Sampling distribution of the sample mean

Take every possible random sample of size $n$ from one population, record $\bar{x}$ (x-bar, the sample mean) from each, and the pile of those values is the sampling distribution of the sample mean. Three separate facts describe it. Its center is $\mu_{\bar{x}} = \mu$ (mu, the population mean), which holds for any population shape and any sample size. Its spread is $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ (sigma over the square root of n), which additionally requires the observations to be independent. Its shape is exactly normal when the population is normal, and approximately normal for a large enough $n$ by the central limit theorem.

Suppose a normal population has $\mu = 500$ and $\sigma = 100$, and you draw samples of $n = 25$. Then $\sigma_{\bar{x}} = 100/\sqrt{25} = 100/5 = 20$. A sample mean of 540 sits $z = \frac{540 - 500}{20} = 2$ standard deviations above center, so about 2.28 percent of samples of 25 produce a mean that high. A single individual measuring 540 is only $z = 0.40$ out, and about 34.46 percent of individuals beat it. Same value, two very different distributions.

That gap is the misreading. "The standard deviation is 20, so about 68 percent of the measurements fall between 480 and 520" is wrong. Individual measurements still have $\sigma = 100$, so about 68 percent of them fall between 400 and 600. The interval 480 to 520 is where sample means of 25 land, and nothing about the population changed when you started averaging.

The second misreading is what large enough means. A right-skewed population still graphs as right-skewed at $n = 400$; the distribution of the mean straightens out, the data do not. The $n \ge 30$ figure is a rule of thumb rather than a switch that flips, since a normal population is exactly normal here at $n = 2$ while a badly skewed one can need far more than 30.

The spread formula assumes independent observations, so it needs sampling with replacement or the 10% condition. In the Fall 2026 course this is topic 4.1, Sampling Distributions for Sample Means, with the central limit theorem in topic 2.12.