Standard deviation of the sampling distribution

This is the actual spread of a statistic over every possible sample of size $n$, and it is computed from parameters. For the sample mean it is $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ (sigma, the population standard deviation, over the square root of n). For the sample proportion it is $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$, using the population proportion $p$. Both are parameters of the sampling distribution, not quantities you can calculate from the data in front of you. Replace $\sigma$ with $s$ or $p$ with $\hat{p}$ and the result is no longer this number: it is the standard error, an estimate of it.

With $\sigma = 15$, samples of $n = 25$ give $15/\sqrt{25} = 15/5 = 3$. Quadruple the sample to $n = 100$ and it halves to $15/10 = 1.5$. Halve it again, to 0.75, and you need $n = 400$. Notice the price: the first halving cost 75 extra observations and the second cost 300. Precision bought under a square root gets expensive fast.

The standing misreading is "the standard deviation is 3, so about 95 percent of the values lie within 6 of the mean." The values do not. Individual observations still have $\sigma = 15$, and nothing about the population changed when you started averaging. The 3 describes where sample means of 25 land, and that distinction is what the whole idea of a sampling distribution exists to make.

The number also has a job: it is the denominator of a standardized statistic. Writing $z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}}$ turns a distance in grams or dollars into a distance in standard deviations of the sampling distribution, which is the scale a normal table reads.

Two boundaries. The centering result, $\mu_{\bar{x}} = \mu$ (mu), needs only random sampling and holds for every $n$ and every population shape. The spread formulas above need something extra, namely independent observations, so they hold when sampling with replacement or under the 10% condition. Shape is a third and separate question, answered by the central limit theorem for means and by Large Counts for proportions.