What is the difference between one-tailed and two-tailed hypothesis tests, and when is each appropriate?
A two-tailed test rejects H0 when the statistic is extreme in either direction; a one-tailed test rejects only in one pre-specified direction. Two-tailed tests are the default because they guard against effects in both directions; one-tailed tests are valid only when a directional hypothesis is theoretically justified and pre-registered before seeing the data.
How to think about it
The choice between one-tailed and two-tailed tests is about the alternative hypothesis, not about convenience. Getting this wrong — in either direction — distorts your Type I error rate.
Two-tailed test
H1: mu ≠ mu0 (the parameter differs from the null value in either direction).
The significance level alpha is split equally between both tails: alpha/2 in each. For alpha = 0.05, the critical z-values are approximately ±1.96. This is appropriate whenever you care about deviations in both directions — which is the vast majority of real analyses.
One-tailed test
H1 is directional: either mu > mu0 or mu < mu0, pre-specified.
All of alpha sits in one tail. The critical z-value for a right-tailed test at alpha = 0.05 is approximately 1.645 — a lower bar than 1.96. This makes the test more powerful for detecting an effect in that direction, but it completely ignores the possibility of an effect in the other direction.
When a one-tailed test is justified
- The research question is inherently directional and was framed that way before data collection.
- An effect in the opposite direction would be treated identically to “no effect” — it would not change any decision.
- Regulatory or scientific precedent supports it (some non-inferiority trial designs).
The arithmetic of the split
For the same observed test statistic, a one-tailed p-value is exactly half the two-tailed p-value. This is the source of abuse: converting a two-tailed result that just missed significance to a one-tailed one to get p < 0.05.
| Test | Rejection region | Power for directional effect |
|---|---|---|
| Two-tailed | Both tails at alpha/2 | Lower |
| One-tailed | One tail at alpha | Higher (for the pre-specified direction) |