Standard error

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.