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Tolerance interval
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==Relation to other intervals== {{further|Interval estimation}} "In the parameters-known case, a 95% tolerance interval and a 95% [[prediction interval]] are the same."<ref name="Ryan2007">{{cite book|author=Thomas P. Ryan|title=Modern Engineering Statistics|url=https://books.google.com/books?id=aZn7XNphKcgC&pg=PA222|access-date=22 February 2013|date=22 June 2007|publisher=John Wiley & Sons|isbn=978-0-470-12843-5|pages=222β}}</ref> If we knew a population's exact parameters, we would be able to compute a range within which a certain proportion of the population falls. For example, if we know a population is [[normal distribution|normally distributed]] with [[mean]] <math>\mu</math> and [[standard deviation]] <math>\sigma</math>, then the interval <math>\mu \pm 1.96\sigma</math> includes 95% of the population (1.96 is the [[z-score]] for 95% coverage of a normally distributed population). However, if we have only a sample from the population, we know only the [[sample mean]] <math>\hat{\mu}</math> and sample standard deviation <math>\hat{\sigma}</math>, which are only estimates of the true parameters. In that case, <math>\hat{\mu} \pm 1.96\hat{\sigma}</math> will not necessarily include 95% of the population, due to variance in these estimates. A tolerance interval bounds this variance by introducing a confidence level <math>\gamma</math>, which is the confidence with which this interval actually includes the specified proportion of the population. For a normally distributed population, a z-score can be transformed into a "''k'' factor" or '''tolerance factor'''<ref>{{cite web|year=2014|title=Statistical interpretation of data β Part 6: Determination of statistical tolerance intervals|url=https://www.iso.org/standard/57191.html|publisher=ISO 16269-6|page=2}}</ref> for a given <math>\gamma</math> via lookup tables or several approximation formulas.<ref>{{cite book | chapter = Tolerance intervals for a normal distribution | chapter-url = http://www.itl.nist.gov/div898/handbook/prc/section2/prc263.htm | title = Engineering Statistics Handbook | publisher = NIST/Sematech | year = 2010 | access-date = 2011-08-26}}</ref> "As the [[Degree of freedom (statistics)|degrees of freedom]] approach infinity, the prediction and tolerance intervals become equal."<ref name=Example2006>{{Cite journal | last1 = De Gryze | first1 = S. | last2 = Langhans | first2 = I. | last3 = Vandebroek | first3 = M. | doi = 10.1016/j.chemolab.2007.03.002 | title = Using the correct intervals for prediction: A tutorial on tolerance intervals for ordinary least-squares regression | journal = Chemometrics and Intelligent Laboratory Systems | volume = 87 | issue = 2 | pages = 147 | year = 2007 }}</ref> The tolerance interval is less widely known than the [[confidence interval]] and [[prediction interval]], a situation some educators have lamented, as it can lead to misuse of the other intervals where a tolerance interval is more appropriate.<ref name=vardeman>{{cite journal | author = Stephen B. Vardeman | title = What about the Other Intervals? | journal = The American Statistician | volume = 46 | issue = 3 | year = 1992 | pages = 193β197 | jstor = 2685212 | doi=10.2307/2685212}}</ref><ref name=nelson>{{cite web | author = Mark J. Nelson | title = You might want a tolerance interval | url = http://www.kmjn.org/notes/tolerance_intervals.html | date = 2011-08-14 | access-date = 2011-08-26}}</ref> The tolerance interval differs from a [[confidence interval]] in that the confidence interval bounds a single-valued population parameter (the [[mean]] or the [[variance]], for example) with some confidence, while the tolerance interval bounds the range of data values that includes a specific proportion of the population. Whereas a confidence interval's size is entirely due to [[sampling error]], and will approach a zero-width interval at the true population parameter as sample size increases, a tolerance interval's size is due partly to sampling error and partly to actual variance in the population, and will approach the population's probability interval as sample size increases.<ref name=vardeman /><ref name=nelson /> The tolerance interval is related to a [[prediction interval]] in that both put bounds on variation in future samples. However, the prediction interval only bounds a single future sample, whereas a tolerance interval bounds the entire population (equivalently, an arbitrary sequence of future samples). In other words, a prediction interval covers a specified proportion of a population ''on average'', whereas a tolerance interval covers it ''with a certain confidence level'', making the tolerance interval more appropriate if a single interval is intended to bound multiple future samples.<ref name=nelson /><ref name=Krishnamoorthy>{{cite book | author = K. Krishnamoorthy| title = Statistical Tolerance Regions: Theory, Applications, and Computation | publisher = John Wiley and Sons | year = 2009 | isbn = 978-0-470-38026-0 | pages = 1β6}}</ref>
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