Definition of sample mean

Question

I've seen two definitions of sample mean on the internet.
One definition defines it as the average of Random variable other defines it as the average of sample values of a sample.
I'm confused which one is correct.
Also if it is defined as the average of random variables why don't we just define it as the average of sample values?

Answer 1

Given samples $\{x_i\}_{i=1}^n$ (presumed to be drawn from an underlying distribution $F$, density $f$), the sample mean $\bar{x}$ defined as:

$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$

is an estimate of the expected value $\mathbb{E}[X]$ of that distribution.

On the other hand, given RVs $\{X_i\}_{i=1}^n$ (drawn independently from distribution $F$), their average $\bar{X}$ defined as:

$\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$

is also a random variable with mean $\mathbb{E}[\bar{X}] = \mathbb{E}[X]$ and distribution given by an n-fold convolution of $f$.

It is not meaningful to compare $\bar{x}$ and $\bar{X}$ directly.

Answer 2

In statistical probability, a single sampled value is a random variable. If you have a sample consisting of $N$ values, then you have $N$ random variables. The words "sample value" are just another name for one of these random variables.

So both methods of obtaining a "sample mean" do exactly the same thing.

The mean value of a random variable is a completely different thing. Be careful not to write "variable" (singular) when the text you are getting your definitions from said "variables" (plural). It changes the possible interpretation of what you write quite dramatically.