Standard error of the mean

Across all samples of size $n$ drawn from one population, the sample mean $\bar{x}$ (x-bar) varies around the population mean $\mu$ (mu) with standard deviation $\frac{\sigma}{\sqrt{n}}$ (sigma over the square root of n), provided the observations are independent. You rarely know $\sigma$, so you estimate it from the sample and report $SE_{\bar{x}} = \frac{s}{\sqrt{n}}$. Either version describes how far a mean lands from $\mu$, never how far one observation lands from the mean.

Five commute times, in minutes: 31, 28, 35, 40, 26. The mean is 32 and the sample standard deviation is $s = 5.6125$, so $SE_{\bar{x}} = \frac{5.6125}{\sqrt{5}} = 2.5100$. Individual commutes scatter about 5.6 minutes wide. The average of five of them typically sits about 2.5 minutes from the true mean commute.

The sentence to catch is "about 68% of commutes fall within one standard error of 32 minutes." That reading is about $s$, not the standard error: if commutes were roughly normal the 68% span would be 26.4 to 37.6, not 29.5 to 34.5. Using the standard error where the sample standard deviation belongs describes the data as $\sqrt{n}$ times more consistent than it is, and it is the usual reason a student's interval comes out absurdly tight.

The square root is where intuition fails. Doubling the sample does not halve the standard error; quadrupling it does. Twenty commutes with the same $s = 5.6125$ give $SE_{\bar{x}} = 1.2550$, exactly half of 2.5100. The standard error is also silent on accuracy: if all five commutes came from the same route on the same morning, collecting more of them only tightens the estimate around the wrong center. It measures precision, and bias is invisible to it.

This is the denominator of every one-sample procedure for a mean in Unit 4 of the AP course, from the sampling distributions in topic 4.1 through the interval in 4.2 and the test in 4.5.