$SEM = \frac{\hat{\sigma}}{\sqrt{n}}$
The formula for the standard error of the mean says that the quality of our measurement involves two quantities: the population variability, and the size of our sample. Because the sample size is the denominator in the formula for SEM, a larger sample size will yield a smaller SEM when holding the population variability constant. We have no control over the population variability, but we do have control over the sample size. Thus, if we wish to improve our sample statistics (by reducing their sampling variability) then we should use larger samples. However, the formula also tells us something very fundamental about statistical sampling – namely, that the utility of larger samples diminishes with the square root of the sample size. This means that doubling the sample size will not double the quality of the statistics; rather, it will improve it by a factor of $\sqrt{2}$. In Section 18.3 we will discuss statistical power, which is intimately tied to this idea.