WordNet

tracing descent from either the paternal or the maternal line only
involving only one part or side; "unilateral paralysis"; "a unilateral decision" (同)one-sided
used informally as an intensifier; "that is one fine dog"
the smallest whole number or a numeral representing this number; "he has the one but will need a two and three to go with it"; "they had lunch at one" (同)1, I, ace, single, unity
a single person or thing; "he is the best one"; "this is the one I ordered"
used of a single unit or thing; not two or more; "`ane is Scottish" (同)1, i, ane
having the indivisible character of a unit; "a unitary action"; "spoke with one voice" (同)unitary
being a single entity made by combining separate components; "three chemicals combining into one solution"
indefinite in time or position; "he will come one day"; "one place or another"
of the same kind or quality; "two animals of one species"
indicates continuity or persistence or concentration; "his spirit lives on"; "shall I read on?"
(of events) planned or scheduled; "the picnic is on, rain or shine"; "we have nothing on for Friday night"
in a state required for something to function or be effective; "turn the lights on"; "get a load on"
in operation or operational; "left the oven on"; "the switch is in the on position"
remove the stalk of fruits or berries
any projection that resembles the tail of an animal (同)tail_end
the rear part of an aircraft (同)tail assembly, empennage
(usually plural) the reverse side of a coin that does not bear the representation of a persons head
the posterior part of the body of a vertebrate especially when elongated and extending beyond the trunk or main part of the body
a spy employed to follow someone and report their movements (同)shadow, shadower
having a tail of a specified kind; often used in combination

PrepTutorEJDIC

一方の,片側だけの
『1個の』,一人の,単一の;《補語にのみ用いて》1歳の / 《時を表す名詞と共に》『ある』… / 《人名に付けて》…という[名の]人 / 同一の,同じ(the same) / 《theまたは所有格を付けて》たった一つの,唯一の / 《another, [the]otherと対比して》一方の / 〈U〉〈C〉『1つ』,1個,一人,1時,1歳 / 〈C〉(数字の)『1』;1の記号(1,Iなど) / 《the O-》超人的なもの / 〈U〉《しばしばa ~》(攻撃・冗談・うそなどの)一発 / 《前に述べた可算名詞を受けて》『それ』 / 《修飾語を伴わないで》 / 《修飾語句を伴って》 / 《後》に「of+複数名詞・代名詞」を伴ってその名詞・代名詞の》『一つ』,一人 / 《後にanother, [the]otherと対照して》一つ,一方 / 《修飾語句を伴って》(特定の)『人』,者 / 《総称的に》『人』 / 私,自分
(また《文》upon)《位置・場所》…『の上に』,の上で,の上の / (また《文》upon)《運動・動作の方向・対象》…『に』[向かって] / 《話》…を身につけて / (また《文》upon)《近接して場所》…の近くに,…に面して,…の側に / 《特定の日時》…『に』 / 《状態》…『して』,…中で,…すると / (また《文》upon)…『について』,…に関して / 《目的》…『で』,…に / (また《文》upon)…に基づいて,…によって,…の支え(助け)で / 《所属を表して》(委員会・チームなどの)一員で / (また《文》upon)…を軸にして,…で / (また《文》upon)《前後に同じ名詞を並べて》…に加えて,…についで / 『上に』,『身につけて』 / (電気・水道・ガス・ラジオなどが)『ついて』,通じて / 行われて,続けて,従事して / 《時の副詞[句]を伴って》前方へ,ずっと
〈C〉『尾』,しっぽ / 〈C〉『尾に似た物』;(飛行機・車の)尾部 / 〈C〉(…の)『後部』,末端,終り(底)の部分《+of+名》 / 〈C〉コインの裏側 / 《複数形で》燕尾(えんび)服(tail coat) / 〈C〉《俗》しり,けつ / 〈C〉《俗》尾行者 / …‘に'尾をつける / 《話》〈人〉‘を'尾行する / 〈物・音が〉しだいに減ってゆく,消えてゆく《+away(off, out)》

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2014/04/30 18:52:33」(JST)

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A two-tailed test, here the normal distribution.

A one-tailed test, showing the p-value as the size of one tail.

In statistical significance testing, a one-tailed test or two-tailed test are alternative ways of computing the statistical significance of a data set in terms of a test statistic, depending on whether only one direction is considered extreme (and unlikely) or both directions are considered equally likely. Alternative names are one-sided and two-sided tests; the terminology "tail" is because the extremes of distributions are often small, as in the normal distribution or "bell curve", pictured above right.

If the test statistic is always positive (or zero), only the one-tailed test is generally applicable, while if the test statistic can assume positive and negative values, both the one-tailed and two-tailed test are of use.

Applications

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.

In the approach of Ronald Fisher, the null hypothesis H₀ 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.^[1] 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 0.5), if the data is in the direction opposite that specified by the test.

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, one is generally interested in whether a treatment results in outcomes that are better than chance, and thus uses a one-tailed test; using a two-tailed test 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.

Coin flipping example

Main article: Checking whether a coin is fair

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) 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 chance of occurring, and thus would have 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 and this would not be significant (not rejecting the null hypothesis) if using 0.05 as the cutoff.

History

p-value of chi-squared distribution for different number of degrees of freedom

The p-value was introduced by Karl Pearson in (Pearson 1900) 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.

Normal distribution, showing two tails

The distinction between one-tailed and two-tailed tests was popularized by Ronald Fisher in the influential (Fisher 1925), 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:^[2]

Some little 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).^[3] 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.

Relation to hypothesis testing

Further information: Statistical hypothesis testing

p-values and one-tailed/two-tailed tests are a concept in the significance testing of Fisher, which only uses a null hypothesis, and either rejects it or not. p-values are not used in the hypothesis testing of Jerzy Neyman and Egon Pearson, which instead compares the null hypothesis to an alternative hypothesis, and chooses between them. However, these approaches are frequently confused and conflated – see statistical hypothesis testing – and thus p-values and one-tailed or two-tailed tests of significance may be incorrectly used in Neyman–Pearson-style hypothesis testing.

This is a mistaken interpretation, but it is a common mistake; this results in a confusing mixture of terminology, as follows – note that "significance level" is used in different senses in Fisher and in Neyman–Pearson, while "alternative hypothesis" is used only in Neyman–Pearson. In this context a one-tailed test is interpreted as using an "alternative hypothesis" that some parameter is greater than it is in the null hypothesis (or less), while a two-tailed test is interpreted as using as "alternative hypothesis" that the parameter is different from what it is in the null hypothesis. For example, if the null hypothesis is that the mean is some value then the one-tailed test "corresponds to" the alternative hypothesis (or ), while the two-tailed test "corresponds to" the alternative hypothesis While Fisher rejected the notion of an alternative hypothesis, Neyman accused him of subconsciously harboring an alternative hypothesis when choosing how to evaluate the null hypothesis, of which this one-tailed/two-tailed choice is one example.

Further, since in the Neyman–Pearson approach "significance levels" (in the sense of false positive/type I error rate, rather than in the Fisher sense of p-value of the test statistic), which are denoted by α, share the "significance level" name and are also conventionally 0.05, theses two concepts may be confused. In this case the cut-offs in the tails are denoted by α, and then compared with the p-value of the data, using α/2 at each end in the two-tailed test. This is incorrect – p-values are not simply related to false positives and cannot be compared with α, as discussed at p-value – but this notation is very common.

Specific tests

If the test statistic follows a Student's t distribution in the null hypothesis – which is common where the underlying variable follows a normal distribution with unknown scaling factor, then the test is referred to as a one-tailed or two-tailed t-test. If the test is performed using the actual population mean and variance, rather than an estimate from a sample, it would be called a one-tailed or two-tailed or Z-test.

The statistical tables for t and for Z provide critical values for both one- and two-tailed tests. That is, they provide the critical values that cut off an entire region at one or the other end of the sampling distribution as well as the critical values that cut off the regions (of half the size) at both ends of the sampling distribution.

References

^ John E. Freund, (1984) Modern Elementary Statistics, sixth edition. Prentice hall. ISBN 0-13-593525-3 (Section "Inferences about Means", chapter "Significance Tests", page 289.)
^ Fisher 1925, p. 47.
^ Fisher 1971, Section 7. The Test of Significance.

Pearson, Karl (1900). "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". Philosophical Magazine Series 5 50 (302): 157–175. doi:10.1080/14786440009463897. edit
Fisher, Ronald (1925). Statistical Methods for Research Workers. Edinburgh: Oliver & Boyd. ISBN 0-05-002170-2. edit
Fisher, Ronald A. (1971) [1935]. The Design of Experiments (9th ed.). Macmillan. ISBN 0-02-844690-9. edit

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