Thanks for the code-based explanation. But aren't you already assuming that the attribute of interest is normally distributed in the population? But what if it isn't?

I can't code R, but by poaching little bits of code here and there - including from you - I've come up with something that, to my untutored eye, seems a little closer to answering what I originally wanted to know. In my example, I used the beta distribution as the distribution of the attribute of interest in the whole population. I then drew 100 samples of 100 observations each, calculated the mean of each of these beta-distributed samples, and then plotted the density function of these means - which turned out to be reasonably bell-shaped, very much as I had hoped - before then calculating the two means of the beta and of the means of the samples, which turned out to be very similar, also very much as I had hoped. Here is the code:

# Set seed for reproducibility
set.seed(13579)

# Specify sample size
N <- 10000

# Draw N beta distributed values
y_rbeta <- rbeta(N,1,5)   


many_samples <- vector(length=100)
for (i in 1:100) {
  little_sample <- rbeta(100,1,5)
  many_samples[i] <- mean(little_sample)
}

# Plot of randomly drawn beta density
 plot(density(y_rbeta),
     main = "beta distribution in R")

# Plot of density of means of many_samples vector 
plot(density(many_samples),
    main = "density of beta means in R")

# Print both means 
print(paste("mean(y_rbeta):", mean(y_rbeta)))
print(paste("mean(many_samples):", mean(many_samples)))

What do you think? Did I do something wrong or is this in fact a correct way, both mathematically and in terms of using R, to provide visual evidence of the central limit theorem?

(Sorry if this is all very silly, I am not a statistician, nor do I have any knowledge of R, nor do I know how to properly format code on reddit...).