Q: Which distribution should I use?

Z for large-sample means or proportions with known σ, t for small-sample means with unknown σ (needs df = n − 1), chi-square for variance and goodness-of-fit / independence tests (needs df), and F for variance ratios and ANOVA (needs two df values, numerator and denominator).

Q: Are critical value and p-value approaches equivalent?

Yes. 'Reject if the test statistic is in the rejection region' and 'reject if p ≤ α' always give the same decision when applied to the same data and same test. Some fields lean on tabled critical values (older textbooks); modern software leans on exact p-values.

Q: Why does my chi-square or F critical value only appear in the right tail?

Because χ² and F are non-negative and right-skewed, the classic goodness-of-fit, independence and variance tests are naturally one-tailed on the upper side. Two-tailed χ² / F tests exist but are uncommon and require using both α/2 tails.

Q: What if I get 0 or infinity as a critical value?

That usually means α is at or beyond a boundary (e.g. α = 0 or α = 1), or the degrees of freedom are too small to be meaningful. Check your inputs — typical α values are 0.10, 0.05 and 0.01.

Question 1

What is a critical value?

Accepted Answer

The cutoff a test statistic must exceed (or fall below) in order to reject the null hypothesis at your chosen significance level α. It is the 'threshold' companion to the p-value approach — if the observed test statistic lies in the shaded rejection region, you reject H₀.

Question 2

How do I choose between one-tailed and two-tailed?

Accepted Answer

Two-tailed tests any difference (H₁: parameter ≠ value) and splits α between both tails. One-tailed tests a specific direction (H₁: parameter > or < value) and puts all of α in one tail. Decide before seeing the data.

Question 3

Which distribution should I use?

Accepted Answer

Z for large-sample means or proportions with known σ, t for small-sample means with unknown σ (needs df = n − 1), chi-square for variance and goodness-of-fit / independence tests (needs df), and F for variance ratios and ANOVA (needs two df values, numerator and denominator).

Question 4

Are critical value and p-value approaches equivalent?

Accepted Answer

Yes. 'Reject if the test statistic is in the rejection region' and 'reject if p ≤ α' always give the same decision when applied to the same data and same test. Some fields lean on tabled critical values (older textbooks); modern software leans on exact p-values.

Question 5

Why does my chi-square or F critical value only appear in the right tail?

Accepted Answer

Because χ² and F are non-negative and right-skewed, the classic goodness-of-fit, independence and variance tests are naturally one-tailed on the upper side. Two-tailed χ² / F tests exist but are uncommon and require using both α/2 tails.

Question 6

What if I get 0 or infinity as a critical value?

Accepted Answer

That usually means α is at or beyond a boundary (e.g. α = 0 or α = 1), or the degrees of freedom are too small to be meaningful. Check your inputs — typical α values are 0.10, 0.05 and 0.01.

Critical Value Calculator

Critical values explained, step by step

1. The rejection region — cutoff at α

2. Symmetric distributions — Z and t use α/2

3. Right-skewed distributions — χ² and F

4. Critical value vs p-value — same decision

Features of this calculator