One skewed parent and, under it on the same axis, the bells that sample means make — wide for n = 4, narrow for n = 25, the means themselves as dots. The population never becomes normal; the averages do, and their spread is σ/√n. Evidence: the standard error falling with √n against the variance falling with n; the ladder is how far the noise drops for each sample size.
One sweet is a surprise; a handful is not
A jar has mostly small sweets and a few giant ones. Pick one and you never know what you'll get. Now grab a whole handful and weigh it: the giants and the smalls balance out, so every handful weighs about the same. Bigger handfuls are steadier still. That is the whole theorem — handfuls behave, even when single sweets don't.
Averages are calmer than the things they average.
x̄ ~ N(μ, σ/√n)
One value
n = 1
σ
A handful
n = 4
σ / 2
A class
n = 25
σ / 5
A survey
n = 100
σ / 10
A census
n = 10,000
σ / 100
Bars are log₁₀ n — the standard error halves only when the sample quadruples