What is the difference between $\sigma, \sigma_{\bar{x}}, S, s,$ and $s_{\bar{x}}$?

Question

What is the difference between $\sigma, \sigma_{\bar{x}}, S, s,$ and $s_{\bar{x}}$?

My textbook uses lots of different symbols, and it's not clear to me what the difference between all of them are. Are they just the same?

Answer 1

$\sigma$ is the population standard deviation, which is generally unknown. Typically, X (a capital letter) represents a value from the population. This is a random variable (it could take any of , typically, many values). A particular value of X is called x (lower case) and has a definite value. If you have n different such sample values you might label them $x_1, x_2, ..., x_n$. The average of n values from the population is also a random variable $\bar{X}$ and it in turn might have a particular value for n particular sample values, $\bar{x}$. Given a random sample average $\bar{X}$, that sample will have a random sample standard deviation S. A particular value for this random variable for a known sample of n values is then s. The random variable $\bar{X}$ for n sample values has a smaller standard deviation than the original X variable. Its population standard deviation is ${{\sigma }_{{\bar{X}}}}=\frac{\sigma }{\sqrt{n}}$ with sample standard deviation ${{s}_{{\bar{x}}}}=\frac{s}{\sqrt{n}}$.