# Why use the standard deviation of sample means for a specific sample?

Working on notes in my Elementary Statistics course, I am trying to figure out why the standard deviation for a single sample used the standard deviation of the sample means. An example of what I am referring to is:

"A random sample size of 36 is taken from a population with mean µ = 17 and standard deviation σ = 6. What is the probability that the sample mean is greater than 18?"

The process being finding the z-score and using a standard normal distribution with $z = \frac{18-17}{6/\sqrt{36}}$

I understand that the standard deviation of all samples of size n is $\sigma_\bar{x} = \frac{\sigma}{\sqrt{n}}$. What I don't get is, why are we using $\sigma_\bar{x}$ for the standard deviation of that specific sample, when it is the standard deviation of all the sample means?

## 1 Answer

"A random sample size of 36 [...] What is the probability that the sample mean is greater than 18?"

You are NOT inferring information about a single sample: you are inferring information about the sample mean, so you use the dispersion of the sample means, which depends on the population parameters and also on the sample size.