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 (where p is the confidence level), a confidence interval 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 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 regions generalise the confidence interval concept to deal with multiple quantities. 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 applied practice, confidence intervals are typically stated at the 95% confidence level. However, when presented graphically, confidence intervals can show several confidence levels, for example 50%, 95% and 99%.
Meaning and interpretation
How to understand confidence intervals
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." Note that this need not be repeated sampling from the same population, just repeated sampling.
- 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 probability associated with a confidence interval may also be considered from a pre-experiment 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.
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.
Users of Bayesian methods, if they produced an interval estimate, would by contrast want to say "My degree of belief that the parameter is in fact in this interval is 90%". 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.
Meaning of the term confidence
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.
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 researcher conducting a study performs 40 statistical tests at 95% confidence, she can expect about two of the tests to return false positives. If she in fact finds 3 links, the confidence associated with those links 'as the result of the survey' is actually about 32%; it's what she should expect to see two-thirds of the time.
Reference;
http://en.wikipedia.org/wiki/Confidence_interval
