What a critical value is
The critical value is the fence on the test statistic's scale. Set your significance level α, and the critical value marks the point beyond which results count as evidence against the null: statistic past the fence, reject; inside the fence, fail to reject. It is the same decision the p-value gives you, read from the other direction. Tests were reported this way for a century because tables of critical values fit in a book appendix and p-values did not.
At α = 0.05 two-tailed: z* = 1.9598; t* (df 10) = 2.2281; chi-square (df 10, right tail) = 18.307; F (3, 20, right tail) = 3.0984. Compare the z and t fences: estimating σ from 11 observations widens the fence from 1.96 to 2.23, the small-sample tax, made visible.
Critical values vs p-values
They never disagree: p < α exactly when the statistic passes the critical fence for that α. Exams and older papers often ask for the critical-value comparison; modern software reports the p-value. This tool gives you the fence for any of the four families (z, t, chi-square, F) at any α, in one or two tails.
Using the z fence (1.96) for a small-sample t-test. At df 7 the true fence is 2.365; using 1.96 rejects the null far more often than the promised 5% when nothing is going on.
Frequently asked questions
What is the critical value for a 95% confidence interval?
The two-tailed α = 0.05 value: 1.96 for z, or t* on your df. Confidence intervals and two-tailed tests use the same fence.
Why does the t critical value shrink as df grows?
More data pins down the standard deviation estimate, so the t curve slims toward the normal. By df 30 the fence is 2.042; by df 120, 1.980, nearly 1.96.