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Confidence interval

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Range to estimate an unknown parameter
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Each row of points is a sample from the same normal distribution. The colored lines are 50% confidence intervals for the population meanμ. At the center of each interval is the sample meanx¯{\textstyle {\bar {x}}}, marked with a diamond. The blue intervals containμ, and the red ones do not.

Instatistics, aconfidence interval (CI) is a range of values used to estimate an unknownstatistical parameter, such as a populationmean.[1] Rather than reporting a single point estimate (e.g. "the average screen time is 3 hours per day"), a confidence interval provides a range, such as 2 to 4 hours, along with a specifiedconfidence level, typically 95%.

A 95% confidence level does not imply a 95% probability that the true parameter lies within a particular calculated interval. The confidence level instead reflects the long-run reliability of the method used to generate the interval.[2][3] In other words, if the same sampling procedure were repeated 100 times from the same population, approximately 95 of the resulting intervals would be expected to contain the true population mean.

Definition

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LetX{\displaystyle X} be arandom sample from aprobability distribution withstatistical parameter(θ,φ){\displaystyle (\theta ,\varphi )}. Here,θ{\displaystyle \theta } is the quantity to be estimated, whileφ{\displaystyle \varphi } includes other parameters (if any) that determine the distribution. A confidence interval for the parameterθ{\displaystyle \theta }, with confidence level or coefficientγ{\displaystyle \gamma }, is an interval(u(X),v(X)){\displaystyle (u(X),v(X))} determined byrandom variablesu(X){\displaystyle u(X)} andv(X){\displaystyle v(X)} with the property:

P(u(X)<θ<v(X))=γfor all (θ,φ).{\displaystyle P(u(X)<\theta <v(X))=\gamma \quad {\text{for all }}(\theta ,\varphi ).}

The numberγ{\displaystyle \gamma }, which is typically large (e.g. 0.95), is sometimes given in the form1−α{\displaystyle 1-\alpha } (or as a percentage100%⋅(1−α){\displaystyle 100\%\cdot (1-\alpha )}), whereα{\displaystyle \alpha } is a small positive number, often 0.05. It means that the interval(u(X),v(X)){\textstyle (u(X),v(X))} has a probabilityγ{\textstyle \gamma } of covering the value ofθ{\textstyle \theta } in repeated sampling.

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 if

P(u(X)<θ<v(X))≈ γ{\displaystyle P(u(X)<\theta <v(X))\approx \ \gamma }

to an acceptable level of approximation. Alternatively, some authors[4] simply require that

P(u(X)<θ<v(X))≥ γ{\displaystyle P(u(X)<\theta <v(X))\geq \ \gamma }

When it is known that thecoverage probability can be strictly larger thanγ{\displaystyle \gamma } for some parameter values, the confidence interval is called conservative, i.e., it errs on the safe side; which also means that the interval can be wider than need be.

Methods of derivation

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There are many ways of calculating confidence intervals, and the best method depends on the situation. Two widely applicable methods arebootstrapping and thecentral limit theorem.[2] The latter method works only if the sample is large, since it entails calculating the sample meanX¯{\displaystyle {\bar {X}}} and sample standard deviationS{\displaystyle S} and using the asymptotically standard normal quantity

X¯−μS/n{\displaystyle {\frac {{\bar {X}}-\mu }{S/{\sqrt {n}}}}}

whereμ{\textstyle \mu } andn{\displaystyle n} are the population mean and the sample size, respectively.

Example

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In thisbar chart, the top ends of the brown bars indicate observed means and the red line segments ("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).

SupposeX1,…,Xn{\displaystyle X_{1},\ldots ,X_{n}} is anindependent sample from anormally distributed population with unknown parametersmeanμ{\displaystyle \mu } andvarianceσ2.{\displaystyle \sigma ^{2}.} Define thesample meanX¯{\displaystyle {\bar {X}}} andunbiased sample varianceS2{\displaystyle S^{2}} as

X¯=X1+⋯+Xnn,{\displaystyle {\bar {X}}={\frac {X_{1}+\cdots +X_{n}}{n}},}
S2=1n−1∑i=1n(Xi−X¯)2.{\displaystyle S^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}.}

Then the value

T=X¯−μS/n{\displaystyle T={\frac {{\bar {X}}-\mu }{S/{\sqrt {n}}}}}

has aStudent'st distribution withn−1{\textstyle n-1} degrees of freedom.[5] This value is useful because its distribution does not depend on the values of the unobservable parametersμ{\textstyle \mu } andσ2{\textstyle \sigma ^{2}}; i.e., it is apivotal quantity.

Suppose we wanted to calculate a 95% confidence interval forμ.{\textstyle \mu .} First, letc{\textstyle c} be the 97.5thpercentile of the distribution ofT{\textstyle T}. Then there is a 2.5% chance thatT{\textstyle T} will be less than−c{\textstyle -c} and a 2.5% chance that it will be larger than+c{\textstyle +c} (as thet distribution is symmetric about 0). In other words,

PT(−c≤T≤c)=0.95.{\displaystyle P_{T}(-c\leq T\leq c)=0.95.}

Consequently, by replacingT{\textstyle T} withX¯−μS/n{\displaystyle {\frac {{\bar {X}}-\mu }{S/{\sqrt {n}}}}} and re-arranging terms,

PX(X¯−cSn≤μ≤X¯+cSn)=0.95{\displaystyle P_{X}{\left({\bar {X}}-{\frac {cS}{\sqrt {n}}}\leq \mu \leq {\bar {X}}+{\frac {cS}{\sqrt {n}}}\right)}=0.95}

wherePX{\displaystyle P_{X}} is the probability measure for the sampleX1,…,Xn{\displaystyle X_{1},\ldots ,X_{n}}.

It means that there is 95% probability with which this conditionX¯−cSn≤μ≤X¯+cSn{\displaystyle {\bar {X}}-{\frac {cS}{\sqrt {n}}}\leq \mu \leq {\bar {X}}+{\frac {cS}{\sqrt {n}}}} occurs in repeated sampling. After observing a sample, we find valuesx¯{\displaystyle {\bar {x}}} forX¯{\displaystyle {\bar {X}}} ands{\displaystyle s} forS,{\displaystyle S,} from which we compute the below interval, and we say it is a 95% confidence interval for the mean.

[x¯−cSn,x¯+cSn].{\displaystyle \left[{\bar {x}}-{\frac {cS}{\sqrt {n}}},{\bar {x}}+{\frac {cS}{\sqrt {n}}}\right].}

Interpretation

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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 along-run frequency inrepeated samples (or inresampling): "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%."[6]
  • 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%probability that the 95% confidence interval calculated from a given future sample will cover the true value of the population parameter."[7] This essentially reframes the "repeated samples" interpretation as a probability rather than a frequency.
  • The confidence interval can be expressed in terms of statistical significance, e.g.: "The 95% confidence interval represents values that are notstatistically significantly different from the point estimate at the .05 level."[8]
Interpretation of the 95% confidence interval in terms of statistical significance

Common misunderstandings

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A 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.[9]

  • 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.[10][11]
  • A 95% confidence level does not mean that 95% of the sample data lie within the confidence interval.[1]
  • 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.[11]

For example, suppose a factory produces metal rods. A random sample of 25 rods gives a 95% confidence interval for the population mean length of 36.8 to 39.0 mm.[12]

  • It is incorrect to say that there is a 95% probability that the true population mean lies within this interval, because the true mean is fixed, not random. For example, it might be 37 mm, which is within the confidence interval, or 40 mm, which is not; in any case, whether it falls between 36.8 and 39.0 mm is a matter of fact, not probability.
  • It is not necessarily true that the lengths of 95% of the sampled rods lie within this interval. In this case, it cannot be true: 95% of 25 is not an integer.
  • It is incorrect to say that if we took a second sample, there is a 95% probability that the sample mean length (an estimate of the population mean length) would fall within this interval. In fact, if the true mean length is far from this specific confidence interval, it could be very unlikely that the next sample mean falls within the interval.

Instead, the 95% confidence level means that if we took 100 such samples, we would expect the true population mean to lie within approximately 95 of the calculated intervals.[1][10][11][12]

Comparison withprediction intervals

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A confidence interval is used to estimate a population parameter, such as the mean. For example, the expected value of a fair six-sided die is 3.5. Based on repeated sampling, after computing many 95% confidence intervals, roughly 95% of them will contain 3.5 (and the width of the confidence interval shrinks with sample size).

A prediction interval, on the other hand, provides a range within which a future individual observation is expected to fall with a certain probability. In the case of a single roll of a fair six-sided die, an exact 95% prediction interval does not exist. However, there are exact 95% prediction intervals for rolling a twenty-sided die. One such interval is[1,19]{\displaystyle [1,19]}, since 95% of the time the roll will result in a 19 or less, and the remaining 5% will result in a 20.

The key distinction is that confidence intervals quantify uncertainty in estimating parameters, while prediction intervals quantify uncertainty in forecasting future observations.

Comparison withcredible intervals

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In many common settings, such as estimating the mean of a normal distribution with known variance,[13] confidence intervals coincide with credible intervals under non-informative priors. In such cases, common misconceptions about confidence intervals (e.g. interpreting them as probability statements about the parameter) may yield practically correct conclusions.

Examples of how naïve interpretation of confidence intervals can be problematic

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Confidence procedure for uniform location

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Ten examples of the 50% Welch and Bayesian intervals are shown in contrasting white and gray rows. The examples are sorted top-to-bottom by decreasing distance betweenX1{\displaystyle X_{1}} andX2{\displaystyle X_{2}}.

Welch[14] presented an example which clearly shows the difference between the theory of confidence intervals and other theories of interval estimation (including Fisher'sfiducial intervals and objectiveBayesian intervals). Robinson[15] 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 thatX1,X2{\displaystyle X_{1},X_{2}} are independent observations from auniform(θ−1/2,θ+1/2){\displaystyle (\theta -1/2,\theta +1/2)} distribution. Then the optimal 50% confidence procedure forθ{\displaystyle \theta } is[16]

X¯±{|X1−X2|2if |X1−X2|<1/21−|X1−X2|2if |X1−X2|≥1/2.{\displaystyle {\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}}}

A fiducial or objective Bayesian argument can be used to derive the interval estimate

X¯±1−|X1−X2|4,{\displaystyle {\bar {X}}\pm {\frac {1-|X_{1}-X_{2}|}{4}},}

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 everyθ1≠θ{\displaystyle \theta _{1}\neq \theta }, the probability that the first procedure containsθ1{\displaystyle \theta _{1}} isless than or equal to the probability that the second procedure containsθ1{\displaystyle \theta _{1}}. 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|X1−X2|≥1/2{\displaystyle |X_{1}-X_{2}|\geq 1/2}, intervals from the first procedure areguaranteed to contain the true valueθ{\displaystyle \theta }: 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 thatX1,X2{\displaystyle X_{1},X_{2}} 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 whenX1,X2{\displaystyle X_{1},X_{2}} are far apart and almost 0% coverage whenX1,X2{\displaystyle X_{1},X_{2}} 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 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 forω2

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Steiger[17] suggested a number of confidence procedures for commoneffect size measures inANOVA. Morey et al.[10] point out that several of these confidence procedures, including the one forω2, have the property that as theF statistic becomes increasingly small—indicating misfit with all possible values ofω2—the confidence interval shrinks and can even contain only the single valueω2 = 0; that is, the CI is infinitesimally narrow (this occurs whenp≥1−α/2{\displaystyle p\geq 1-\alpha /2} for a100(1−α)%{\displaystyle 100(1-\alpha )\%} CI).

This behavior is consistent with the relationship between the confidence procedure andsignificance testing: asF 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 ofω2. Hence the interval will be very narrow or even empty (or, by a convention suggested by Steiger, containing only 0). However, this doesnot indicate that the estimate ofω2 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

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Methods for calculating confidence intervals for the binomial proportion appeared from the 1920s.[18][19] The main ideas of confidence intervals in general were developed in the early 1930s,[20][21][22] and the first thorough and general account was given byJerzy Neyman in 1937.[7]

Neyman described the development of the ideas as follows (reference numbers have been changed):[22]

[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.[23] It so happened that, somewhat earlier, Fisher published his first paper[24] 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,[20] 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 probabilitiesa priori. At the same time I mildly suggested that Fisher's approach to the problem involved a minor misunderstanding.

In medical journals, confidence intervals were promoted in the 1970s but only became widely used in the 1980s.[25] By 1988, medical journals were requiring the reporting of confidence intervals.[26]

Confidence interval for specific distributions

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See also

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References

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  1. ^abcHazra, Avijit (2017)."Using the confidence interval confidently".Journal of Thoracic Disease.9 (10):4124–4129.doi:10.21037/jtd.2017.09.14.PMC 5723800.PMID 29268424.
  2. ^abDekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Meester, Ludolf Erwin (2005)."A Modern Introduction to Probability and Statistics".Springer Texts in Statistics.doi:10.1007/1-84628-168-7.ISBN 978-1-85233-896-1.ISSN 1431-875X.
  3. ^"Confidence Intervals".Yale Department of Statistics. Retrieved2025-04-05.
  4. ^Roussas, George G. (1997).A Course in Mathematical Statistics (2nd ed.). Academic Press. p. 397.
  5. ^Rees, D.G. (2001).Essential Statistics, 4th Edition, Chapman and Hall/CRC.ISBN 1-58488-007-4 (Section 9.5)
  6. ^Cox D.R., Hinkley D.V. (1974)Theoretical Statistics, Chapman & Hall, p49, p209
  7. ^abNeyman, J. (1937)."Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability".Philosophical Transactions of the Royal Society A.236 (767):333–380.Bibcode:1937RSPTA.236..333N.doi:10.1098/rsta.1937.0005.JSTOR 91337.
  8. ^Cox D.R., Hinkley D.V. (1974) Theoretical Statistics, Chapman & Hall, pp. 214, 225, 233
  9. ^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.[1]
  10. ^abcMorey, R. D.; Hoekstra, R.; Rouder, J. N.; Lee, M. D.; Wagenmakers, E.-J. (2016)."The Fallacy of Placing Confidence in Confidence Intervals".Psychonomic Bulletin & Review.23 (1):103–123.doi:10.3758/s13423-015-0947-8.PMC 4742505.PMID 26450628.
  11. ^abcGreenland, Sander; Senn, Stephen J.; Rothman, Kenneth J.; Carlin, John B.; Poole, Charles; Goodman, Steven N.; Altman, Douglas G. (April 2016)."Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations".European Journal of Epidemiology.31 (4):337–350.doi:10.1007/s10654-016-0149-3.ISSN 0393-2990.PMC 4877414.PMID 27209009.
  12. ^abTan, Sze Huey; Tan, Say Beng (2010-09-01)."The Correct Interpretation of Confidence Intervals".Proceedings of Singapore Healthcare.19 (3):276–278.doi:10.1177/201010581001900316.ISSN 2010-1058.
  13. ^Bolstad, William M. (2007).Introduction to Bayesian statistics (2nd ed.). Hoboken, N.J: John Wiley. pp. 223–236.ISBN 978-0-470-14115-1.OCLC 122338165.
  14. ^Welch, B. L. (1939)."On Confidence Limits and Sufficiency, with Particular Reference to Parameters of Location".The Annals of Mathematical Statistics.10 (1):58–69.doi:10.1214/aoms/1177732246.JSTOR 2235987.
  15. ^Robinson, G. K. (1975). "Some Counterexamples to the Theory of Confidence Intervals".Biometrika.62 (1):155–161.doi:10.2307/2334498.JSTOR 2334498.
  16. ^Pratt, J. W. (1961). "Book Review: Testing Statistical Hypotheses. by E. L. Lehmann".Journal of the American Statistical Association.56 (293):163–167.doi:10.1080/01621459.1961.10482103.JSTOR 2282344.
  17. ^Steiger, J. H. (2004). "Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis".Psychological Methods.9 (2):164–182.doi:10.1037/1082-989x.9.2.164.PMID 15137887.
  18. ^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
  19. ^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
  20. ^abNeyman, 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)
  21. ^J. Neyman (1935), Ann. Math. Statist. 6(3): 111-116 (September, 1935).https://doi.org/10.1214/aoms/1177732585
  22. ^abNeyman, 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
  23. ^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.
  24. ^Fisher, R. A. (1930). "Inverse Probability".Mathematical Proceedings of the Cambridge Philosophical Society.26 (4):528–535.Bibcode:1930PCPS...26..528F.doi:10.1017/S0305004100016297.hdl:2440/15206.
  25. ^Altman, Douglas G. (1991)."Statistics in medical journals: Developments in the 1980s".Statistics in Medicine.10 (12):1897–1913.doi:10.1002/sim.4780101206.ISSN 1097-0258.PMID 1805317.
  26. ^Gardner, Martin J.; Altman, Douglas G. (1988)."Estimating with confidence".British Medical Journal.296 (6631):1210–1211.doi:10.1136/bmj.296.6631.1210.PMC 2545695.PMID 3133015.

Bibliography

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