One-tailed and two-tailed p-values. When to use them in statistics

🔹 Definition

P-value in statistics tells us how extreme our observed data is, assuming the null hypothesis (H₀) is true.

The difference between one-tailed and two-tailed p-values depends on what kind of difference or effect we are testing for.

---

🔸 1. Two-Tailed Test

You use a two-tailed test when you’re testing for any significant difference —
either greater than or less than a certain value.

✅ Example:

Suppose the average blood pressure (BP) in a population is 120 mmHg.
You want to test whether a new drug changes BP (it could increase or decrease).

• Null hypothesis (H₀): μ = 120

• Alternative hypothesis (H₁): μ ≠ 120

If your sample mean = 126, and the calculated p-value = 0.04 (two-tailed),
then this means there’s a 4% chance of observing such an extreme difference (either direction) if the true mean were really 120.

• If α = 0.05, since 0.04 < 0.05 → Reject H₀.

• Conclusion: The drug significantly changes BP (either up or down).

---

🔸 2. One-Tailed Test

You use a one-tailed test when you only care about one direction —
either increase or decrease, not both.

✅ Example:

You believe the drug reduces BP (not increases).

• Null hypothesis (H₀): μ = 120

• Alternative hypothesis (H₁): μ < 120

If your sample mean = 114, and the two-tailed p-value = 0.04,
then the one-tailed p-value = 0.02 (half of it, since only one direction is tested).

• If α = 0.05, 0.02 < 0.05 → Reject H₀.

• Conclusion: The drug significantly reduces BP.

But if the drug increased BP to 126, the one-tailed p-value (for “μ < 120”) would be > 0.05 → Not significant,
because the effect is in the opposite direction.

---

🔹 In Summary

Aspect	One-Tailed Test	Two-Tailed Test
Direction tested	One (>, or <)	Both (≠)
p-value	Area in one tail	Area in both tails
Hypothesis	H₁: μ > μ₀ or μ < μ₀	H₁: μ ≠ μ₀
When used	Effect in a known direction	Any difference possible
Example	“Drug lowers BP”	“Drug changes BP”

---

💡 Tip:

• Use two-tailed when you’re exploring or uncertain of direction.

• Use one-tailed only when a specific direction is logically or clinically justified before testing.

---