<P> The p - value was introduced by Karl Pearson 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 . </P> <P> The distinction between one - tailed and two - tailed tests was popularized by Ronald Fisher in the influential book Statistical Methods for Research Workers, 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: </P> <P> 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 . </P> <P> 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). 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 . </P>

Two tailed and one tailed test of hypothesis