One - and two - tailed tests - wikipedia One - and two - tailed tests A two - tailed test applied to 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 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. A two - tailed test is appropriate if the estimated value may be more than or less than the reference value, for example, whether a test taker may score above or below the historical average. A one - tailed test is appropriate if the estimated value may depart from the reference value in only one direction, for example, whether a machine produces more than one - percent defective products. 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 or `` bell curve '', pictured on the right. Contents 1 Applications 2 Coin flipping example 3 History 4 Specific tests 5 See also 6 References 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 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. 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. 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 ) 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. History p - value of chi - squared distribution for different number of degrees of freedom 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. Normal distribution, showing two tails 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 : 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. -- 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 ). 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 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 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, when two samples are being compared References Jump up ^ Kock, N. ( 2015 ). One - tailed or two - tailed P values in PLS - SEM ? International Journal of e-Collaboration, 11 ( 2 ), 1 - 7. Jump up ^ Mundry, R., & Fischer, J. ( 1998 ). Use of statistical programs for nonparametric tests of small samples often leads to incorrect P values : Examples from Animal Behaviour. Animal behaviour, 56 ( 1 ), 256 - 259. Jump up ^ Pillemer, D.B. ( 1991 ). One - versus two - tailed hypothesis tests in contemporary educational research. Educational Researcher, 20 ( 9 ), 13 - 17. Jump up ^ 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. ) Jump up ^ JM Bland, DG Bland ( BMJ, 1994 ) Statistics Notes : One and two sided tests of significance Jump up ^ 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 '' ( PDF ). Philosophical Magazine. Series 5. 50 ( 302 ) : 157 -- 175. doi : 10.1080 / 14786440009463897. ^ Jump up to : Fisher, Ronald ( 1925 ). Statistical Methods for Research Workers. Edinburgh : Oliver & Boyd. ISBN 0 - 05 - 002170 - 2. Jump up ^ Fisher, Ronald A. ( 1971 ) ( 1935 ). The Design of Experiments ( 9th ed. ). Macmillan. ISBN 0 - 02 - 844690 - 9. hide Statistics Outline Index Descriptive statistics Continuous data Center Mean arithmetic geometric harmonic Median Mode Dispersion Variance Standard deviation Coefficient of variation Percentile Range Interquartile range Shape Central limit theorem Moments Skewness Kurtosis L - moments Count data Index of dispersion Summary tables Grouped data Frequency distribution Contingency table Dependence Pearson product - moment correlation Rank correlation Spearman 's rho Kendall 's tau Partial correlation Scatter plot Graphics Bar chart Biplot Box plot Control chart Correlogram Fan chart Forest plot Histogram Pie chart Q -- Q plot Run chart Scatter plot Stem - and - leaf display Radar chart Data collection Study design Population Statistic Effect size Statistical power Sample size determination Missing data Survey methodology Sampling stratified cluster Standard error Opinion poll Questionnaire Controlled experiments Design control optimal Controlled trial Randomized Random assignment Replication Blocking Interaction Factorial experiment Uncontrolled studies Observational study Natural experiment Quasi-experiment hide Statistical inference Statistical theory Population Statistic Probability distribution Sampling distribution Order statistic Empirical distribution Density estimation Statistical model L space Parameter location scale shape Parametric family Likelihood ( monotone ) Location -- scale family Exponential family Completeness Sufficiency Statistical functional Bootstrap U V Optimal decision loss function Efficiency Statistical distance divergence Asymptotics Robustness Frequentist inference Point estimation Estimating equations Maximum likelihood Method of moments M - estimator Minimum distance Unbiased estimators Mean - unbiased minimum - variance Rao -- Blackwellization Lehmann -- Scheffé theorem Median unbiased Plug - in Interval estimation Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife Testing hypotheses 1 - & 2 - tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons Parametric tests Likelihood - ratio Wald Score Specific tests Z - test ( normal ) Student 's t - test F - test Goodness of fit Chi - squared G - test Kolmogorov -- Smirnov Anderson -- Darling Lilliefors Jarque -- Bera Normality ( Shapiro -- Wilk ) Likelihood - ratio test Model selection Cross validation AIC BIC Rank statistics Sign Sample median Signed rank ( Wilcoxon ) Hodges -- Lehmann estimator Rank sum ( Mann -- Whitney ) Nonparametric anova 1 - way ( Kruskal -- Wallis ) 2 - way ( Friedman ) Ordered alternative ( Jonckheere -- Terpstra ) Bayesian inference Bayesian probability prior posterior Credible interval Bayes factor Bayesian estimator Maximum posterior estimator Correlation Regression analysis Correlation Pearson product - moment Partial correlation Confounding variable Coefficient of determination Regression analysis Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines ( MARS ) Linear regression Simple linear regression Ordinary least squares General linear model Bayesian regression Non-standard predictors Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity Generalized linear model Exponential families Logistic ( Bernoulli ) / Binomial / Poisson regressions Partition of variance Analysis of variance ( ANOVA, anova ) Analysis of covariance Multivariate ANOVA Degrees of freedom Categorical / Multivariate / Time - series / Survival analysis Categorical Cohen 's kappa Contingency table Graphical model Log - linear model McNemar 's test Multivariate Regression Manova Principal components Canonical correlation Discriminant analysis Cluster analysis Classification Structural equation model Factor analysis Multivariate distributions Elliptical distributions Normal Time - series General Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality Specific tests Dickey -- Fuller Johansen Q - statistic ( Ljung -- Box ) Durbin -- Watson Breusch -- Godfrey Time domain Autocorrelation ( ACF ) partial ( PACF ) Cross-correlation ( XCF ) ARMA model ARIMA model ( Box -- Jenkins ) Autoregressive conditional heteroskedasticity ( ARCH ) Vector autoregression ( VAR ) Frequency domain Spectral density estimation Fourier analysis Wavelet Whittle likelihood Survival Survival function Kaplan -- Meier estimator ( product limit ) Proportional hazards models Accelerated failure time ( AFT ) model First hitting time Hazard function Nelson -- Aalen estimator Test Log - rank test Applications Biostatistics Bioinformatics Clinical trials / studies Epidemiology Medical statistics Engineering statistics Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification Social statistics Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population statistics Psychometrics Spatial statistics Cartography Environmental statistics Geographic information system Geostatistics Kriging Portal Commons WikiProject Retrieved from `` https://en.wikipedia.org/w/index.php?title=One-_and_two-tailed_tests&oldid=808354488 '' Categories : Statistical tests Talk Contents Interaction About Wikipedia Português Edit links This page was last edited on 2 November 2017, at 11 : 10 ( UTC ). 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Question: how do you know if you have a two tailed test
Answer: if the estimated value may be more than or less than the reference value