# Confidence interval

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The Confidence interval (CI) is a "range of values for a variable of interest, e.g., a rate, constructed so that this range has a specified probability of including the true value of the variable."[1]

In large samples, calculations for the CI for rates and proportions may be based on the normal distribution.[2][3]

The equation using the normal distribution is:[4]

{\displaystyle {\begin{aligned}{\mbox{CI lower limit}}&={\bar {X}}-{\mbox{Z}}*{\mbox{SE}}\\{\mbox{CI upper limit}}&={\bar {X}}+{\mbox{Z}}*{\mbox{SE}}\end{aligned}}}

Where

${\displaystyle {\bar {X}}={\mbox{sample mean}}\,\!}$
${\displaystyle {\mbox{Z}}=1.96\,({\mbox{if}}\,\alpha =0.05\,{\mbox{for}}\,95\%\,{\mbox{confidence intervals}})\,\!}$
${\displaystyle {\mbox{Z}}=3.29\,({\mbox{if}}\,\alpha =0.001\,{\mbox{for}}\,99.9\%\,{\mbox{confidence intervals}})\,\!}$
${\displaystyle {\mbox{SE}}={\mbox{standard error}}={\frac {\sigma }{\sqrt {n}}}}$

${\displaystyle \sigma ={\mbox{standard deviation}}\,\!}$

For small samples, calculations should be made using the binomial distribution or the Poisson distribution.

## References

1. Anonymous (2022), Confidence interval (English). Medical Subject Headings. U.S. National Library of Medicine.
2. Fleiss, Joseph L. (1973). Statistical methods for rates and proportions. New York: Wiley, 13. ISBN 0-471-26370-2.
3. Cochran, William Cox; Snedecor, George W. (1980). Statistical methods. Ames: Iowa State University Press, 118. ISBN 0-8138-1560-6.
4. Gardner MJ, Altman DG (March 1986). "Confidence intervals rather than P values: estimation rather than hypothesis testing". Br Med J (Clin Res Ed) 292 (6522): 746–50. PMID 3082422. [e]