Editing One- and two-tailed tests (section)

== History ==
[[File:Chi-square distributionCDF-English.png|thumb|''p''-value of [[chi-squared distribution]] for different number of degrees of freedom]]
The ''p''-value was introduced by [[Karl Pearson]]<ref>{{Cite journal | last = Pearson | first = Karl | author-link = Karl Pearson | title = On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling | doi = 10.1080/14786440009463897 | journal = Philosophical Magazine |series=Series 5 | volume = 50 | issue = 302 | pages = 157–175 | year = 1900 | url = http://www.economics.soton.ac.uk/staff/aldrich/1900.pdf }}</ref> in the [[Pearson's chi-squared test]], where he defined P (original notation) as the probability that the statistic would be at or above a given level. This is a one-tailed definition, and the chi-squared distribution is asymmetric, only assuming positive or zero values, and has only one tail, the upper one. It measures [[goodness of fit]] of data with a theoretical distribution, with zero corresponding to exact agreement with the theoretical distribution; the ''p''-value thus measures how likely the fit would be this bad or worse.

[[File:Standard deviation diagram.svg|thumb|[[Normal distribution]], showing two tails]]
The distinction between one-tailed and two-tailed tests was popularized by [[Ronald Fisher]] in the influential book [[Statistical Methods for Research Workers]],<ref name=fisher>{{cite book |title=Statistical Methods for Research Workers |last=Fisher |first=Ronald |author-link1=Ronald Fisher |year=1925 |publisher=Oliver & Boyd |location=Edinburgh |isbn=0-05-002170-2 |title-link=Statistical Methods for Research Workers }}</ref> where he applied it especially to the [[normal distribution]], which is a symmetric distribution with two equal tails. The normal distribution is a common measure of location, rather than goodness-of-fit, and has two tails, corresponding to the estimate of location being above or below the theoretical location (e.g., sample mean compared with theoretical mean). In the case of a symmetric distribution such as the normal distribution, the one-tailed ''p''-value is exactly half the two-tailed ''p''-value:<ref name=fisher />
{{quotation|
Some confusion is sometimes introduced by the fact that in some cases we wish to know the probability that the deviation, known to be positive, shall exceed an observed value, whereas in other cases the probability required is that a deviation, which is equally frequently positive and negative, shall exceed an observed value; the latter probability is always half the former.
|[[Ronald Fisher]]|[[Statistical Methods for Research Workers]]}}

Fisher emphasized the importance of measuring the tail – the observed value of the test statistic and all more extreme – rather than simply the probability of specific outcome itself, in his ''[[The Design of Experiments]]'' (1935).<ref>{{cite book |title=The Design of Experiments |edition=9th |last=Fisher |first=Ronald A. |author-link=Ronald Fisher |orig-year=1935 |year=1971 |publisher=Macmillan |isbn=0-02-844690-9 }}</ref> He explains this as because a ''specific'' set of data may be unlikely (in the null hypothesis), but more extreme outcomes likely, so seen in this light, the specific but not extreme unlikely data should not be considered significant.