Editing Confidence interval (section)

=== Common misunderstandings ===
[[File:Neyman_Construction_Confidence_Intervals.png|thumb|A plot of 50 confidence intervals from 50 samples generated from a normal distribution]]
Confidence intervals and levels are frequently misunderstood, and published studies have shown that even professional scientists often misinterpret them.<ref>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. [http://www.ejwagenmakers.com/inpress/HoekstraEtAlPBR.pdf]</ref>

* A 95% confidence level does not mean that for a given realized interval there is a 95% probability that the population parameter lies within the interval.<ref name="Morey" /><ref name=":2"/>
* A 95% confidence level does not mean that 95% of the sample data lie within the confidence interval.<ref name="using_confidently"/>
* A 95% confidence level does not mean that there is a 95% probability of the parameter estimate from a repeat of the experiment falling within the confidence interval computed from a given experiment.<ref name=":2">{{Cite journal |last1=Greenland |first1=Sander |last2=Senn |first2=Stephen J. |last3=Rothman |first3=Kenneth J. |last4=Carlin |first4=John B. |last5=Poole |first5=Charles |last6=Goodman |first6=Steven N. |last7=Altman |first7=Douglas G. |date=April 2016 |title=Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations |journal=European Journal of Epidemiology |volume=31 |issue=4 |pages=337–350 |doi=10.1007/s10654-016-0149-3 |issn=0393-2990 |pmc=4877414 |pmid=27209009}}</ref>

For example, suppose a factory produces metal rods. A random sample of 25 rods gives a 95% confidence interval for the population mean length of 36.8 to 39.0&nbsp;mm.<ref name="Tan">{{Cite journal |last1=Tan |first1=Sze Huey |last2=Tan |first2=Say Beng |date=2010-09-01 |title=The Correct Interpretation of Confidence Intervals |journal=Proceedings of Singapore Healthcare |language=EN |volume=19 |issue=3 |pages=276–278 |doi=10.1177/201010581001900316 |issn=2010-1058|doi-access=free }}</ref>
* It is incorrect to say that there is a 95% probability that the true population mean lies within this interval, because the true mean is fixed, not random. For example, it might be 37&nbsp;mm, which is within the confidence interval, or 40&nbsp;mm, which is not; in any case, whether it falls between 36.8 and 39.0&nbsp;mm is a matter of fact, not probability.
* It is not necessarily true that the lengths of 95% of the sampled rods lie within this interval. In this case, it cannot be true: 95% of 25 is not an integer.
* It is incorrect to say that if we took a second sample, there is a 95% probability that the sample mean length (an estimate of the population mean length) would fall within this interval. In fact, if the true mean length is far from this specific confidence interval, it could be very unlikely that the next sample mean falls within the interval.
Instead, the 95% confidence level means that if we took 100 such samples, we would expect the true population mean to lie within approximately 95 of the calculated intervals.<!-- refs justifying the conclusions; [[WP:SCG]] --><ref name="using_confidently"/><ref name="Morey" /><ref name=":2"/><ref name="Tan"/>