math>\sqrt{n}</math>.<ref name=Dekking/>

== Example ==
[[File:Confidenceinterval.svg|thumb|200px|In this [[Bar chart|bar chart]], the top ends of the brown bars indicate observed means and the red line segments ("[[Error bar|error bars]]") represent the confidence intervals around them. Although the error bars are shown as symmetric around the means, that is not always the case. In most graphs, the error bars do not represent confidence intervals (e.g., they often represent standard errors or standard deviations).]]

Suppose <math>{X_1, \ldots, X_n}</math> is an independent sample from a [[Normal distribution|normally distributed]] population with unknown parameters [[Mean|mean]] <math>\mu</math> and [[Variance|variance]] <math>\sigma^2.</math> Let

: <math>\bar{X} = \frac{{X_1 + \cdots + X_n}}{{n}},</math>

: <math>S^2 = \frac{{1}}{{n-1}}\sum_{i=1}^n (X_i - \bar{X})^2.</math>

Where <math>\bar{X}</math> is the [[Sample mean and covariance|sample mean]], and <math>S^2</math> is the [[Variance#Sample variance|sample variance]]. Then

: <math>T = \frac{{\bar{X} - \mu}}{{S/\sqrt{n}}}</math>

has a [[Student's t-distribution|Student's ''t'' distribution]] with <math>n - 1</math> degrees of freedom.<ref>Rees, D.G. (2001). ''Essential Statistics'', 4th Edition, Chapman and Hall/CRC. {{isbn|1-58488-007-4}} (Section 9.5)</ref> Note that the distribution of <math>T</math> does not depend on the values of the unobservable parameters <math>\mu</math> and <math>\sigma^2</math>; i.e., it is a [[Pivotal quantity|pivotal quantity]]. Suppose we wanted to calculate a 95% confidence interval for <math>\mu.</math> Then, denoting <math>c</math> as the 97.5th [[Percentile|percentile]] of this distribution,

: <math>P(-c \leq T \leq c) = 0.95.</math>

Note that "97.5th" and "0.95" are correct in the preceding expressions. There is a 2.5% chance that <math>T</math> will be less than <math>-c</math> and a 2.5% chance that it will be larger than <math>+c.</math> Thus, the probability that <math>T</math> will be between <math>-c</math> and <math>+c</math> is 95%.

Consequently,

: <math>P\left(\bar{X} - \frac{{cS}}{{\sqrt{n}}} \leq \mu \leq \bar{X} + \frac{{cS}}{{\sqrt{n}}}\right) = 0.95,</math>

and we have a theoretical (stochastic) 95% confidence interval for <math>\mu.</math>

After observing the sample we find values <math>\bar{x}</math> for <math>\bar{X}</math> and <math>s</math> for <math>S,</math> from which we compute the confidence interval

: <math>\left[\bar{x} - \frac{cs}{\sqrt{n}}, \bar{x} + \frac{cs}{\sqrt{n}}\right].</math>

== Interpretation ==
Various interpretations of a confidence interval can be given (taking the 95% confidence interval as an example in the following).
* The confidence interval can be expressed in terms of a [[Frequentist probability|long-run frequency]] in [[Replication (statistics)|repeated samples]] (or in [[Resampling (statistics)|resampling]]): "Were this procedure to be repeated on numerous samples, the proportion of calculated 95% confidence intervals that encompassed the true value of the population parameter would tend toward 95%."<ref name="CH" />
* The confidence interval can be expressed in terms of probability with respect to a single theoretical (yet to be realized) sample: "There is a 95% [[Coverage probability|probability]] that the 95% confidence interval calculated from a given future sample will cover the true value of the population parameter."<ref name="Neyman" /> This essentially reframes the "repeated samples" interpretation as a probability rather than a frequency. {{See also|Neyman construction}}
* The confidence interval can be expressed in terms of statistical significance, e.g.: "The 95% confidence interval represents values that are not statistically significantly different from the point estimate at the .05 level."<ref>Cox D.R., Hinkley D.V. (1974) Theoretical Statistics, Chapman & Hall, pp. 214, 225, 233</ref>
[[File:95% confidence interval.svg|thumb|Interpretation of the 95% confidence interval in terms of statistical significance.|none|369x369px]]

=== Common misunderstandings ===
[[File:Neyman_Construction_Confidence_Intervals.png|thumb|Plot of 50 confidence intervals from 50 samples generated from a normal distribution.]]
Confidence intervals and levels are frequently misunderstood, and published studies have shown that even professional scientists often misinterpret them.<ref>{{cite web|last=Kalinowski|first=Pawel|title=Identifying Misconceptions about Confidence Intervals|url=http://iase-web.org/documents/papers/icots8/ICOTS8_C104_KALINOWSKI.pdf|date=2010|access-date=2021-12-22}}</ref><ref>{{Cite web |url=http://irt.com.ne.kr/data/researchers%20misunderstand%20ci%20and%20error%20bars.pdf |title=Archived copy |access-date=2014-09-16 |archive-url=https://web.archive.org/web/20160304043241/http://irt.com.ne.kr/data/researchers%20misunderstand%20ci%20and%20error%20bars.pdf |archive-date=2016-03-04 |url-status=dead }}</ref><ref>Hoekstra, R., R. D. Morey, J. N. Rouder, and E-J. Wagenmakers, 2014. Robust misinterpretation of confidence intervals. Psychonomic Bulletin & Review Vol. 21, No. 5, pp. 1157-1164. [http://www.ejwagenmakers.com/inpress/HoekstraEtAlPBR.pdf]</ref><ref>[https://www.sciencenews.org/blog/context/scientists-grasp-confidence-intervals-doesnt-inspire-confidence Scientists' grasp of confidence intervals doesn't inspire confidence], Science News, July 3, 2014</ref><ref name=":2">{{Cite journal|last1=Greenland|first1=Sander| last2=Senn|first2=Stephen J.|last3=Rothman|first3=Kenneth J.|last4=Carlin|first4=John B.|last5=Poole|first5=Charles| last6=Goodman|first6=Steven N.|last7=Altman|first7=Douglas G.|date= April 2016 |title=Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations|journal=European Journal of Epidemiology |volume=31|issue=4|pages=337–350| doi=10.1007/s10654-016-0149-3|issn=0393-2990|pmc=4877414|pmid=27209009}}</ref><ref name="Helske Helske Cooper Ynnerman pp. 3397–3409">{{cite journal | last1=Helske | first1=Jouni | last2=Helske | first2=Satu | last3=Cooper | first3=Matthew | last4=Ynnerman | first4=Anders | last5=Besancon | first5=Lonni | title=Can Visualization Alleviate Dichotomous Thinking? Effects of Visual Representations on the Cliff Effect | journal=IEEE Transactions on Visualization and Computer Graphics | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=27 | issue=8 | date=2021-08-01 | issn=1077-2626 | doi=10.1109/tvcg.2021.3073466|arxiv=2002.07671 | pages=3397–3409| pmid=33856998 | s2cid=233230810 | url=http://urn.fi/URN:NBN:fi:jyu-202105213110 }}</ref>

* A 95% confidence level does not mean that for a given realized interval there is a 95% probability that the population parameter lies within the interval (i.e., a 95% probability that the interval covers the population parameter).<ref name="Morey" /> According to the  frequentist interpretation, once an interval is calculated, this interval either covers the parameter value or it does not; it is no longer a matter of probability. The 95% probability relates to the reliability of the estimation procedure, not to a specific calculated interval.<ref>{{cite web|url=http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm|title=1.3.5.2. Confidence Limits for the Mean|work=nist.gov|access-date=2014-09-16|archive-url=https://web.archive.org/web/20080205120031/http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm|archive-date=2008-02-05|url-status=dead}}</ref> [[Biography:Jerzy Neyman|Neyman]] himself (the original proponent of confidence intervals) made this point in his original paper:<ref name="Neyman" /><blockquote>It will be noticed that in the above description, the probability statements refer to the problems of estimation with which the statistician will be concerned in the future. In fact, I have repeatedly stated that the frequency of correct results will tend to ''α''. Consider now the case when a sample is already drawn, and the calculations have given [particular limits]. Can we say that in this particular case the probability of the true value [falling between these limits] is equal to ''α''? The answer is obviously in the negative. The parameter is an unknown constant, and no probability statement concerning its value may be made...</blockquote>

* A 95% confidence level does not mean that 95% of the sample data lie within the confidence interval.
* A 95% confidence level does not mean that there is a 95% probability of the parameter estimate from a repeat of the experiment falling within the confidence interval computed from a given experiment.<ref name=":2" />

== Counterexamples ==
Since confidence interval theory was proposed, a number of counter-examples to the theory have been developed to show how the interpretation of confidence intervals can be problematic, at least if one interprets them naïvely.

=== Confidence procedure for uniform location ===
Welch<ref>{{cite journal |last= Welch|first=B. L. |date= 1939|title= On Confidence Limits and Sufficiency, with Particular Reference to Parameters of Location|jstor= 2235987 |journal= The Annals of Mathematical Statistics|volume= 10|issue= 1|pages= 58–69| doi= 10.1214/aoms/1177732246|doi-access= free}}</ref> presented an example which clearly shows the difference between the theory of confidence intervals and other theories of interval estimation (including Fisher's [[Fiducial inference|fiducial]] intervals and objective [[Bayesian inference|Bayesian]] intervals). Robinson<ref>{{cite journal |last= Robinson|first=G. K. |date= 1975|title= Some Counterexamples to the Theory of Confidence Intervals|jstor= 2334498 |journal= Biometrika|volume= 62| issue= 1|pages= 155–161|doi=10.2307/2334498}}</ref> called this example "[p]ossibly the best known counterexample for Neyman's version of confidence interval theory." To Welch, it showed the superiority of confidence interval theory; to critics of the theory, it shows a deficiency. Here we present a simplified version.

Suppose that <math>X_1,X_2</math> are independent observations from a [[Continuous uniform distribution|uniform]] <math>(\theta - 1/2, \theta + 1/2)</math> distribution. Then the optimal 50% confidence procedure for <math>\theta</math> is<ref>{{cite journal |last= Pratt|first=J. W. |date= 1961| title= Book Review: Testing Statistical Hypotheses. by E. L. Lehmann|jstor= 2282344 |journal= Journal of the American Statistical Association|volume= 56|issue= 293|pages= 163–167|doi= 10.1080/01621459.1961.10482103}}</ref>

: <math>\bar{X} \pm \begin{cases}
\dfrac{|X_1-X_2|}{2} & \text{if } |X_1-X_2| < 1/2 \\[8pt]
\dfrac{1-|X_1-X_2|}{2} &\text{if } |X_1-X_2| \geq 1/2.
\end{cases}
</math>

A fiducial or objective Bayesian argument can be used to derive the interval estimate
: <math>\bar{X} \pm \frac{1-|X_1-X_2|}{4},</math>
which is also a 50% confidence procedure. Welch showed that the first confidence procedure dominates the second, according to desiderata from confidence interval theory; for every <math>\theta_1\neq\theta</math>, the probability that the first procedure contains <math>\theta_1</math> is ''less than or equal to'' the probability that the second procedure contains <math>\theta_1</math>. The average width of the intervals from the first procedure is less than that of the second. Hence, the first procedure is preferred under classical confidence interval theory.

However, when <math>|X_1-X_2| \geq 1/2</math>, intervals from the first procedure are ''guaranteed'' to contain the true value <math>\theta</math>: Therefore, the nominal 50% confidence coefficient is unrelated to the uncertainty we should have that a specific interval contains the true value. The second procedure does not have this property.

Moreover, when the first procedure generates a very short interval, this indicates that <math>X_1,X_2</math> are very close together and hence only offer the information in a single data point. Yet the first interval will exclude almost all reasonable values of the parameter due to its short width. The second procedure does not have this property.

The two counter-intuitive properties of the first procedure – 100% [[Coverage probability|coverage]] when <math>X_1,X_2</math> are far apart and almost 0% coverage when <math>X_1,X_2</math> are close together – balance out to yield 50% coverage on average. However, despite the first procedure being optimal, its intervals offer neither an assessment of the precision of the estimate nor an assessment of the uncertainty one should have that the interval contains the true value.

This counter-example is used to argue against naïve interpretations of confidence intervals. If a confidence procedure is asserted to have properties beyond that of the nominal coverage (such as relation to precision, or a relationship with Bayesian inference), those properties must be proved; they do not follow from the fact that a procedure is a confidence procedure.

=== Confidence procedure for ''ω''<sup>2</sup> ===
Steiger<ref>{{cite journal |last= Steiger|first=J. H. |date= 2004|title= Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis|journal= Psychological Methods|volume= 9|issue= 2|pages= 164–182|doi= 10.1037/1082-989x.9.2.164|pmid=15137887 }}</ref> suggested a number of confidence procedures for common [[Effect size#Omega-squared (ω2)|effect size]] measures in [[Analysis of variance|ANOVA]]. Morey et al.<ref name=Morey /> point out that several of these confidence procedures, including the one for ''ω''<sup>2</sup>, have the property that as the ''F'' statistic becomes increasingly small—indicating misfit with all possible values of ''ω''<sup>2</sup>—the confidence interval shrinks and can even contain only the single value ''ω''<sup>2</sup>&nbsp;=&nbsp;0; that is, the CI is infinitesimally narrow (this occurs when <math>p\geq1-\alpha/2</math> for a <math>100(1-\alpha)\%</math> CI).

This behavior is consistent with the relationship between the confidence procedure and [[Statistical hypothesis testing|significance testing]]: as ''F'' becomes so small that the group means are much closer together than we would expect by chance, a significance test might indicate rejection for most or all values of ''ω''<sup>2</sup>. Hence the interval will be very narrow or even empty (or, by a convention suggested by Steiger, containing only 0). However, this does ''not'' indicate that the estimate of ''ω''<sup>2</sup> is very precise. In a sense, it indicates the opposite: that the trustworthiness of the results themselves may be in doubt. This is contrary to the common interpretation of confidence intervals that they reveal the precision of the estimate.

== History ==
Methods for calculating confidence intervals for the binomial proportion appeared from the 1920s.<ref>Edwin B. Wilson (1927) Probable Inference, the Law of Succession, and Statistical Inference, Journal of the American Statistical Association, 22:158, 209-212, https://doi.org/10.1080/01621459.1927.10502953</ref><ref>C.J. Clopper, E.S. Pearson, The use of confidence or fiducial limits illustrated in the case of the binomial, Biometrika 26(4), 1934, pages 404–413, https://doi.org/10.1093/biomet/26.4.404</ref> The main ideas of confidence intervals in general were developed in the early 1930s,<ref name="Neyman1934">Neyman, J. (1934). On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection. Journal of the Royal Statistical Society, 97(4), 558–625. https://doi.org/10.2307/2342192 (see Note I in the appendix)</ref><ref>J. Neyman (1935), Ann. Math. Statist. 6(3): 111-116 (September, 1935). https://doi.org/10.1214/aoms/1177732585</ref><ref name="Neyman1970">Neyman, J. (1970). A glance at some of my personal experiences in the process of research. In Scientists at Work: Festschrift in honour of Herman Wold. Edited by T. Dalenius, G. Karlsson, S. Malmquist. Almqvist & Wiksell, Stockholm. https://worldcat.org/en/title/195948</ref> and the first thorough and general account was given by [[Biography:Jerzy Neyman|Jerzy Neyman]] in 1937.<ref name="Neyman" /> 

Neyman described the development of the ideas as follows (reference numbers have been changed):<ref name="Neyman1970" />

<blockquote>
[My work on confidence intervals] originated about 1930 from a simple question of Waclaw Pytkowski, then my student in Warsaw, engaged in an empirical study in farm economics. The question was: how to characterize non-dogmatically the precision of an estimated regression coefficient? ...

Pytkowski's monograph ... appeared in print in 1932.<ref>Pytkowski, W., The dependence of the income in small farms upon their area, the outlay and the capital invested in cows. (Polish, English summary) Bibliotaka Palawska, 1932.</ref> It so happened that, somewhat earlier, Fisher published his first paper<ref>Fisher, R. (1930). Inverse Probability. Mathematical Proceedings of the Cambridge Philosophical Society, 26(4), 528-535. https://doi.org/10.1017/S0305004100016297</ref> concerned with fiducial distributions and fiducial argument. Quite unexpectedly, while the conceptual framework of fiducial argument is entirely different from that of confidence intervals, the specific solutions of several particular problems coincided. Thus, in the first paper in which I presented the theory of confidence intervals, published in 1934,<ref name="Neyman1934" /> I recognized Fisher's priority for the idea that interval estimation is possible without any reference to Bayes' theorem and with the solution being independent from probabilities ''a priori''. At the same time I mildly suggested that Fisher's approach to the problem involved a minor misunderstanding.
</blockquote>

In medical journals, confidence intervals were promoted in the 1970s but only became widely used in the 1980s.<ref>{{Cite journal|last=Altman|first=Douglas G.|date=1991|title=Statistics in medical journals: Developments in the 1980s|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.4780101206|journal=Statistics in Medicine|language=en|volume=10|issue=12|pages=1897–1913|doi=10.1002/sim.4780101206|pmid=1805317 |issn=1097-0258}}</ref> By 1988, medical journals were requiring the reporting of confidence intervals.<ref>{{Cite journal|last1=Gardner|first1=Martin J.|last2=Altman|first2=Douglas G.|date=1988|title=Estimating with confidence|url=https://doi.org/10.1136/bmj.296.6631.1210|journal=British Medical Journal|language=en|volume=296|issue=6631|pages=1210–1211|doi=10.1136/bmj.296.6631.1210 |pmid=3133015 |pmc=2545695 }}</ref>

== See also ==
*{{Annotated link|CLs upper limits (particle physics)}}
*[[68–95–99.7 rule|68–95–99.7 rule]]
*Confidence band, an interval estimate for a curve
*{{Annotated link|Confidence distribution}}
*{{Annotated link|Confidence region}}, a higher dimensional generalization
*{{Annotated link|Credence (statistics)}}
*[[Credible interval]], a Bayesian alternative for interval estimation
*{{Annotated link|CDF-based nonparametric confidence interval|Cumulative distribution function-based nonparametric confidence interval}}
*{{Annotated link|Error bar}}
*{{Annotated link|Estimation statistics}}
*[[Margin of error]], the CI halfwidth
*{{Annotated link|p-value}}
*[[Prediction interval]], an interval estimate for a random variable
*[[Probable error]]
*{{Annotated link|Robust confidence intervals}}

=== Confidence interval for specific distributions ===
* [[Binomial proportion confidence interval|Confidence interval for binomial distribution]]
* [[Power law#Estimating the exponent from empirical data|Confidence interval for exponent of the power law distribution]]
* [[Exponential distribution#Confidence intervals|Confidence interval for mean of the exponential distribution]]
* [[Poisson distribution#Confidence interval|Confidence interval for mean of the Poisson distribution]]
* [[Normal distribution#Confidence intervals|Confidence intervals for mean and variance of the normal distribution]]

== References ==
{{Reflist|30em|refs=

<ref name=CH>Cox D.R., Hinkley D.V. (1974) ''Theoretical Statistics'', Chapman & Hall, p49, p209</ref>

<ref name=Morey>{{cite journal |last1= Morey|first1=R. D. |last2=Hoekstra|first2= R.|last3=Rouder|first3=J. N. | last4=Lee|first4=M. D.|last5=Wagenmakers|first5=E.-J.|date= 2016|title= The Fallacy of Placing Confidence in Confidence Intervals |journal= Psychonomic Bulletin & Review|volume= 23|issue= 1|pages= 103–123|doi= 10.3758/s13423-015-0947-8|pmid=26450628 |pmc=4742505 }}</ref>

<ref name=Neyman>{{cite journal | last1 = Neyman | first1 = J. |  year = 1937 | title = Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability | jstor = 91337 | journal = Philosophical Transactions of the Royal Society A | volume = 236 | issue = 767| pages = 333–380 | doi=10.1098/rsta.1937.0005| bibcode = 1937RSPTA.236..333N | doi-access = free }}</ref>



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<ref name=Mayo>Mayo, D. G. (1981) [http://www.phil.vt.edu/dmayo/PhilStatistics/In%20Defense%20of%20the%20Neyman-Pearson%20Theory%20of%20Confidence%20Intervals.pdf "In defence of the Neyman–Pearson theory of confidence intervals"], ''Philosophy of Science'', 48 (2), 269–280. {{JSTOR|187185}}</ref>
<ref name="Jiang He">Jiang He, et al., ''Passive Smoking and the Risk of Coronary Heart Disease — A Meta-Analysis of Epidemiologic Studies'', N Engl J Med 1999; 340:920–926.</ref>
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}}

== Bibliography ==
{{refbegin}}
{{divcol}}
* {{springer|title=Confidence estimation|id=p/c024620}}
* [[Biography:Ronald Fisher|Fisher, R.A.]] (1956) ''Statistical Methods and Scientific Inference.'' Oliver and Boyd, Edinburgh. (See p.&nbsp;32.)
* Freund, J.E. (1962) ''Mathematical Statistics'' Prentice Hall, Englewood Cliffs, NJ. (See pp.&nbsp;227–228.)
* [[Biography:Ian Hacking|Hacking, I.]] (1965) ''Logic of Statistical Inference.'' Cambridge University Press, Cambridge. {{isbn|0-521-05165-7}}
* Keeping, E.S. (1962) ''Introduction to Statistical Inference.'' D. Van Nostrand, Princeton, NJ.
* {{cite journal | last1 = Kiefer | first1 = J. |  year = 1977 | title = Conditional Confidence Statements and Confidence Estimators (with discussion) | jstor = 2286460 | journal = Journal of the American Statistical Association | volume = 72 | issue = 360a| pages = 789–827 | doi=10.1080/01621459.1977.10479956}}
* Mayo, D. G. (1981) [http://www.phil.vt.edu/dmayo/PhilStatistics/In%20Defense%20of%20the%20Neyman-Pearson%20Theory%20of%20Confidence%20Intervals.pdf "In defence of the Neyman–Pearson theory of confidence intervals"], ''Philosophy of Science'', 48 (2), 269–280. {{JSTOR|187185}}.
* Mehta, S. (2014) ''Statistics Topics''. {{isbn|978-1-4992-7353-3}}.
* {{cite journal|last1=Morey|first1=R. D.|last2=Hoekstra|first2=R.|last3=Rouder|first3=J. N.|last4=Lee|first4=M. D.|last5=Wagenmakers|first5=E.-J.|title=The fallacy of placing confidence in confidence intervals|journal=Psychonomic Bulletin & Review|volume=23|issue=1|year=2016|pages=103–123|doi=10.3758/s13423-015-0947-8|pmid=26450628|pmc=4742505}}
* [[Biography:Jerzy Neyman|Neyman, J.]] (1937) [https://www.jstor.org/stable/91337 "Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability"] ''Philosophical Transactions of the Royal Society of London A,'' '''236,''' 333–380. (Seminal work)
* {{cite journal | last1 = Robinson | first1 = G.K. | year = 1975 | title = Some Counterexamples to the Theory of Confidence Intervals | jstor = 2334498 | journal = Biometrika | volume = 62 | issue = 1| pages = 155–161 | doi=10.1093/biomet/62.1.155}}
* Savage, L. J. (1962), ''The Foundations of Statistical Inference''. Methuen, London.
* Smithson, M. (2003) ''Confidence intervals''. Quantitative Applications in the Social Sciences Series, No. 140. Belmont, CA: SAGE Publications. {{isbn|978-0-7619-2499-9}}.
{{refend}}

== External links ==
* [https://web.archive.org/web/20111122062933/http://www.latrobe.edu.au/psy/esci/ The Exploratory Software for Confidence Intervals tutorial programs that run under Excel]
* Confidence interval calculators for [https://web.archive.org/web/20090703130234/http://www.danielsoper.com/statcalc/calc28.aspx R-Squares], [https://web.archive.org/web/20090806085113/http://www.danielsoper.com/statcalc/calc26.aspx Regression Coefficients], and [https://web.archive.org/web/20090404113030/http://www.danielsoper.com/statcalc/calc27.aspx Regression Intercepts]
* {{MathWorld | urlname = ConfidenceInterval | title = Confidence Interval}}
* [http://www.causeweb.org/ CAUSEweb.org] Many resources for teaching statistics including Confidence Intervals.
* [http://www.usablestats.com/tutorials/CI An interactive introduction to Confidence Intervals]
* ''[http://demonstrations.wolfram.com/ConfidenceIntervalsConfidenceLevelSampleSizeAndMarginOfError/ Confidence Intervals: Confidence Level, Sample Size, and Margin of Error]'' by Eric Schulz, the [[Wolfram Demonstrations Project]].
* ''[http://health.utah.gov/opha/IBIShelp/ConfInts.pdf Confidence Intervals in Public Health] {{Webarchive|url=https://web.archive.org/web/20160809093746/http://health.utah.gov/opha/IBIShelp/ConfInts.pdf |date=2016-08-09 }}.'' Straightforward description with examples and what to do about small sample sizes or rates near 0.

{{Statistics}}


[[Category:Statistical intervals]]

{{Sourceattribution|Confidence interval}}