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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?

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    $\begingroup$ Can you give us a bit more context on the level of study you are on so any answer could be tailored to that level? Often $X_i$ represents a random variable which is an independent, identical copy of the population distribution. On the other hand $x_i$ (with little $x$) represents a realisation (single value) from the above random variable, and hence does not have a variance. $\endgroup$
    – B.Liu
    Jun 23 '21 at 14:31
  • $\begingroup$ @B.Liu I got my master's degree in IT a few years ago. Now I try to study some maths on my own. This question connected to sevenkul's anwser in this thread where you can find the end part of the equality again. I don't understand how we can replace a sample size with the populstion size when we replace $Var(X_{i})$ by the $\sigma^{2}$. $\endgroup$
    – Tolfel
    Jun 23 '21 at 15:35
  • $\begingroup$ Everything follows from the mantra "variances add." $\endgroup$
    – whuber
    Jun 23 '21 at 17:06
  • $\begingroup$ I have to admit I am still not entirely clear on what exactly you are confused on. Question to tease that out: What do you think $Var(X_i)$ should be in this case? $\endgroup$
    – B.Liu
    Jun 23 '21 at 17:12
  • $\begingroup$ @whuber I have a feeling that the OP might be confusing $\sigma^2$ with $S^2$ (or $s^2$) with their mention of sample and population. But hopefully the clarification question would help our understanding. $\endgroup$
    – B.Liu
    Jun 23 '21 at 17:14
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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.

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    $\begingroup$ In other words $X_i$ represents drawing a ticket out of the hat, while $x_i$ represents the ticket. $\endgroup$
    – Dave
    Jun 23 '21 at 14:41
  • $\begingroup$ @mhdadk in some articles I see formulas for the expected value like $E(X)=\sum_{i=1}^{n}X_{i}$. In other articles this formula can look like $E(X)=\sum_{i=1}^{n}x_{i}$. When should I use random variable and when sampled value? $\endgroup$
    – Tolfel
    Jun 25 '21 at 7:16
  • $\begingroup$ @Tolfel you need to review your basic probability theory. Here is a good book to do this: probabilitycourse.com $\endgroup$
    – mhdadk
    Jun 25 '21 at 11:12
  • $\begingroup$ @mhdadk Thank you $\endgroup$
    – Tolfel
    Jun 27 '21 at 17:39

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