# Which sample standard deviation do you use for estimating standard error of the mean?

I know that estimating the standard error of the mean has the following formula: $${\mathop{\sigma}}_{\bar x}= \frac{{s}}{\sqrt{n}}$$ where s is the sample standard deviation and n is the sample size. I would like to know which sample standard deviation is used? Because we are taking multiple samples of size n anyways to find the sampling distribution of the sample mean, do we just find the standard deviation of one random sample of size n and use that to estimate the standard error of the mean?

• What task are you doing where you’re drawing multiple samples from the population?
– Dave
Commented Jun 22, 2020 at 2:10
• I would draw multiple samples of size n to find the sampling distribution of the sample mean. It is more of a conceptual activity than a practical one for me, but the question is asking if the sample standard deviation used in the formula for estimating the standard error of the mean is simply from a random sample of size n in the population. Or are there any other nuances to picking a sample standard deviation for this situation? Commented Jun 22, 2020 at 2:14
• You use the sample standard deviation of your data.
– Dave
Commented Jun 22, 2020 at 2:19
• Alright, I understand now. Thanks so much for the help! Commented Jun 22, 2020 at 2:20
• But I still don’t get why you have multiple samples from this population, and that you do makes me wonder what you’re doing. Some kind of simulation?
– Dave
Commented Jun 22, 2020 at 2:22

If you have a sufficient number of samples of the same size $$n$$ from the population, a theoretical sampling distribution can be drawn and corresponding to the central limit theorem, you will observe that this distribution is symmetrical and bell shaped. In this case we don't need to calculate the standard error via the formula because since we are able to observe $$\sigma_\bar{x}$$ directly from the sampling distribution of the sample means (empirical standard error estimate).
In "real life research" you usually get only one sample from the population. There, the sample standard deviation $$s$$ is calculated and used in the formula $$\sigma_\bar{x}=\frac{s}{\sqrt{n}}$$ to estimate the standard error of the mean.