Your first paragraph is how I had understood the CLT as well. I’m wondering now if the author of the post knows more than me and is correct about what he’s talking about, or if he’s wrong.
More formally, if we have x ~ N(m, s) then nx ~ N(nm, sqrt(n)s).
Edit to add: Here's a small python program so you can run it in a jupyter notebook or something and see for yourself:
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
broken_dice_probabilities = [int(x!=3)*0.2 for x in range(6)]
broken_dice_sides = [x+1 for x in range(6)]
n=1000000
population = np.random.choice(broken_dice_sides, replace=True, p=broken_dice_probabilities, size=n)
ax=sns.histplot(population)
ax.set_title("Underlying population")
plt.show()
n_samples = 100000
sample_size = 20
clt_samples=np.zeros(n_samples)
for i in range(n_samples):
this_sample = np.random.choice(population, replace=True, size=sample_size)
clt_samples[i]=np.mean(this_sample)
ax=sns.histplot(clt_samples)
ax.set_title("Sample means")
plt.show()
So if the original X_i are drawn from any distribution with defined with mean m and variance s², then the CLT says that in the limit, the sample mean is distributed N(m, s), and, equivalently, the sample sum is distributed N(nm, sqrt(n)s).
The question is when the "in the limit" results starts being practically useful, and that depends on the distribution of the original X_i.