I can't quite understand what this formula means:
$$\sigma_{\overline{x}}=\frac{\sigma}{\sqrt n}$$
I know what standard deviation $\sigma$ is - it's the average distance of my data points (samples) from the mean. But this part is confusing:
For example, suppose the random variable $X$ records a randomly selected student's score on a national test, where the population distribution for the score is normal with mean $70$ and standard deviation $5$ ($N(70,5)$). Given a simple random sample (SRS) of $200$ students, the distribution of the sample mean score has mean $70$ and standard deviation $$\frac{5}{\sqrt{200}} \approx \frac{5}{14.14} \approx 0.35$$
I thought the standard deviation $\sigma = 5$ means that if I take the scores of all students and calculate the mean, then the average distance of a score from that mean will be equal to $5$. The set of all scores is called the 'population', right? But here it says the more students' scores I take, the lower the standard deviation - thus the closer the number of samples gets to the size of population, the lower the standard deviation (and its get further from $5$).