Confidence intervals (CIs)

These are used to indicate the uncertainty around an estimated or observed association, difference, or effect. Informally, a CI indicates the range of possible true values that are not too inconsistent with the observed data. Thus, we usually claim to have strong evidence against any value outside the CI, but not those inside it. In particular, negative results (claims for evidence against any substantial association, difference, or effect) should be supported by noting that one or both ends of the CI are too small to be of substantial biological, clinical, or scientific importance. See Estimation versus Hypothesis Testing and Limitations of P-values.

Although alternative levels are possible, 95% confidence intervals are most commonly used. These are (by definition) constructed by a method that will include the true value 95% percent of the time. Note, however, that the "95%" property pertains to the method and not to any specific interval that it produces. It is therefore incorrect to claim that there is a 95% chance that the true value falls within your CI. So use less specific, more informal phrasing when interpreting CIs. See also Bayesian vs Frequentist Statistics.

Overview

In statistics, a confidence interval (CI) is an interval 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.

For example, a CI can be used to describe how reliable survey results are. In a poll of election voting-intentions, 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 estimates more generally have applications across the whole range of quantitative studies.

Brief explanation

For a given proportion p (or confidence level), a CI for a population parameter is an 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 semi-infinite intervals, and it is possible that an outcome of a confidence-interval calculation could be the set of all values from minus infinity to plus infinity.

Confidence intervals are the most prevalent form of interval estimation. Interval estimates may be contrasted with point 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.

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.

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 regions. Such regions can indicate not only the extent of likely estimation 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. See also confidence bands.

In modern applied practice, confidence intervals are often stated at the 95% level.^[2] However, when presented graphically, confidence intervals or confidence regions may be shown for several confidence levels, for example 50%, 90% and 99%.

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