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{{StatsPsy}} 

{{StatsPsy}} 

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In [[statistics]], a '''confidence interval (CI) or confidence bound''' is an [[interval estimationinterval estimate]] of a [[population parameter]]. Instead of estimating the parameter by a single value, an interval likely to include the parameter is given. Thus, confidence intervals are used to indicate the reliability of an estimate. How likely the interval is to contain the parameter is determined by the '''confidence level''' or confidence coefficient. Increasing the desired confidence level will widen the confidence interval. 




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[[Image:Confidenceinterval.pngthumb200pxIn this diagram, the horizontal lines forming the tops of the bars represent observation [[mean]]s and the red lines represent the confidence intervals surrounding them. The difference between the two populations on the left is [[Statistical significancesignificant]]. However, "[i]t is a common statistical misconception to suppose that two quantities whose 95% confidence intervals just fail to overlap are significantly different at the 5% level"<ref name="gh95">Goldstein, H., & Healey, M.J.R. (1995). [http://www.jstor.org/view/09641998/di993086/99p0458v/0 "The graphical presentation of a collection of means."] ''Journal of the Royal Statistical Society'', '''158''', 17577. </ref>.]] 
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For example, a CI can be used to describe how reliable survey results are. In a poll of election votingintentions, the result might be that 40% of respondents intend to vote for a certain party. A 95% confidence interval for the proportion in the whole population having the same intention on the survey date might be 36% to 44%. All other things being equal, a survey result with a small CI is more reliable than a result with a large CI and one of the main things controlling this width in the case of population surveys is the size of the sample questioned. Confidence intervals and [[interval estimationinterval estimate]]s more generally have applications across the whole range of quantitative studies. 




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In [[statistics]], a '''confidence interval (CI)''' is an [[interval estimationinterval estimate]] of a [[population parameter]]. Instead of estimating the parameter by a single value, an interval likely to include the parameter is given. How likely the interval is to contain the parameter is determined by the '''confidence level''' or confidence coefficient. Increasing the desired confidence level will widen the confidence interval. 
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If a statistic is presented with a confidence interval, and is claimed to be statistically significant, the underlying test leading to that claim will have been performed at a [[significance levelsignificance level]] of 100% minus the confidence level of the interval. If that test has produced a [[type 1 errortype I error]], the statistic and its confidence interval will bear no relationship to the underlying parameter. 




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Confidence intervals are used to indicate the reliability of an estimate. For example, a CI can be used to describe how reliable survey results are. All other things being equal, a survey result with a small CI is more reliable than a result with a large CI. 
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== Brief explanation == 




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More specifically, for a given associated proportion ''p'' (or [[confidence level]]), a CI for a [[population parameter]] is an [[Interval (mathematics)interval]] that is calculated from a random sample of an underlying population such that, if the sampling was repeated numerous times and the confidence interval recalculated from each sample according to the same method, a proportion ''p'' of the confidence intervals would contain the [[population parameter]] in question. In unusual cases, a confidence set may consist of a collection several separate intervals, which may include semiinfinite intervals, and it is possible that an outcome of a confidenceinterval calculation could be the set of all values from minus infinity to plus infinity. 
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[[Image:Confidenceinterval.pngthumb200pxIn this [[bar chart]], the top ends of the bars indicate observation [[mean]]s and the red [[line segment]]s represent the confidence intervals surrounding them. The difference between the two populations on the left is [[Statistical significancesignificant]]. However, it is a common misconception to suppose that two parameters whose 95% confidence intervals fail to overlap are significantly different at the 5% level.<ref name="gh95">Goldstein, H., & Healey, M.J.R. (1995). [http://www.jstor.org/stable/view/2983411 "The graphical presentation of a collection of means."] ''Journal of the Royal Statistical Society'', '''158''', 17577.</ref><ref>{{cite journal author=Wolfe R, Hanley J title=If we're so different, why do we keep overlapping? When 1 plus 1 doesn't make 2 journal=CMAJ volume=166 issue=1 pages=65–6 year=2002 month=Jan pmid=11800251 pmc=99228 doi= url=http://www.cmaj.ca/cgi/pmidlookup?view=long&pmid=11800251}}</ref>]] 




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Confidence intervals are the most prevalent form of [[interval estimation]]. Interval estimates may be contrasted with [[point estimationpoint estimates]] and have the advantage over these as summaries of a dataset in that more information is conveyed – not just a "best estimate" of a parameter but an indication of the accuracy with which the parameter is known. 
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For a given proportion ''p'' (where ''p'' is the confidence level), a confidence interval for a [[population parameter]] is an [[Interval (mathematics)interval]] that is calculated from a random sample of an underlying population such that, if the sampling was repeated numerous times and the confidence interval recalculated from each sample according to the same method, a proportion ''p'' of the confidence intervals would contain the [[population parameter]] in question. In unusual cases, a confidence set may consist of a collection of several separate intervals, which may include semiinfinite intervals, and it is possible that an outcome of a confidenceinterval calculation could be the set of all values from minus infinity to plus infinity. 

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Confidence intervals are the most prevalent form of [[interval estimation]]. Interval estimates may be contrasted with [[point estimationpoint estimates]] and have the advantage over these as summaries of a dataset in that they convey more information – not just a "best estimate" of a parameter but an indication of the precision with which the parameter is known. 





Confidence intervals play a similar role in [[frequentist statistics]] to the [[credibility interval]] in [[Bayesian statistics]]. However, confidence intervals and credibility intervals are not only mathematically different; they have radically different interpretations. 

Confidence intervals play a similar role in [[frequentist statistics]] to the [[credibility interval]] in [[Bayesian statistics]]. However, confidence intervals and credibility intervals are not only mathematically different; they have radically different interpretations. 




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The concept of a confidence interval for a single quantity can be generalised to be able to deal with several quantities simultaneously, in which case they are called [[confidence region]]s. Such regions can indicate not only the extent of likely [[sampling errorestimation errors]] but can also reveal whether (for example) if the estimate for one quantity is too large then the other is also likely to be too large. 
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[[Confidence region]]s generalise the confidence interval concept to deal with multiple quantities. Such regions can indicate not only the extent of likely [[sampling errorestimation errors]] but can also reveal whether (for example) the estimate for one quantity is too large then the other is also likely to be too large. See also [[confidence band]]s. 




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In modern applied practice, confidence intervals are often stated at the 95% level.<ref>Zar, J.H. (1984) ''Biostatistical Analysis.'' Prentice Hall International, New Jersey. pp 4345</ref> However, when presented graphically, confidence intervals or [[confidence region]]s may be shown for several confidence levels, for example 50%, 90% and 99%. 
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In applied practice, confidence intervals are typically stated at the 95% confidence level.<ref>Zar, J.H. (1984) ''Biostatistical Analysis.'' Prentice Hall International, New Jersey. pp 4345</ref> However, when presented graphically, confidence intervals can show several confidence levels, for example 50%, 95% and 99%. 





==Theoretical basis== 

==Theoretical basis== 

===Definition=== 

===Definition=== 
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====CI's as random intervals==== 
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====Confidence Intervals as random intervals==== 





Confidence intervals are constructed on the basis of a given dataset: ''x'' denotes the set of observations in the dataset, and ''X'' is used when considering the outcomes that might have been observed from the same population, where ''X'' is treated as a [[random variable]] whose observed outcome is ''X'' = ''x''. A confidence interval is specified by a pair of functions ''u''(.) and ''v''(.) and the confidence interval for the given data set is defined as the interval (''u''(''x''), ''v''(''x'')). To complete the definition of a confidence interval, there needs to be a clear understanding of the quantity for which the CI provides an interval estimate. Suppose this quantity is ''w''. The property of the rules ''u''(.) and ''v''(.) that makes the interval (''u''(''x''),''v''(''x'')) closest to what a confidence interval for ''w'' would be, relates to the properties of the set of random intervals given by (''u''(''X''),''v''(''X'')): that is treating the endpoints as random variables. This property is the ''coverage probability'' or the probability ''c'' that the random interval includes ''w'', 

Confidence intervals are constructed on the basis of a given dataset: ''x'' denotes the set of observations in the dataset, and ''X'' is used when considering the outcomes that might have been observed from the same population, where ''X'' is treated as a [[random variable]] whose observed outcome is ''X'' = ''x''. A confidence interval is specified by a pair of functions ''u''(.) and ''v''(.) and the confidence interval for the given data set is defined as the interval (''u''(''x''), ''v''(''x'')). To complete the definition of a confidence interval, there needs to be a clear understanding of the quantity for which the CI provides an interval estimate. Suppose this quantity is ''w''. The property of the rules ''u''(.) and ''v''(.) that makes the interval (''u''(''x''),''v''(''x'')) closest to what a confidence interval for ''w'' would be, relates to the properties of the set of random intervals given by (''u''(''X''),''v''(''X'')): that is treating the endpoints as random variables. This property is the ''coverage probability'' or the probability ''c'' that the random interval includes ''w'', 
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For the above to provide a viable means to statistical inference, something further is required: a tie between the quantity being estimated and the [[probability distribution]] of the outcome ''X''. Suppose that this [[probability distribution]] is characterised by the unobservable [[parameter]] θ, which is a quantity to be estimated, and by other unobservable parameters φ which are not of immediate interest. These other quantities φ in which there is no immediate interest are called [[nuisance parameter]]s, as statistical theory still needs to find some way to deal with them. 

For the above to provide a viable means to statistical inference, something further is required: a tie between the quantity being estimated and the [[probability distribution]] of the outcome ''X''. Suppose that this [[probability distribution]] is characterised by the unobservable [[parameter]] θ, which is a quantity to be estimated, and by other unobservable parameters φ which are not of immediate interest. These other quantities φ in which there is no immediate interest are called [[nuisance parameter]]s, as statistical theory still needs to find some way to deal with them. 




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The definition of a confidence interval for θ is, for a given α, 
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The definition of a confidence interval for θ for any number α between 0 and 1 is an interval 

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: <math> (u(X), v(X)) </math> 

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for which 

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: <math>\Pr_{X;\theta,\varphi}(u(X)<\theta<v(X)) = 1  \alpha\text{ for all }(\theta,\varphi)\,</math> 




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:<math>{\Pr}_{X;\theta,\phi}(u(X)<\theta<v(X))=\alpha</math> for all <math>(\theta,\phi).\,</math> 
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and ''u''(''X'') and ''v''(''X'') are ''observable'' random variables, i.e. one need not know the value of the unobservable quantities ''θ'', ''φ'' in order to know the values of ''u''(''X'') and ''v''(''X''). 




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The number <math>\alpha</math> (sometimes reported as a percentage (100%·<math>\alpha</math>)) is called the confidence level or confidence coefficient. Here <math>{\Pr}_{X;\theta,\phi}</math> is used to indicate the probability when the random variable ''X'' has the distribution characterised by <math>(\theta,\phi)</math>. An important part of this specification is that the random interval (''U'', ''V'') covers the unknown value θ with a high probability no matter what the true value of θ actually is. 
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The number 1 − ''α'' (sometimes reported as a percentage 100%·(1 − ''α'')) is called the '''confidence level''' or '''confidence coefficient'''. Most standard books adopt this convention, where ''α'' will be a small number. Here <math>{\Pr}_{X;\theta,\varphi}</math> is used to indicate the probability when the random variable ''X'' has the distribution characterised by <math>(\theta,\phi)</math>. An important part of this specification is that the random interval (''U'', ''V'') covers the unknown value θ with a high probability no matter what the true value of θ actually is. 





Note that here <math>{\Pr}_{X;\theta,\phi}</math> need not refer to an explicitly given parameterised family of distributions, although it often does. Just as the random variable ''X'' notionally corresponds to other possible realisations of ''x'' from the same population or from the same version of reality, the parameters <math>(\theta,\phi)</math> indicate that we need to consider other versions of reality in which the distribution of ''X'' might have different characteristics. 

Note that here <math>{\Pr}_{X;\theta,\phi}</math> need not refer to an explicitly given parameterised family of distributions, although it often does. Just as the random variable ''X'' notionally corresponds to other possible realisations of ''x'' from the same population or from the same version of reality, the parameters <math>(\theta,\phi)</math> indicate that we need to consider other versions of reality in which the distribution of ''X'' might have different characteristics. 
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====Intervals for random outcomes==== 

====Intervals for random outcomes==== 




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Confidence intervals can be defined for random quantities as well as for fixed quantities as in the above. See [[prediction interval]]. For this, consider an additional singlevalued random variable ''Y'' which may or may not be statistically dependent on ''X''. Then the rule for for constructing the interval(''u''(''x''), ''v''(''x'')) provides a confidence interval for the asyettobe observed value ''y'' of ''Y'' if 
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Confidence intervals can be defined for random quantities as well as for fixed quantities as in the above. See [[prediction interval]]. For this, consider an additional singlevalued random variable ''Y'' which may or may not be statistically dependent on ''X''. Then the rule for constructing the interval(''u''(''x''), ''v''(''x'')) provides a confidence interval for the asyettobe observed value ''y'' of ''Y'' if 




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:<math>{\Pr}_{X,Y;\theta,\phi}(u(X)<Y<v(X))=\alpha</math> for all <math>(\theta,\phi).\,</math> 
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:<math>\Pr_{X,Y;\theta,\varphi}(u(X)<Y<v(X)) = 1\alpha\text{ for all }(\theta,\varphi).\,</math> 




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Here <math>{\Pr}_{X,Y;\theta,\phi}</math> is used to indicate the probability over the joint distribution of the random variables (''X,Y'') when this is characterised by parameters <math>(\theta,\phi)</math>. 
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Here <math>{\Pr}_{X,Y;\theta,\varphi}</math> is used to indicate the probability over the joint distribution of the random variables (''X'', ''Y'') when this is characterised by parameters <math>(\theta,\varphi)</math>. 





==== Approximate confidence intervals==== 

==== Approximate confidence intervals==== 
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and the rule for constructing the interval may be accepted as providing a confidence interval if 

and the rule for constructing the interval may be accepted as providing a confidence interval if 




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:<math>c(\theta,\phi)\approxeq \alpha</math> for all <math>(\theta,\phi)</math> 
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:<math>c(\theta,\phi)\approxeq 1\alpha</math> for all <math>(\theta,\phi)</math> 





to an acceptable level of approximation. 

to an acceptable level of approximation. 
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==== Comparison to Bayesian interval estimates ==== 

==== Comparison to Bayesian interval estimates ==== 




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A Bayesian interval estimate is called a [[credible interval]]. Using much of the same notation as above, the definition of a credible interval for the unknown true value of θ is, for a given α, 
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A Bayesian interval estimate is called a [[credible interval]]. Using much of the same notation as above, the definition of a credible interval for the unknown true value of θ is, for a given α<ref>{{cite book author=Bernardo JE, Smith, Adrian title=Bayesian theory publisher=Wiley location=New York year=2000 pages=259 isbn=047149464X oclc= doi= accessdate=}}</ref>, 




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:<math>{\Pr}_{\ThetaX=x}(u(x)<\Theta<v(x))=\alpha.</math> 
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:<math>{\Pr}_{\ThetaX=x}(u(x)<\Theta<v(x))=1\alpha.</math> 




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Here Θ is used to emhasize that the unknown value of <math>\theta</math> is being treated as a random variable. The definitions of the two types of intervals may be compared as follows. 
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Here Θ is used to emphasize that the unknown value of <math>\theta</math> is being treated as a random variable. The definitions of the two types of intervals may be compared as follows. 

*The definition of a confidence interval involves probabilities calculated from the distribution of ''X'' for given <math>(\theta,\phi)</math> (or conditional on these values) and the condition needs to hold for all values of <math>(\theta,\phi)</math>. 

*The definition of a confidence interval involves probabilities calculated from the distribution of ''X'' for given <math>(\theta,\phi)</math> (or conditional on these values) and the condition needs to hold for all values of <math>(\theta,\phi)</math>. 

*The definition of a credible interval involves probabilities calculated from the distribution of Θ conditional on the observed values of ''X''=''x'' and marginalised (or averaged) over the values of <math>\Phi</math>, where this last quantity is the random variable corresponding to the uncertainty about the [[nuisance parameter]]s in <math>\phi</math>. 

*The definition of a credible interval involves probabilities calculated from the distribution of Θ conditional on the observed values of ''X''=''x'' and marginalised (or averaged) over the values of <math>\Phi</math>, where this last quantity is the random variable corresponding to the uncertainty about the [[nuisance parameter]]s in <math>\phi</math>. 
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For nonstandard 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 estimationpoint estimate]] of the quantity being considered. 

For nonstandard 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 estimationpoint estimate]] of the quantity being considered. 




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;Sample statistics: This is closely related to the [[method of moments]] for estimation. A simple example arises where the quantity to be estimated is the mean, in which case an natural estimate is the sample mean. The usual arguments indicate that the sample variance can be used to estimate the variance of the sample mean. A naive confidence interval for the true mean can be constructed centred on the sample mean with a width which is a multiple of the square root of the sample variance. 
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;Sample statistics: This is closely related to the [[method of moments]] for estimation. A simple example arises where the quantity to be estimated is the mean, in which case a natural estimate is the sample mean. The usual arguments indicate that the sample variance can be used to estimate the variance of the sample mean. A naive 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: Where estimates are constructed using the [[maximum likelihood principle]], the theory for this provides two ways of constructing confidence intervals or confidence regions for the estimates. 

;Likelihood theory: Where estimates are constructed using the [[maximum likelihood principle]], the theory for this provides two ways of constructing confidence intervals or confidence regions for the estimates. 




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;Estimating equations: The estimation approach here can be considered as both a generalisation of the method of moments and a generalisation of the maximum likelihood approach. There are corresponding generalisations of the results of maximum likelihood theory that allow confidence intervals to be constructed based on estimates derived from [[estimating equations]]. 
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;Estimating equations: The estimation approach here can be considered as both a generalization of the 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]]. 





;Via [[significance testing]]: If significance tests are available for general values of a parameter, then confidence intervals/regions can be constructed by including in the 100p% confidence region all those points for which the significance test of the null hypothesis that the true value is the given value is not rejected at a significance level of (1p). 

;Via [[significance testing]]: If significance tests are available for general values of a parameter, then confidence intervals/regions can be constructed by including in the 100p% confidence region all those points for which the significance test of the null hypothesis that the true value is the given value is not rejected at a significance level of (1p). 




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==Practical example== 
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==Examples== 

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===Practical example=== 

A machine fills cups with margarine, and is supposed to be adjusted so that the mean content of the cups is close to 250 grams of margarine. Of course it is not possible to fill every cup with exactly 250 grams of margarine. Hence the weight of the filling can be considered to be a random variable ''X''. The distribution of ''X'' is assumed here to be a normal distribution with unknown expectation μ and (for the sake of simplicity) known standard deviation σ = 2.5 grams. To check if the machine is adequately adjusted, a sample of ''n'' = 25 cups of margarine is chosen at random and the cups weighed. The weights of margarine are <math>X_1,\dots,X_{25}</math>, a random sample from ''X''. 

A machine fills cups with margarine, and is supposed to be adjusted so that the mean content of the cups is close to 250 grams of margarine. Of course it is not possible to fill every cup with exactly 250 grams of margarine. Hence the weight of the filling can be considered to be a random variable ''X''. The distribution of ''X'' is assumed here to be a normal distribution with unknown expectation μ and (for the sake of simplicity) known standard deviation σ = 2.5 grams. To check if the machine is adequately adjusted, a sample of ''n'' = 25 cups of margarine is chosen at random and the cups weighed. The weights of margarine are <math>X_1,\dots,X_{25}</math>, a random sample from ''X''. 




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:<math>Z = \frac {\bar X\mu}{\sigma/\sqrt{n}} =\frac {\bar X\mu}{0.5} </math> 

:<math>Z = \frac {\bar X\mu}{\sigma/\sqrt{n}} =\frac {\bar X\mu}{0.5} </math> 




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dependent on μ, but with a standard normal distribution independent of the parameter μ to be estimated. Hence it is possible to find numbers −''z'' and ''z'', independent of μ, where ''Z'' lies in between with probability 1 − α, a measure of how confident we want to be. We take 1 − α = 0.95. So we have: 
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The above expression, standardizes your varible. This allows you to do this analysis, to calculate the 95% confidence interval. μ is some future measurement, sigma is your standard deviation, N is your sample size (in this case 25), and X bar is your sample mean (in this case 250.2). In order to calculate a confidence interval we first need to pick an α varible. Since we are interested in the 95% confidence interval, we set α = 0.05. Hence it is possible to find numbers −''z'' and ''z'', independent of μ, where ''Z'' lies in between with probability 1 − α. We take 1 − α = 0.95. So we have: 





:<math>P(z\le Z\le z) = 1\alpha = 0.95.</math> 

:<math>P(z\le Z\le z) = 1\alpha = 0.95.</math> 




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The number ''z'' follows from: 
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The number ''z'' follows from the [[cumulative distribution function]], which gives us are ''z'' value, and is valid becuase we standardized our big Z. (Also see [[probit]]). Therefore: 





:<math>\Phi(z) = P(Z \le z) = 1  \frac{\alpha}2 = 0.975\,,</math> 

:<math>\Phi(z) = P(Z \le z) = 1  \frac{\alpha}2 = 0.975\,,</math> 

::<math>z=\Phi^{1}(\Phi(z)) = \Phi^{1}(0.975) = 1.96\,,</math> 

::<math>z=\Phi^{1}(\Phi(z)) = \Phi^{1}(0.975) = 1.96\,,</math> 




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(see [[probit]] and [[cumulative distribution function]]), and we get: 
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and then we get: 





:<math>0.95 = 1\alpha=P(z \le Z \le z)=P \left(1.96 \le \frac {\bar X\mu}{\sigma/\sqrt{n}} \le 1.96 \right) </math> 

:<math>0.95 = 1\alpha=P(z \le Z \le z)=P \left(1.96 \le \frac {\bar X\mu}{\sigma/\sqrt{n}} \le 1.96 \right) </math> 
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:<math>(\bar x  0.98;\bar x + 0.98) = (250.2  0.98; 250.2 + 0.98) = (249.22; 251.18).\,</math> 

:<math>(\bar x  0.98;\bar x + 0.98) = (250.2  0.98; 250.2 + 0.98) = (249.22; 251.18).\,</math> 




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This interval has fixed endpoints, where μ might be in between (or not). There is no probability of such an event. We '''cannot''' say: "with probability (1 − α) the parameter μ lies in the confidence interval." We only know that by repetition in 100(1 − α) % of the cases μ will be in the calculated interval. In 100α % of the cases however it doesn't. And unfortunately we don't know in which of the cases this happens. That's why we say: with '''confidence level''' 100(1 − α) % μ lies in the confidence interval." 
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[[Image:NYWconfidenceinterval.svgthumb300pxThe vertical [[line segment]]s represent 50 realisations of a confidence interval for μ.]] 




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The following picture shows 50 realisations of a confidence interval for μ. 
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This interval has fixed endpoints, where μ might be in between (or not). There is no probability of such an event. We '''cannot''' say: "with probability (1 − α) the parameter μ lies in the confidence interval." We only know that by repetition in 100(1 − α) % of the cases μ will be in the calculated interval. In 100α % of the cases however it doesn't. And unfortunately we don't know in which of the cases this happens. That's why we say: "with '''confidence level''' 100(1 − α) % μ lies in the confidence interval." 
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[[Image:NYWconfidenceinterval.pngcenter]] 

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Observation of the sample means we choose from the population of all realisations. There the probability is 95% we end up having chosen an interval that contains the parameter. After realisation we just have our chosen interval. As seen from the picture there was a fair chance we choose an interval containing μ; however we may be unlucky and have picked the wrong one. We'll never know; we're stuck with our interval. 





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==Theoretical example== 
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The figure on the right shows 50 realisations of a confidence interval for a given population mean μ. If we randomly choose one realisation, the probability is 95% we end up having chosen an interval that contains the parameter; however we may be unlucky and have picked the wrong one. We'll never know; we're stuck with our interval. 

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===Theoretical example=== 

Suppose ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are an [[statistical independenceindependent]] sample from a [[normal distributionnormally distributed]] population with [[mean]] μ and [[variance]] σ<sup>2</sup>. Let 

Suppose ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are an [[statistical independenceindependent]] sample from a [[normal distributionnormally distributed]] population with [[mean]] μ and [[variance]] σ<sup>2</sup>. Let 




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an interval with fixed numbers as endpoints, of which we can no more say there is a certain probability it contains the parameter μ. Either μ is in this interval or isn't. 

an interval with fixed numbers as endpoints, of which we can no more say there is a certain probability it contains the parameter μ. Either μ is in this interval or isn't. 




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==Meaning and interpretation== 
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==Relation to hypothesis testing== 
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=== Confidence intervals in measurement === 





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The results of measurements are often accompanied by confidence intervals. For instance, suppose a scale is known to yield the actual mass of an object plus a [[normal distributionnormally distributed]] random error with mean 0 and known [[standard deviation]] ''σ''. If we weigh 100 objects of known mass on this scale and report the values ±σ, then we can expect to find that around 68% of the reported ranges include the actual mass. 
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While the formulations of the notions of confidence intervals and of [[statistical hypothesis testing]] are distinct they are in some senses related and to some extent complementary. While not all confidence intervals are constructed in this way, one general purpose approach to constructing confidence intervals is to define a 100(1−α)% confidence interval to consist of all those values θ<sub>0</sub> for which a test of the hypothesis θ=θ<sub>0</sub> is not rejected at a significance level of 100α%. Such an approach may not always be available since it presupposes the practical availability of an appropriate significance test. Naturally, any assumptions required for the significance test would carry over to the confidence intervals. 




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If we wish to report values with a smaller [[standard error (statistics)standard error]] value, then we repeat the measurement ''n'' times and average the results. Then the 68.2% confidence interval is <math>\pm \sigma/\sqrt{n}</math>. For example, repeating the measurement 100 times reduces the confidence interval to 1/10 of the original width. 
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It may be convenient to make the general correspondence that parameter values within a confidence interval are equivalent to those values that would not be rejected by an hypothesis test, but this would be dangerous. In many instances the confidence intervals that are quoted are only approximately valid, perhaps derived from "plus or minus twice the standard error", and the implications of this for the supposedly corresponding hypothesis tests are usually unknown. 




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Note that when we report a 68.2% confidence interval (usually termed standard error) as ''v'' ± ''σ'', this does not mean that the true mass has a 68.2% chance of being in the reported range. In fact, the true mass is either in the range or not. How can a value outside the range be said to have any chance of being in the range? Rather, our statement means that 68.2% of the ranges we report using ± ''σ'' are likely to include the true mass. 
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==Meaning and interpretation== 
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This is not just a quibble. Under the incorrect interpretation, each of the 100 measurements described above would be specifying a different range, and the true mass supposedly has a 68% chance of being in each and every range. Also, it supposedly has a 32% chance of being outside each and every range. If two of the ranges happen to be disjoint, the statements are obviously inconsistent. Say one range is 1 to 2, and the other is 2 to 3. Supposedly, the true mass has a 68% chance of being between 1 and 2, but only a 32% chance of being less than 2 or more than 3. The incorrect interpretation reads more into the statement than is meant. 

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On the other hand, under the correct interpretation, each and every statement we make is really true, because the statements are not about any specific range. We could report that one mass is 10.2 ± 0.1 grams, while really it is 10.6 grams, and not be lying. But if we report fewer than 1000 values and more than two of them are that far off, we will have some explaining to do. 

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− 
It is also possible to estimate a confidence interval without knowing the standard deviation of the random error. This is done using the [[Student's tdistribution#Confidence intervals derived from Student's tdistributiont distribution]], or by using [[Nonparametric statisticsnonparametric]] [[Resampling (statistics)resampling methods]] such as the [[Resampling (statistics)#Bootstrapbootstrap]], which do not require that the error have a normal distribution. 

− 


− 
===How to understand confidence intervals=== 






For users of frequentist methods, various interpretations of a confidence interval can be given. 

For users of frequentist methods, various interpretations of a confidence interval can be given. 




− 
*The confidence interval can be expressed in terms of samples (or repeated samples): "''Were this procedure to be repeated on multiple samples, the calculated confidence interval (which would differ for each sample) would encompass the true population parameter 90% of the time."'' 
+ 
*The confidence interval can be expressed in terms of samples (or repeated samples): "''Were this procedure to be repeated on multiple samples, the calculated confidence interval (which would differ for each sample) would encompass the true population parameter 90% of the time."'' <ref>Cox DR, Hinkley DV. (1974) Theoretical Statistics, Chapman & Hall, p49, 209</ref> Note that this need not be repeated sampling from the same population, just repeated sampling <ref> Kendall, M.G. and Stuart, D.G. (1973) The Advanced Theory of Statistics. Vol 2: Inference and Relationship, Griffin, London. Section 20.4</ref>. 
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*The explanation of a confidence interval can amount to something like: "''The confidence interval represents values for the population parameter for which the difference between the parameter and the observed estimate is not [[statistically significant]] at the 10% level''". In fact, this relates to one particular way in which a confidence interval may be constructed. 
+ 
*The explanation of a confidence interval can amount to something like: "''The confidence interval represents values for the population parameter for which the difference between the parameter and the observed estimate is not [[statistically significant]] at the 10% level''"<ref>Cox DR, Hinkley DV. (1974) Theoretical Statistics, Chapman & Hall, p214, 225, 233</ref>. In fact, this relates to one particular way in which a confidence interval may be constructed. 

*The probability associated with a confidence interval may also be considered from a preexperiment point of view, in the same context in which arguments for the random allocation of treatments to study items are made. Here the experimenter sets out the way in which they intend to calculate a confidence interval and know, before they do the actual experiment, that the interval they will end up calculating has a certain chance of covering the true but unknown value. This is very similar to the "repeated sample" interpretation above, except that it avoids relying on considering hypothetical repeats of a sampling procedure that may not be repeatable in any meaningful sense. 

*The probability associated with a confidence interval may also be considered from a preexperiment point of view, in the same context in which arguments for the random allocation of treatments to study items are made. Here the experimenter sets out the way in which they intend to calculate a confidence interval and know, before they do the actual experiment, that the interval they will end up calculating has a certain chance of covering the true but unknown value. This is very similar to the "repeated sample" interpretation above, except that it avoids relying on considering hypothetical repeats of a sampling procedure that may not be repeatable in any meaningful sense. 




− 
Critics of frequentist methods dislike at least the first two of the three interpretations given above. Users of Bayesian methods, if they produced an [[interval estimationinterval estimate]], would by contrast want to say "''My degree of belief that the parameter is in fact in this interval is 90%''". Disagreements about these issues are not disagreements about solutions to mathematical problems. Rather they are disagreements about the ways in which mathematics is to be applied. 
+ 
In each of the above, the following applies. If the true value of the parameter lies outside the 90% confidence interval once it has been calculated, then an event has occurred which had a probability of 10% (or less) of happening by chance. 




− 
<!I will add an example of a "recognizable subset" here; i.e., a case in which the data themselves make the epistemic conclusion dubious. Who, what would this be and where would this go, given that the comment has been moved?> 






+ 
Users of Bayesian methods, if they produced an [[interval estimationinterval estimate]], would by contrast want to say "''My degree of belief that the parameter is in fact in this interval is 90%''" <ref>Cox DR, Hinkley DV. (1974) Theoretical Statistics, Chapman & Hall, p390</ref>. See [[Credible interval]]. 

+ 
Disagreements about these issues are not disagreements about solutions to mathematical problems. Rather they are disagreements about the ways in which mathematics is to be applied. 




− 
{{Unreferencedsectiondate=February 2008}} 
+ 
==Meaning of the term ''confidence''== 
− 
{{Disputedsectiondate=March 2008}} 
+ 
There is a difference in meaning between the common usage of the word 'confidence' and its statistical usage, which is often confusing to the layman. In common usage, a claim to 95% confidence in something is normally taken as indicating virtual certainty. In statistics, a claim to 95% confidence simply means that the researcher has seen something occur that only happens one time in twenty or less. If one were to roll two dice and get double six, few would claim this as proof that the dice were fixed, although statistically speaking one could have 97% confidence that they were. Similarly, the finding of a statistical link at 95% confidence is not proof, nor even very good evidence, that there is any real connection between the things linked. 




− 
Confidence levels are typically given alongside statistics resulting from sampling. 
+ 
When a study involves multiple statistical tests, some laymen assume that the confidence associated with individual tests is the confidence one should have in the results of the study itself. In fact, the results of all the statistical tests conducted during a study must be judged as a whole in determining what confidence one may place in the positive links it produces. If a study involving 40 statistical tests at 95% confidence was performed, about two of the tests can be expected to return false positives. If 3 links are found, the confidence associated with those links 'as the result of the survey' is actually about 32%; it's what should be expected twothirds of the time. 




− 
In a statement "we are 90% confident that between 35% and 45% of voters favor Candidate A", 90% is our confidence level and 35%45% is our confidence interval. 
+ 
== Confidence intervals in measurement == 




− 
It is very tempting to misunderstand this statement in the following way. We used capital letters ''U'' and ''V'' for random variables; it is conventional to use lowercase letters ''u'' and ''v'' for their observed values in a particular instance. The misunderstanding is the conclusion that{{Factdate=February 2008}} 
+ 
{{Unreferencedsectiondate=April 2008}} 

+ 
{{Disputedsectiondate=April 2008}} 




− 
:<math>\Pr(u<\theta<v)=0.9,\,</math> 
+ 
The results of measurements are often accompanied by confidence intervals. For instance, suppose a scale is known to yield the actual mass of an object plus a [[normal distributionnormally distributed]] random error with mean 0 and known [[standard deviation]] ''σ''. If we weigh 100 objects of known mass on this scale and report the values ±σ, then we can expect to find that around 68% of the reported ranges include the actual mass. 




− 
so that after the data has been observed, a conditional probability distribution of θ, given the data, is inferred. For example, suppose ''X'' is [[normal distributionnormally distributed]] with [[expected value]] θ and [[variance]] 1. (It is grossly unrealistic to take the variance to be known while the expected value must be inferred from the data, but it makes the example simple.) The random variable ''X'' is observable. (The random variable ''X'' − θ is ''not'' observable, since its value depends on θ.) Then ''X'' − θ is normally distributed with expectation 0 and variance 1. Given that 90% of the standard normal distribution lies between −1.645 and 1.645, we know: 
+ 
If we wish to report values with a smaller [[standard error (statistics)standard error]] value, then we repeat the measurement ''n'' times and average the results. Then the 68.2% confidence interval is <math>\pm \sigma/\sqrt{n}</math>. For example, repeating the measurement 100 times reduces the confidence interval to 1/10 of the original width. 




− 
:<math>\Pr(1.645<X\theta<1.645)=0.9.\,</math> 
+ 
Note that when we report a 68.2% confidence interval (usually termed standard error) as ''v'' ± ''σ'', this does not mean that the true mass has a 68.2% chance of being in the reported range. In fact, the true mass is either in the range or not. How can a value outside the range be said to have any chance of being in the range? Rather, our statement means that 68.2% of the ranges we report using ± ''σ'' are likely to include the true mass. 




− 
Consequently 
+ 
This is not just a quibble. Under the incorrect interpretation, each of the 100 measurements described above would be specifying a different range, and the true mass supposedly has a 68% chance of being in each and every range. Also, it supposedly has a 32% chance of being outside each and every range. If two of the ranges happen to be disjoint, the statements are obviously inconsistent. Say one range is 1 to 2, and the other is 2 to 3. Supposedly, the true mass has a 68% chance of being between 1 and 2, but only a 32% chance of being less than 2 or more than 3. The incorrect interpretation reads more into the statement than is meant. 




− 
:<math>\Pr(X1.645<\theta<X+1.645)=0.9,\,</math> 
+ 
On the other hand, under the correct interpretation, each and every statement we make is really true, because the statements are not about any specific range. We could report that one mass is 10.2 ± 0.1 grams, while really it is 10.6 grams, and not be lying. But if we report fewer than 1000 values and more than two of them are that far off, we will have some explaining to do. 




− 
so the interval from ''X'' − 1.645 to ''X'' + 1.645 is a 90% confidence interval for θ. But when ''X'' = 82 is observed, can we then say that 
+ 
It is also possible to estimate a confidence interval without knowing the standard deviation of the random error. This is done using the [[Student's tdistribution#Confidence intervals derived from Student's tdistributiont distribution]], or by using [[Nonparametric statisticsnonparametric]] [[Resampling (statistics)resampling methods]] such as the [[Resampling (statistics)#Bootstrapbootstrap]], which do not require that the error have a normal distribution. 
− 


− 
:<math>\Pr(821.645<\theta<82+1.645)=0.9\ \mbox{?}</math> 

− 


− 
This conclusion does not follow from the laws of probability because θ is not a "random variable"; i.e., no probability distribution has been assigned to it.{{Factdate=February 2008}} Confidence intervals are generally a [[frequentismfrequentist]] method, i.e., employed by those who interpret "90% probability" as "occurring in 90% of all cases".{{Factdate=February 2008}} Suppose, for example, that θ is the [[mass]] of the [[planet]] [[Neptune (planet)Neptune]], and the randomness in our [[measurement]] error means that 90% of the time our statement that the mass is between this number and that number will be correct. The mass is not what is random. Therefore, given that we have measured it to be 82 units, we cannot say that in 90% of all cases, the mass is between 82 − 1.645 and 82 + 1.645. There are no such cases; there is, after all, only one planet Neptune. 

− 


− 
But if probabilities are construed as degrees of belief rather than as relative frequencies of occurrence of random events, i.e., if we are [[Bayesian probabilityBayesians]] rather than frequentists, can we ''then'' say we are 90% sure that the mass is between 82 − 1.645 and 82 + 1.645? Many answers to this question have been proposed, and are philosophically controversial. The answer will not be a mathematical theorem, but a philosophical tenet. Less controversial are Bayesian [[credible interval]]s, in which one starts with a prior probability distribution of θ, and finds a posterior probability distribution, which is the conditional probability distribution of θ given the data. 

− 


− 
== [[Robust statisticsRobust]] confidence intervals == 

− 


− 
In the process of weighing 1000 objects, under practical conditions, it is easy to believe that the operator might make a mistake in procedure and so report an incorrect mass (thereby making one type of [[systematic error]]). Suppose he has 100 objects and he weighed them all, one at a time, and repeated the whole process ten times. Then he can calculate a sample standard deviation for each object, and look for [[outlier]]s. Any object with an unusually large standard deviation probably has an outlier in its data. These can be removed by various nonparametric techniques. If he repeated the process only three times, he would simply take the [[median]] of the three measurements and use σ to give a confidence interval. The 200 extra weighings served only to detect and correct for operator error and did nothing to improve the confidence interval. With more repetitions, he could use a [[truncated mean]], discarding say the largest and smallest values and averaging the rest. He could then use a [[Bootstrapping (statistics)bootstrap]] calculation to determine a confidence interval narrower than that calculated from σ, and so obtain some benefit from a large amount of extra work. 

− 


− 
These procedures are [[Robust statisticsrobust]] against procedural errors which are not modeled by the assumption that the balance has a fixed known standard deviation σ. In practical applications where the occasional operator error can occur, or the balance can malfunction, the assumptions behind simple statistical calculations cannot be taken for granted. Before trusting the results of 100 objects weighed just three times each to have confidence intervals calculated from σ, it is necessary to test for and remove a reasonable number of outliers (testing the assumption that the operator is careful and correcting for the fact that he is not perfect), and to test the assumption that the data really have a normal distribution with standard deviation σ. 

− 


− 
The theoretical analysis of such an experiment is complicated, but it is easy to set up a [[spreadsheet]] which draws random numbers from a normal distribution with standard deviation σ to simulate the situation (use =norminv(rand(),0,σ)). See for example Wittwer, J.W., [http://vertex42.com/ExcelArticles/mc/ "Monte Carlo Simulation in Excel: A Practical Guide"], June 1, 2004. These techniques also work in Open Office and gnumeric. 

− 


− 
After removing obvious outliers, one could subtract the median from the other two values for each object, and examine the distribution of the 200 resulting numbers. It should be normal with mean near zero and standard deviation a little larger than σ. A simple [[Monte Carlo methodMonte Carlo]] spreadsheet calculation would reveal typical values for the standard deviation (around 105 to 115% of σ). Or, one could subtract the mean of each triplet from the values, and examine the distribution of 300 values. The mean is identically zero, but the standard deviation should be somewhat smaller (around 75 to 85% of σ). 






==Confidence intervals for proportions and related quantities== 

==Confidence intervals for proportions and related quantities== 

{{seealsoMargin of error}} 

{{seealsoMargin of error}} 

+ 
{{seealsoBinomial proportion confidence interval}} 

An approximate confidence interval for a population mean can be constructed for random variables that are not normally distributed in the population, relying on the [[central limit theorem]], if the [[sample size]]s and counts are big enough. The formulae are identical to the case above (where the sample mean is actually normally distributed about the population mean). The approximation will be quite good with only a few dozen observations in the sample if the [[probability distribution]] of the random variable is not too different from the [[normal distribution]] (e.g. its [[cumulative distribution function]] does not have any [[discontinuities]] and its [[skewness]] is moderate). 

An approximate confidence interval for a population mean can be constructed for random variables that are not normally distributed in the population, relying on the [[central limit theorem]], if the [[sample size]]s and counts are big enough. The formulae are identical to the case above (where the sample mean is actually normally distributed about the population mean). The approximation will be quite good with only a few dozen observations in the sample if the [[probability distribution]] of the random variable is not too different from the [[normal distribution]] (e.g. its [[cumulative distribution function]] does not have any [[discontinuities]] and its [[skewness]] is moderate). 




− 
One type of sample mean is the mean of an [[indicator variable]], which takes on the value 1 for true and the value 0 for false. (Statisticians often call indicator variables "dummy variables", but that term is also frequently used by mathematicians for the concept of a [[bound variable]].) The mean of such a variable is equal to the proportion that have the variable equal to one (both in the population and in any sample). Thus, the sample mean for a variable labeled MALE in data is just the proportion of sampled observations who have MALE = 1, i.e. the proportion who are male. This is a useful property of [[indicator variable]]s, especially for hypothesis testing. 
+ 
One type of sample mean is the mean of an [[indicator variable]], which takes on the value 1 for true and the value 0 for false. The mean of such a variable is equal to the proportion that have the variable equal to one (both in the population and in any sample). This is a useful property of [[indicator variable]]s, especially for hypothesis testing. To apply the [[central limit theorem]], one must use a large enough sample. A rough rule of thumb is that one should see at least 5 cases in which the indicator is 1 and at least 5 in which it is 0. Confidence intervals constructed using the above formulae may include negative numbers or numbers greater than 1, but proportions obviously cannot be negative or exceed 1. Additionally, sample proportions can only take on a finite number of values, so the [[central limit theorem]] and the [[normal distribution]] are not the best tools for building a confidence interval. See "[[Binomial proportion confidence interval]]" for better methods which are specific to this case. 
− 


− 
To apply the [[central limit theorem]], one must use a large enough sample. A rough rule of thumb is that one should see at least 5 cases in which the indicator is 1 and at least 5 in which it is 0. Confidence intervals constructed using the above formulae may include negative numbers or numbers greater than 1, but proportions obviously cannot be negative or exceed 1. The probability assigned to negative numbers and numbers greater than 1 is usually small when the [[sample size]] is large and the proportion being estimated is not too close to 0 or 1. 

− 


− 
Confidence intervals for cases where the method above assigns a substantial probability to (−∞, 0) or to (1, ∞) may be constructed by inverting hypothesis tests. If we think of conducting hypothesis tests over the whole feasible range of parameter values, and including any values for which a single hypothesis test would not reject the null hypothesis that the true value was that value, given our sample value, we can make a confidence interval based on the [[central limit theorem]] that does not violate the basic properties of proportions. 

− 


− 
On the other hand, sample proportions can only take on a finite number of values, so the [[central limit theorem]] and the [[normal distribution]] are not the best tools for building a confidence interval. A better method would rely on the [[binomial distribution]] or the [[beta distribution]], and there are a number of better methods in widespread use. For details on advantages and disadvantages of each, see: 

− 
* "Interval Estimation for a Binomial Proportion", Lawrence D. Brown, T. Tony Cai, Anirban DasGupta, ''Statistical Science'', volume 16, number 2 (May, 2001), pages 101117. 






==See also== 

==See also== 

* [[Analysis of variance]] 

* [[Analysis of variance]] 

+ 
* [[Binomial proportion confidence interval]] 

+ 
* [[Bootstrapping (statistics)]] 

* [[Confidence region]] 

* [[Confidence region]] 

+ 
* [[Cumulative frequency]] 

+ 
* [[Effect size (statistical)]] 

+ 
* [[Hypothesis testing]] 

+ 
* [[Predictability (measurement)]] 

* [[Prediction interval]] 

* [[Prediction interval]] 

+ 
* [[Tolerance interval]] 

* [[Regression analysis]] 

* [[Regression analysis]] 

+ 
* [[Robust confidence intervals]] 

* [[Segmented regression]] 

* [[Segmented regression]] 
− 
* [[Cumulative frequency]] 
+ 
* [[Statistical sample parameters]] 
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* [[Bootstrapping (statistics)]] 
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* [[Statistical significance]] 
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* [[Binomial proportion confidence interval]] 
+ 
* [[Statistical tests]] 

+ 






== Online calculators == 

== Online calculators == 
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{{refbegin}} 

{{refbegin}} 

* [[Ronald FisherFisher, R.A.]] (1956) ''Statistical Methods and Scientific Inference.'' Oliver and Boyd, Edinburgh. (See p. 32.) 

* [[Ronald FisherFisher, R.A.]] (1956) ''Statistical Methods and Scientific Inference.'' Oliver and Boyd, Edinburgh. (See p. 32.) 
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* Freund, J.E. (1962) ''Mathematical Statistics'' Prentice Hall, Englewood Cliffs, NJ. (See pp. 227228.) 
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* Freund, J.E. (1962) ''Mathematical Statistics'' Prentice Hall, Englewood Cliffs, NJ. (See pp. 227–228.) 

* [[Ian HackingHacking, I.]] (1965) ''Logic of Statistical Inference.'' Cambridge University Press, Cambridge 

* [[Ian HackingHacking, I.]] (1965) ''Logic of Statistical Inference.'' Cambridge University Press, Cambridge 

* Keeping, E.S. (1962) ''Introduction to Statistical Inference.'' D. Van Nostrand, Princeton, NJ. 

* Keeping, E.S. (1962) ''Introduction to Statistical Inference.'' D. Van Nostrand, Princeton, NJ. 
− 
* [[Jack Kiefer (mathematician)Kiefer, J.]] (1977) [http://links.jstor.org/sici?sici=01621459%28197712%2972%3A360%3C789%3ACCSACE%3E2.0.CO%3B29 "Conditional Confidence Statements and Confidence Estimators (with discussion)"] ''Journal of the American Statistical Association,'' '''72,''' 789827. 
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* [[Jack Kiefer (mathematician)Kiefer, J.]] (1977) [http://links.jstor.org/sici?sici=01621459%28197712%2972%3A360%3C789%3ACCSACE%3E2.0.CO%3B29 "Conditional Confidence Statements and Confidence Estimators (with discussion)"] ''Journal of the American Statistical Association,'' '''72,''' 789–827. 
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* [[Jerzy NeymanNeyman, J.]] (1937) [http://links.jstor.org/sici?sici=00804614%2819370830%29236%3A767%3C333%3AOOATOS%3E2.0.CO%3B26 "Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability"] ''Philosophical Transactions of the Royal Society of London A,'' '''236,''' 333380. (Seminal work.) 
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* [[Jerzy NeymanNeyman, J.]] (1937) [http://links.jstor.org/sici?sici=00804614%2819370830%29236%3A767%3C333%3AOOATOS%3E2.0.CO%3B26 "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.) 
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* Robinson, G.K. (1975) [http://links.jstor.org/sici?sici=00063444%28197504%2962%3A1%3C155%3ASCTTTO%3E2.0.CO%3B24 "Some Counterexamples to the Theory of Confidence Intervals."] ''Biometrika,'' '''62,''' 155161. 
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* Robinson, G.K. (1975) [http://links.jstor.org/sici?sici=00063444%28197504%2962%3A1%3C155%3ASCTTTO%3E2.0.CO%3B24 "Some Counterexamples to the Theory of Confidence Intervals."] ''Biometrika,'' '''62,''' 155–161. 

{{refend}} 

{{refend}} 
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==Further reading== 

− 
*Abelson, R. P. (1997). On the surprising longevity of flogged horses: Why there is a case for the significance test: Psychological Science Vol 8(1) Jan 1997, 1215. 

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*Aiken, L. R. (1983). An interactive program for testing the significance of various correlation coefficients: Educational and Psychological Measurement Vol 43(1) Spr 1983, 181182. 

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*Alf, E. F., Jr., & Graf, R. G. (1999). Asymptotic confidence limits for the difference between two squared multiple correlations: A simplified approach: Psychological Methods Vol 4(1) Mar 1999, 7075. 

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*Alf, E. F., & Grossberg, J. M. (1979). The geometric mean: Confidence limits and significance tests: Perception & Psychophysics Vol 26(5) Nov 1979, 419421. 

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*Algina, J. (1999). A comparison of methods for constructing confidence intervals for the squared multiple correlation coefficient: Multivariate Behavioral Research Vol 34(4) 1999, 493504. 

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*Algina, J., & Keselman, H. J. (2003). Approximate confidence intervals for effect sizes: Educational and Psychological Measurement Vol 63(4) Aug 2003, 537553. 

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*Algina, J., Keselman, H. J., & Penfield, R. D. (2006). Confidence Interval Coverage for Cohen's Effect Size Statistic: Educational and Psychological Measurement Vol 66(6) Dec 2006, 945960. 

− 
*Algina, J., & Moulder, B. C. (2001). Sample sizes for confidence intervals on the increase in the squared multiple correlation coefficient: Educational and Psychological Measurement Vol 61(4) Aug 2001, 633649. 

− 
*Anderson, A. J. (2003). Utility of a dynamic termination criterion in the ZEST adaptive threshold method: Vision Research Vol 43(2) Jan 2003, 165170. 

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*Andrade, A. G., & Scalco, M. Z. (1989). Concepts of validity and confidence in an instrument for evaluating treatment for alcoholics: Jornal Brasileiro de Psiquiatria Vol 38(1) JanFeb 1989, 4145. 

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*Aoshima, M., & Mukhopadhyay, N. (1998). FixedWidth Simultaneous Confidence Intervals for Multinormal Means in Several Intraclass Correlation Models: Journal of Multivariate Analysis Vol 66(1) Jul 1998, 4663. 

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*Appel, V., & Kipnis, D. (1954). The use of levels of confidence in item analysis: Journal of Applied Psychology Vol 38(4) Aug 1954, 256259. 

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*Atkinson, L. (1990). Standard errors of prediction for the Vineland Adaptive Behavior Scales: Journal of School Psychology Vol 28(4) Win 1990, 355359. 

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*Attorresi, H. F., Aguerri, M. E., Lozzia, G. S., & Galibert, M. S. (2004). Confidence intervals or true scores. Their assumptions made explicit: Interdisciplinaria Revista de Psicologia y Ciencias Afines Vol 21(1) 2004, 2951. 

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*Baranski, J. V., & Petrusic, W. M. (1994). The calibration and resolution of confidence in perceptual judgments: Perception & Psychophysics Vol 55(4) Apr 1994, 412428. 

− 
*Barchard, K. A., & Hakstian, A. R. (1997). The robustness of confidence intervals for coefficient alpha under violation of the assumption of essential parallelism: Multivariate Behavioral Research Vol 32(2) 1997, 169191. 

− 
*Bedrick, E. J. (1991). Approximate confidence intervals for the correlation from data in twobytwo tables: British Journal of Mathematical and Statistical Psychology Vol 44(2) Nov 1991, 369378. 

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*Belia, S., Fidler, F., Williams, J., & Cumming, G. (2005). Researchers Misunderstand Confidence Intervals and Standard Error Bars: Psychological Methods Vol 10(4) Dec 2005, 389396. 

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*Bell, J. F. (1986). Simultaneous confidence intervals for the linear functions of expected mean squares used in generalizability theory: Journal of Educational Statistics Vol 11(3) Fal 1986, 197205. 

− 
*Berger, M. P. (1977). STP: A subroutine for simultaneous test procedures and confidence intervals: Behavior Research Methods & Instrumentation Vol 9(4) Aug 1977, 385. 

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*Bergstrom, B. A., & Lunz, M. E. (1992). Confidence in pass/fail decisions for computer adaptive and paper and pencil examinations: Evaluation & the Health Professions Vol 15(4) Dec 1992, 453464. 

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*Bernbach, H. A. (1972). Confidence ratings for individual items in recall: Psychological Review Vol 79(6) Nov 1972, 536537. 

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*Berry, K. J., & Mielke, P. W. (1976). Large sample confidence limits for Goodman and Kruskal's proportional prediction measure TAUsub(b): Educational and Psychological Measurement Vol 36(3) Fal 1976, 747751. 

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*Berry, K. J., & Mielke, P. W., Jr. (1996). Exact confidence limits for population proportions based on the negative hypergeometric probability distribution: Perceptual and Motor Skills Vol 83(3, Pt 2) Dec 1996, 12161218. 

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*Berry, K. J., Mielke, P. W., & Helmericks, S. G. (1988). Exact confidence limits for proportions: Educational and Psychological Measurement Vol 48(3) Fal 1988, 713716. 

− 
*Bieliauskas, L. A., Fastenau, P. S., Lacy, M. A., & Roper, B. L. (1997). Use of the odds ratio to translate neuropsychological test scores into realworld outcomes: From statistical significance to clinical significance: Journal of Clinical and Experimental Neuropsychology Vol 19(6) Dec 1997, 889896. 

− 
*Bird, K. D. (2002). Confidence intervals for effect sizes in analysis of variance: Educational and Psychological Measurement Vol 62(2) Apr 2002, 197226. 

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*Bivens, H., & Slotnick, B. (2000). Decrement in the horizontalvertical illusion: Are subjects aware of their increased accuracy? : Perceptual and Motor Skills Vol 90(2) Apr 2000, 403412. 

− 
*Blackhouse, G., Briggs, A. H., & O'Brien, B. J. (2002). A note on the estimation of confidence intervals for costeffectiveness when costs and effects are censored: Medical Decision Making Vol 22(2) MarApr 2002, 173177. 

− 
*Blouin, D. C., & Riopelle, A. J. (2005). On Confidence Intervals for WithinSubjects Designs: Psychological Methods Vol 10(4) Dec 2005, 397412. 

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*Bobko, P. (1983). An analysis of correlations corrected for attenuation and range restriction: Journal of Applied Psychology Vol 68(4) Nov 1983, 584589. 

− 
*Bobko, P., Sapinkopf, R., & Anderson, N. S. (1978). A lack of confidence about formulae for regression confidence intervals: Teaching of Psychology Vol 5(2) Apr 1978, 102103. 

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*Bond, C. F., Jr., & Richardson, K. (2004). Seeing the fisher Ztransformation: Psychometrika Vol 69(2) Jun 2004, 291303. 

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*Bonett, D. G. (2006). Robust Confidence Interval for a Ratio of Standard Deviations: Applied Psychological Measurement Vol 30(5) Sep 2006, 432439. 

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*Bonett, D. G., & Price, R. M. (2002). Statistical inference for a linear function of medians: Confidence intervals, hypothesis testing, and sample size requirements: Psychological Methods Vol 7(3) Sep 2002, 370383. 

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*Bonett, D. G., & Price, R. M. (2005). Inferential Methods for the Tetrachoric Correlation Coefficient: Journal of Educational and Behavioral Statistics Vol 30(2) Sum 2005, 213225. 

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*Borenstein, M. (1994). A note on the use of confidence intervals in psychiatric research: Psychopharmacology Bulletin Vol 30(2) 1994, 235238. 

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*Borgmeier, C., & Homer, R. H. (2006). An evaluation of the predictive validity of confidence ratings in identifying functional behavioral assessment hypothesis statements: Journal of Positive Behavior Interventions Vol 8(2) Spr 2006, 100105. 

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*Brandstatter, E. (1999). Confidence intervals as an alternative to significance testing: Methods of Psychological Research Vol 4(2) 1999, 3346. 

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*Braun, C., Kaiser, S., Kincses, W.E., & Elbert, T. (1997). Confidence interval of single dipole locations based on EEG data: Brain Topography Vol 10(1) Fal 1997, 3139. 

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==External links== 

==External links== 
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* Confidence interval calculators for [http://www.danielsoper.com/statcalc/calc28.aspx RSquares], [http://www.danielsoper.com/statcalc/calc26.aspx Regression Coefficients], and [http://www.danielsoper.com/statcalc/calc27.aspx Regression Intercepts] 

* Confidence interval calculators for [http://www.danielsoper.com/statcalc/calc28.aspx RSquares], [http://www.danielsoper.com/statcalc/calc26.aspx Regression Coefficients], and [http://www.danielsoper.com/statcalc/calc27.aspx Regression Intercepts] 

* {{mathworld  urlname = ConfidenceInterval  title = Confidence Interval}} 

* {{mathworld  urlname = ConfidenceInterval  title = Confidence Interval}} 
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* [http://www.giacomo.lorenzoni.name/arganprobstat/ Analytical argumentations of probability and statistics] 

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* Free download of software for segmented linear regression and cumulative frequency analysis with [http://www.waterlog.info/software.htm confidence intervals] 


* [http://www.causeweb.org CAUSEweb.org] Many resources for teaching statistics including Confidence Intervals. 

* [http://www.causeweb.org CAUSEweb.org] Many resources for teaching statistics including Confidence Intervals. 

* [http://www.measuringusability.com/stats/ci/ci_instr1.php An interactive introduction to Confidence Intervals ] 

* [http://www.measuringusability.com/stats/ci/ci_instr1.php An interactive introduction to Confidence Intervals ] 

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* ''[http://demonstrations.wolfram.com/ConfidenceIntervalsConfidenceLevelSampleSizeAndMarginOfError/ Confidence Intervals: Confidence Level, Sample Size, and Margin of Error]'' by Eric Schulz, [[The Wolfram Demonstrations Project]]. 

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{{statistics}} 

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