One-Tailed vs Two-Tailed Tests

A two-tailed critical value at α=0.05 is 1.96 for Z. A one-tailed critical value at the same α is 1.645. That 0.315 difference changes whether you reject the null. The choice between a one tailed vs two tailed test is determined entirely by the alternative hypothesis. OpenIntro Statistics (4th ed.) and Moore, McCabe & Craig both state the rule: the alternative hypothesis tells you where the rejection region goes.

Open the alternative hypothesis (H₁ or Hₐ). If it uses “≠” (not equal to), you run a two-tailed test. The rejection region sits in both tails of the sampling distribution. If it uses “>” (greater than), you run a right-tailed test. If it uses “<” (less than), you run a left-tailed test. The direction in the inequality names the tail.

Left-Tailed Test

For the alternative “μ < 100”, the rejection region is in the left tail. The critical value is negative. For Z at α=0.05, the left-tailed critical value is -1.645. The test statistic must fall below that threshold to reject the null. This is a left tailed test.

Right-Tailed Test

For “μ > 100”, the rejection region is in the right tail. The critical value is positive. For Z at α=0.05, the right-tailed critical value is +1.645. The test statistic must exceed that threshold. This is a right tailed test.

Two-Tailed Test

For “μ ≠ 100”, the rejection region is split. Half of α goes to the left tail, half to the right tail. The two-tailed critical value for Z at α=0.05 is ±1.96. The test statistic must be either below -1.96 or above +1.96. The failure case: using a two-tailed critical value for a one-tailed test inflates Type II error; using a one-tailed critical value for a two-tailed test inflates Type I error.

For a one-tailed test, the entire α (say 0.05) sits in one tail. The critical value is the quantile that cuts off that α in the chosen tail. For a two-tailed test, you split α equally: α/2 in each tail. At α=0.05, each tail gets 0.025. The critical value is the quantile that cuts off 0.025 in each tail, which is the same as the quantile that cuts off 0.975 from the left for the right tail. This is why the two-tailed critical value (1.96) is larger than the one-tailed value (1.645) at the same α. The statistical term alpha divided by 2 is the area in each tail for a two-tailed test. The TI-84 invNorm function takes a left-tail area. For a two-tailed test at α=0.05, input 1 - α/2 = 0.975 to get the right-tail critical value 1.96. For a one-tailed test, input 1 - α = 0.95 to get 1.645.

Alpha	Two-Tailed Z	One-Tailed Z	Two-Tailed t (df=30)	One-Tailed t (df=30)
0.10	1.645	1.282	1.697	1.310
0.05	1.960	1.645	2.042	1.697
0.01	2.576	2.326	2.750	2.457

The Z critical value is fixed for a given α. The t critical value changes with degrees of freedom. At α=0.05 two-tailed, Z=1.96; t with df=2 is 4.303; t with df=30 is 2.042; t with df=100 is 1.984. The t critical value approaches the Z critical value as degrees of freedom increase. The choice between Z and t depends on whether the population standard deviation is known or estimated from the sample. The “n ≥ 30 means use Z” rule is incorrect: use Z when sigma is known, t when sigma is estimated. A two tailed critical value for t is always larger than the Z value at the same α because the t-distribution has heavier tails. NIST/SEMATECH e-Handbook of Statistical Methods provides exact tables for both distributions.

Do not look at the test statistic and then decide to run a one-tailed test instead of a two-tailed test. This is p-hacking. If the alternative hypothesis originally stated “μ ≠ 100” and the test statistic is +1.70, which does not exceed the two-tailed critical value of 1.96, switching to a right-tailed test (critical value 1.645) to get a significant result inflates the Type I error rate beyond the stated α. The ASA Statement on Statistical Significance and P-Values (Wasserstein & Lazar, 2016) clarifies that such post-hoc decisions undermine the validity of the test. The tail choice must be made before the data are collected, based on the alternative hypothesis alone.

Chi-square and F distributions are asymmetric, they have no negative values. Hypothesis tests using these distributions always use the right tail. A chi-square critical value at α=0.05 with 5 degrees of freedom is 11.07 from the right tail of the distribution. Using the left tail would give a value near zero, which is useless for a hypothesis test. The chi-square and F inverse functions in Excel and Google Sheets are named CHISQ.INV.RT and F.INV.RT, the RT stands for right tail. The TI-84 Plus CE does not have built-in inverse functions for chi-square or F; you must use the chi-square cdf and solve manually or use a different tool. For an F-test with numerator df=3 and denominator df=20 at α=0.05, the critical value from software is 3.098.

AP Statistics students who need the correct critical value for a one- or two-tailed test and understand why it changes with degrees of freedom will get the exact threshold. College intro stats students choosing among Z, t, chi-square, and F critical values will find the decision rule. Graduate students in psychology, public health, or economics who need exact values for non-standard alpha levels can use the software functions named here. Analysts checking a statistical package output against a hand calculation will find the correct function names. Skip this if you need a p-value instead of a threshold, use a p-value calculator. Skip if you need a full hypothesis testing course, start with OpenIntro Statistics. Skip if you need critical values for non-parametric tests like Mann-Whitney U or Wilcoxon signed-rank; those distributions are not covered. The single thing that most often goes wrong is using the “n ≥ 30” rule instead of the known-sigma rule, which produces a critical value that is too small and inflates Type I error.