Critical value

A critical value is a quantile of a reference distribution, chosen so that a stated area falls beyond it. For a confidence interval at level $C$ it is the number $z^*$ (z-star) or $t^*$ (t-star) with the middle $C$ of the curve lying between $-z^*$ and $z^*$. For a significance test it is the point cutting off $\alpha$ (alpha) in whichever tail the alternative hypothesis points to. One idea, two jobs: it sets a width, or it sets a boundary.

The standard normal values, listed in full on the z-table:

At 95%, 1.960 is the value leaving 0.025 in each tail, so 0.95 of the standard normal curve sits between $-1.960$ and $1.960$. With the population standard deviation unknown you switch to $t^*$ at the right degrees of freedom, and it is larger than the matching $z^*$ at every finite df. At 95% the t-table gives 2.262 at 9 df, 2.064 at 24 df, and 1.984 at 99 df, closing on 1.960 from above without reaching it.

The error worth naming is "the critical value for 95% confidence is 1.96." It is 1.96 only when the reference curve is the standard normal. A mean from a sample of 12 uses $t$ with 11 df, where the critical value is 2.201, and with a normal population, reaching out only 1.96 standard errors captures the true mean 92.4% of the time rather than 95%. The interval is about 11% too narrow and it undercovers by more than two points.

Not every critical value comes in a $\pm$ pair. The chi-square distribution is right skewed and its tests use the upper tail alone, so there is a single cutoff: 7.815 at 3 degrees of freedom and $\alpha = 0.05$, from the chi-square table. A critical value also marks only the boundary. How far past it you landed is what a p-value reports, which is why two tests can share a critical value and carry very different evidence.