Thanks to @sanjayk for answer.
I agree, things seems a bit confusing at times.
See if this description helps.
Think of X_i~'s (i.e. X_1, X_2…etc) ONLY as Random Variables. For e.g. number appearing on a single throw of a dice, or sum of numbers appearing on 5 throws of a dice, or mean of numbers appearing on 3 throws of a dice).
Point is, you have control over defining the experiment and what corresponding R.V. is associated with experiment outcome. Experiment can be simple or complex, single roll or multiple roll, observation/R.V. can be number showing up on the dice or some sum or mean of numbers showing up as a result of multiple rolls etc etc.
Next, you do that experiment n number of times, so that you have observation of n R.V.s i.e. X_1, X_2,...,X_n.
CLT says, if you define another R.V. X such that, its the sum of X_i 's:
then distribution of X will be normal as n\to\infty.
Recall, what distribution of X means. To visualize:
- Do the experiment and find X_i (for i=1 till n)
- next, calculate X i…e the sum of X_i's
- Perform step 1 and step 2 again and again to have several values of X and then plot the values of X. The plot will resemble a normal distribution.
I hope the above description helped you understand what is n.
R.V. X_i can resemble a sample statistic, for e.g. you draw a sample and calculate the sample mean i.e. (X_i). Note that sample size used for calculating sample mean (X_i) here is different from the n used to calculate the R.V X. Right? (n here will be number of samples you will take, such that each will have a sample mean, which you will use for calculating X)
X_i can be also be a R.V. coming from a single roll of a dice and be the number showing up. In this case, you can simply take n observations and use it to calculate the X, the r.v. representing the sum. Here, you might be tempted to say, you draw a sample of size n and its sum is X.
Did I leave you more confused?
As long as you can remember (1)X_i are independent r.v.s, and (2)the sum of n such R.V.s will be another R.V. (3) which follows a normal distribution as (4) n\to\infty, you are good.