<P> One - tailed tests are used for asymmetric distributions that have a single tail, such as the chi - squared distribution, which are common in measuring goodness - of - fit, or for one side of a distribution that has two tails, such as the normal distribution, which is common in estimating location; this corresponds to specifying a direction . Two - tailed tests are only applicable when there are two tails, such as in the normal distribution, and correspond to considering either direction significant . </P> <P> In the approach of Ronald Fisher, the null hypothesis H will be rejected when the p - value of the test statistic is sufficiently extreme (vis - a-vis the test statistic's sampling distribution) and thus judged unlikely to be the result of chance . In a one - tailed test, "extreme" is decided beforehand as either meaning "sufficiently small" or meaning "sufficiently large"--values in the other direction are considered not significant . In a two - tailed test, "extreme" means "either sufficiently small or sufficiently large", and values in either direction are considered significant . For a given test statistic there is a single two - tailed test, and two one - tailed tests, one each for either direction . Given data of a given significance level in a two - tailed test for a test statistic, in the corresponding one - tailed tests for the same test statistic it will be considered either twice as significant (half the p - value), if the data is in the direction specified by the test, or not significant at all (p - value above 0.05), if the data is in the direction opposite that specified by the test . </P> <P> For example, if flipping a coin, testing whether it is biased towards heads is a one - tailed test, and getting data of "all heads" would be seen as highly significant, while getting data of "all tails" would be not significant at all (p = 1). By contrast, testing whether it is biased in either direction is a two - tailed test, and either "all heads" or "all tails" would both be seen as highly significant data . In medical testing, while one is generally interested in whether a treatment results in outcomes that are better than chance, thus suggesting a one - tailed test; a worse outcome is also interesting for the scientific field, therefore one should use a two - tailed test that corresponds instead to testing whether the treatment results in outcomes that are different from chance, either better or worse . In the archetypal lady tasting tea experiment, Fisher tested whether the lady in question was better than chance at distinguishing two types of tea preparation, not whether her ability was different from chance, and thus he used a one - tailed test . </P> <P> In coin flipping, the null hypothesis is a sequence of Bernoulli trials with probability 0.5, yielding a random variable X which is 1 for heads and 0 for tails, and a common test statistic is the sample mean (of the number of heads) X _̄ . (\ displaystyle (\ bar (X)).) If testing for whether the coin is biased towards heads, a one - tailed test would be used--only large numbers of heads would be significant . In that case a data set of five heads (HHHHH), with sample mean of 1, has a 1 / 32 = 0.03125 ≈ 0.03 (\ displaystyle 1 / 32 = 0.03125 \ approx 0.03) chance of occurring, (5 consecutive flips with 2 outcomes - ((1 / 2) ^ 5 = 1 / 32), and thus would have p ≈ 0.03 (\ displaystyle p \ approx 0.03) and would be significant (rejecting the null hypothesis) if using 0.05 as the cutoff . However, if testing for whether the coin is biased towards heads or tails, a two - tailed test would be used, and a data set of five heads (sample mean 1) is as extreme as a data set of five tails (sample mean 0), so the p - value would be 2 / 32 = 0.0625 ≈ 0.06 (\ displaystyle 2 / 32 = 0.0625 \ approx 0.06) and this would not be significant (not rejecting the null hypothesis) if using 0.05 as the cutoff . </P>

One sided or two sided fisher's exact test