Why a variance of a sample mean is the population variance divided by a sample size? On the wikipedia page about variance I have found the following equality
$$ Var(\overline{X})=Var(\frac{1}{n}\sum_{i=1}^{n}X_{i})=\frac{1}{n^{2}}\sum_{i=1}^{n}Var(X_{i})=\frac{1}{n^{2}}n\sigma^{2}=\frac{\sigma^{2}}{n} $$
I am confused because $Var(X_{i})$ is replaced by $\sigma^{2}$ where $\sigma^{2}$ is the variance of the population. I don't understand this because $X_{i}$ is not the population. It is a single value from a sample. Why this replacement is allowed?


I don't understand this because $X_i$ is not the population.

$X_i$ is a random variable.

It is a single value from a sample.

No it is not. $X_i$ is a random variable, but $X_i = x_i$ is a sample.

Why this replacement is allowed?

The description in the preceding paragraph in the link you provided explains this well. Your confusion stems from what a random variable is. I suggest you have a look at this resource to understand random variables better.
