Confidence interval

Informally, in frequentist statistics, a confidence interval (CI) is an interval which is expected to typically contain the parameter being estimated. More specifically, given a confidence level $\gamma$ (95% and 99% are typical values), a CI is a random interval which contains the parameter being estimated $\gamma$ % of the time.^[1]^[2] The confidence level, degree of confidence or confidence coefficient represents the long-run proportion of CIs (at the given confidence level) that theoretically contain the true value of the parameter; this is tantamount to the nominal coverage probability. For example, out of all intervals computed at the 95% level, 95% of them should contain the parameter's true value.^[3]

Factors affecting the width of the CI include the sample size, the variability in the sample, and the confidence level.^[4] All else being the same, a larger sample produces a narrower confidence interval, greater variability in the sample produces a wider confidence interval, and a higher confidence level produces a wider confidence interval.^[5]

The confidence interval can be expressed in terms of a in repeated samples (or in resampling): "Were this procedure to be repeated on numerous samples, the proportion of calculated 95% confidence intervals that encompassed the true value of the population parameter would tend toward 95%."^[9]

long-run frequency

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% that the 95% confidence interval calculated from a given future sample will cover the true value of the population parameter."^[10] This essentially reframes the "repeated samples" interpretation as a probability rather than a frequency.
See also: Neyman construction

probability

The confidence interval can be expressed in terms of statistical significance, e.g.: "The 95% confidence interval represents values that are not different from the point estimate at the .05 level."^[11]

statistically significantly

CLs upper limits (particle physics)

68–95–99.7 rule

an interval estimate for a curve

Confidence band

Confidence distribution

a higher dimensional generalization

Confidence region

– measure of belief strength used in statistics

Credence (statistics)

a Bayesian alternative for interval estimation

Credible interval

Cumulative distribution function-based nonparametric confidence interval

– Graphical representations of the variability of data

Error bar

– Data analysis approach in frequentist statistics

Estimation statistics

the CI halfwidth

Margin of error

– Function of the observed sample results

p-value

an interval estimate for a random variable

Prediction interval

Probable error

– Statistical indicators of the deviation of a sample

Robust confidence intervals

The Exploratory Software for Confidence Intervals tutorial programs that run under Excel

Confidence interval calculators for , Regression Coefficients, and Regression Intercepts

R-Squares

"Confidence Interval". MathWorld.

Weisstein, Eric W.

Many resources for teaching statistics including Confidence Intervals.

CAUSEweb.org

An interactive introduction to Confidence Intervals

by Eric Schulz, the Wolfram Demonstrations Project.

Confidence Intervals: Confidence Level, Sample Size, and Margin of Error

Archived 2016-08-09 at the Wayback Machine. Straightforward description with examples and what to do about small sample sizes or rates near 0.