# One- and two-tailed tests

**one-tailed testtwo-tailed testone-sidedtwo-sided testOne-sided testtwo-sidedone-tailed and two-tailedone-tailed hypothesisone-tailed or two-tailedone-tailed or two-tailed test**

In statistical significance testing, a one-tailed test and a two-tailed test are alternative ways of computing the statistical significance of a parameter inferred from a data set, in terms of a test statistic.wikipedia

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### Statistical significance

**statistically significantsignificantsignificance level**

In statistical significance testing, a one-tailed test and a two-tailed test are alternative ways of computing the statistical significance of a parameter inferred from a data set, in terms of a test statistic.

These 5% can be allocated to one side of the sampling distribution, as in a one-tailed test, or partitioned to both sides of the distribution, as in a two-tailed test, with each tail (or rejection region) containing 2.5% of the distribution.

### Test statistic

**Common test statisticst''-test of test statistics**

In statistical significance testing, a one-tailed test and a two-tailed test are alternative ways of computing the statistical significance of a parameter inferred from a data set, in terms of a test statistic. In the approach of Ronald Fisher, the null hypothesis H 0 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.

Using one of these sampling distributions, it is possible to compute either a one-tailed or two-tailed p-value for the null hypothesis that the coin is fair.

### Null hypothesis

**nullnull hypotheseshypothesis**

This method is used for null hypothesis testing and if the estimated value exists in the critical areas, the alternative hypothesis is accepted over the null hypothesis. In the approach of Ronald Fisher, the null hypothesis H 0 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 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) \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.

The choice of null hypothesis (H 0 ) and consideration of directionality (see "one-tailed test") is critical.

### P-value

**p''-valuepp''-values**

In the approach of Ronald Fisher, the null hypothesis H 0 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.

Thus computing a p-value requires a null hypothesis, a test statistic (together with deciding whether the researcher is performing a one-tailed test or a two-tailed test), and data.

### Z-test

**Z''-teststandardized testingStouffer Z**

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.

which one-tailed and two-tailed p-values can be calculated as Φ(−Z) (for upper/right-tailed tests), Φ(Z) (for lower/left-tailed tests) and 2Φ(−|Z|) (for two-tailed tests) where Φ is the standard normal cumulative distribution function.

### Student's t-test

**t-testt''-testStudent's ''t''-test**

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.

Each of these statistics can be used to carry out either a one-tailed or two-tailed test.

### Paired difference test

**matching methodpaired samples**

* Paired difference test, when two samples are being compared

The power of the unpaired, one-sided test carried out at level

### Statistical hypothesis testing

**hypothesis testingstatistical teststatistical tests**

In statistical significance testing, a one-tailed test and a two-tailed test are alternative ways of computing the statistical significance of a parameter inferred from a data set, in terms of a test statistic.

### Parameter

**parametersparametricargument**

In statistical significance testing, a one-tailed test and a two-tailed test are alternative ways of computing the statistical significance of a parameter inferred from a data set, in terms of a test statistic.

### Normal distribution

**normally distributedGaussian distributionnormal**

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. 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, colored in yellow, or "bell curve", pictured on the right and colored in green. 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. 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.

### Chi-squared distribution

**chi-squaredchi-square distributionchi square distribution**

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.

### Ronald Fisher

**R.A. FisherR. A. FisherFisher**

In the approach of Ronald Fisher, the null hypothesis H 0 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. 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.

### Sampling distribution

**finite sample distributiondistributionsampling**

In the approach of Ronald Fisher, the null hypothesis H 0 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.

### Lady tasting tea

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.

### Bernoulli trial

**Bernoulli trialsBernoulli random variablesBernoulli-distributed**

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) \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.

### Sample mean and covariance

**sample meansample covariancesample covariance matrix**

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) \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.

### Karl Pearson

**PearsonPearson, KarlCarl Pearson**

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.

### Pearson's chi-squared test

**chi-square statisticPearson chi-squared testchi-square**

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.

### Goodness of fit

**goodness-of-fitfitgoodness-of-fit test**

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.

### Statistical Methods for Research Workers

**intraclass correlationmethods**

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 Design of Experiments

**book**

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).

### Student's t-distribution

**Student's ''t''-distributiont-distributiont''-distribution**

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.

### Quantile function

**quantileinverse distribution functionnormal quantile function**

The statistical tables for t and for Z provide critical values for both one- and two-tailed tests.

### Critical value

**critical value setcritical valuesThreshold value**

The statistical tables for t and for Z provide critical values for both one- and two-tailed tests.

### Cochran's C test

In statistics, Cochran's C test, named after William G. Cochran, is a one-sided upper limit variance outlier test.