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Confidence interval

A confidence interval is a pair of lower and upper bounds likely to contain, in repeated sampling, the true value of an unknown statistical parameter, such as a population mean.[1]

A confidence interval is based on a pre-defined coverage probability, called the confidence level, which is typically 95%.[2] A 95% confidence level does not imply a 95% probability that the true parameter lies within a particular interval calculated from a sample.[2] The confidence level instead reflects the long-run performance of the method used to generate the interval.[2]

There are many ways of calculating confidence intervals, and the best method depends on the situation.[3] Two widely applicable methods are bootstrapping and the central limit theorem.[3] For example, a random sample of 25 metal rods might give a 95% confidence interval of 36.8 to 39.0 mm for the population mean length.[4] The 95% confidence level then means that if 100 such samples were taken, the true population mean would be expected to lie within approximately 95 of the calculated intervals.[5]

Interpreting confidence intervals is especially tricky when multiple groups are involved.[6] Two common misconceptions are that two groups are significantly different if their intervals just touch, and that they differ when one group's interval excludes the other's mean.[6] Neither of these is true in general, but even experienced researchers often make these mistakes.[6]

Why it matters to investors

Confidence intervals and levels are frequently misunderstood, and published studies have shown that even professional scientists often misinterpret them.[7] By 1988, medical journals were requiring the reporting of confidence intervals.[8]

Related: P-value, Randomized controlled trial

Sources

  1. Hazra, 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 . ncbi.nlm.nih.gov (via Wikipedia: Confidence interval)
  2. Dekking, 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 . {{ cite journal }} : CS1 maint: periodical has ISBN ( link ) link.springer.com; "Confidence Intervals" . Yale Department of Statistics . Retrieved 2025-04-05 . stat.yale.edu (via Wikipedia: Confidence interval)
  3. Dekking, 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 . {{ cite journal }} : CS1 maint: periodical has ISBN ( link ) link.springer.com (via Wikipedia: Confidence interval)
  4. Tan, 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 . doi.org (via Wikipedia: Confidence interval)
  5. Hazra, 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 . ncbi.nlm.nih.gov; Morey, 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 . ncbi.nlm.nih.gov; Greenland, 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 . ncbi.nlm.nih.gov; Tan, 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 . doi.org (via Wikipedia: Confidence interval)
  6. Belia, Sarah; Fidler, Fiona; Williams, Jennifer; Cumming, Geoff (December 2005). "Researchers misunderstand confidence intervals and standard error bars" . Psychological Methods . 10 (4): 389– 396. doi : 10.1037/1082-989X.10.4.389 . ISSN 1082-989X . PMID 16392994 . pubmed.ncbi.nlm.nih.gov (via Wikipedia: Confidence interval)
  7. 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. ejwagenmakers.com (via Wikipedia: Confidence interval)
  8. 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 . doi.org (via Wikipedia: Confidence interval)

Checked 16 October 2026. How we check sources