Critical Value vs Standard Error

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.

The standard error is an estimate. It plays the same role for a statistic that the standard deviation plays for raw data, and it is called a standard error precisely because the quantity it reports is built from statistics rather than parameters. For a sample mean it is $SE_{\bar{x}} = \frac{s}{\sqrt{n}}$, using $s$, the sample standard deviation. For a sample proportion it is $SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, using $\hat{p}$ (p-hat, the sample proportion). Put the parameters in instead, $\frac{\sigma}{\sqrt{n}}$ (sigma over the square root of n) and $\sqrt{p(1-p)/n}$, and you have the true standard deviation of the sampling distribution, which is the thing the standard error is estimating. That one swap, statistic in place of parameter, is the whole difference between the two names.

Suppose 25 measurements have a sample standard deviation of $s = 4.5$ grams. Then $SE_{\bar{x}} = \frac{4.5}{\sqrt{25}} = \frac{4.5}{5} = 0.9$ grams. Read the two numbers aloud: individual measurements sit about 4.5 grams from the sample mean, while the sample mean itself sits about 0.9 grams from the population mean across repeated samples of 25.

The misreading is "the standard error tells you how spread out the data are." It does not. That is $s$, and $s$ stays about the same as you collect more data, because a larger sample estimates the same population spread. The standard error shrinks toward zero with $\sqrt{n}$ because it describes a statistic, not an observation.

A second wrong reading is treating a small standard error as proof the estimate is close to the truth. It measures precision, not accuracy. A convenience sample of 4,000 produces a tiny standard error around whatever value its method is centered on, and no amount of precision repairs a center that is wrong.

One boundary worth memorizing: a one-sample z-test for a proportion puts the null value $p_0$ (p-naught) where $\hat{p}$ normally sits, giving $\sqrt{p_0(1-p_0)/n}$, because every quantity in a test is computed as though $H_0$ were true. Otherwise the standard error is the denominator of a standardized test statistic and the second factor in a margin of error.