![DisNormal06](https://upload.wikimedia.org/wikipedia/commons/9/96/DisNormal06.svg)
In statistical
significance testing
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...
, a one-tailed test and a two-tailed test are alternative ways of computing the
statistical significance
In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis (simply by chance alone). More precisely, a study's defined significance level, denoted by \alpha, is the p ...
of a
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
inferred from a data set, in terms of a
test statistic
A test statistic is a statistic (a quantity derived from the sample) used in statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specif ...
. A two-tailed test is appropriate if the estimated value is greater or less than a certain range of values, for example, whether a test taker may score above or below a specific range of scores. This method is used for
null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...
testing and if the estimated value exists in the critical areas, the alternative hypothesis is accepted over the null hypothesis.
A one-tailed test is appropriate if the estimated value may depart from the reference value in only one direction, left or right, but not both. An example can be whether a machine produces more than one-percent defective products. In this situation, if the estimated value exists in one of the one-sided critical areas, depending on the direction of interest (greater than or less than), the alternative hypothesis is accepted over the null hypothesis. Alternative names are one-sided and two-sided tests; the terminology "tail" is used because the extreme portions of distributions, where observations lead to rejection of the null hypothesis, are small and often "tail off" toward zero as in the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, colored in yellow, or "bell curve", pictured on the right and colored in green.
Applications
One-tailed tests are used for asymmetric distributions that have a single tail, such as the
chi-squared distribution
In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squa ...
, which are common in measuring
goodness-of-fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...
, or for one side of a distribution that has two tails, such as the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, 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
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
, the
null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...
H
0 will be rejected when the
''p''-value of the
test statistic
A test statistic is a statistic (a quantity derived from the sample) used in statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specif ...
is sufficiently extreme (vis-a-vis the test statistic's
sampling distribution
In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic. If an arbitrarily large number of samples, each involving multiple observations (data points), were s ...
) and thus judged unlikely to be the result of chance. This is usually done by comparing the resulting p-value with the specified significance level, denoted by
, when computing the statistical significance of a parameter''.'' 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. One may report that the left or right tail probability as the one-tailed p-value, which ultimately corresponds to the direction in which the test statistic deviates from H
0. 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. When provided a significance level
, the critical regions would exist on the two tail ends of the distribution with an area of
each for a two-tailed test. Alternatively, the critical region would solely exist on the single tail end with an area of
for a one-tailed test. For a given significance level in a two-tailed test for a test statistic, the corresponding one-tailed tests for the same test statistic 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
) if the data is in the direction opposite of the critical region specified by the test.
For example, if
flipping a coin
Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking which side is showing when it lands, in order to choose between two alternatives, heads or tails, sometimes used to resolve a dispute betwe ...
, 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
In the design of experiments in statistics, the lady tasting tea is a randomized experiment devised by Ronald Fisher and reported in his book ''The Design of Experiments'' (1935). The experiment is the original exposition of Fisher's notion of a ...
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
In coin flipping, the
null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...
is a sequence of
Bernoulli trial
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is c ...
s 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
The sample mean (or "empirical mean") and the sample covariance are statistics computed from a Sample (statistics), sample of data on one or more random variables.
The sample mean is the average value (or mean, mean value) of a sample (statistic ...
(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, (5 consecutive flips with 2 outcomes - ((1/2)^5 =1/32). This would have
and would be significant (rejecting the null hypothesis) if the test was analyzed at a significance level of
(the significance level corresponding to the cutoff bound). 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). As a result, the ''p''-value would be
and this would not be significant (not rejecting the null hypothesis) if the test was analyzed at a significance level of
.
History
![Chi-square distributionCDF-English](https://upload.wikimedia.org/wikipedia/commons/8/8e/Chi-square_distributionCDF-English.png)
The ''p''-value was introduced by
Karl Pearson
Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university st ...
in the
Pearson's chi-squared test
Pearson's chi-squared test (\chi^2) is a statistical test applied to sets of categorical data to evaluate how likely it is that any observed difference between the sets arose by chance. It is the most widely used of many chi-squared tests (e.g., ...
, 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
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...
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.
![Standard deviation diagram](https://upload.wikimedia.org/wikipedia/commons/8/8c/Standard_deviation_diagram.svg)
The distinction between one-tailed and two-tailed tests was popularized by
Ronald Fisher
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
in the influential book
Statistical Methods for Research Workers
''Statistical Methods for Research Workers'' is a classic book on statistics, written by the statistician R. A. Fisher. It is considered by some to be one of the 20th century's most influential books on statistical methods, together with his ''The ...
,
where he applied it especially to the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, 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:
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
''The Design of Experiments'' is a 1935 book by the English statistician Ronald Fisher about the design of experiments and is considered a foundational work in experimental design. Among other contributions, the book introduced the concept of the ...
'' (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.
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
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
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
''Z''-test.
The
statistical tables for ''t'' and for ''Z'' provide
critical values
Critical value may refer to:
*In differential topology, a critical value of a differentiable function between differentiable manifolds is the image (value of) ƒ(''x'') in ''N'' of a critical point ''x'' in ''M''.
*In statistical hypothe ...
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.
See also
*
Paired difference test In statistics, a paired difference test is a type of location test that is used when comparing two sets of measurements to assess whether their expected value, population means differ. A paired difference test uses additional information about the ...
, when two samples are being compared
References
{{DEFAULTSORT:One-Tailed Test
Statistical tests