The Universe Loves Bells
The CLT explains why so many natural phenomena are bell-shaped: when many small independent factors sum up, you get a bell. That's most of the world. So we assume Gaussian everywhere by default — sensor noise, prediction errors, model residuals, latent variables in VAEs.
But Don't Worship It
Some real distributions are NOT normal:
- Income, wealth, city populations — power laws / lognormal. Long heavy tail.
- Word frequencies — Zipf's law, also a power law.
- Stock market returns — fat tails. Black swans don't follow a normal distribution; pretending they do is what blew up Long-Term Capital Management in 1998.
- Cluster centers in mixture models — bimodal or multimodal.
Always ask: does my data actually look like a bell, or am I assuming it does because the math is easier?
Risk modeling history. A lot of famous financial blow-ups (1998, 2008) involved assuming Gaussian distributions for things that had fat tails. Three-sigma "once in a thousand years" events kept happening yearly. The math was correct given the assumption; the assumption was wrong.
Track Reward
The bell is the universe's default shape — for good reason. The 68-95-99.7 rule, z-scores for comparing apples to oranges, the CLT for why averages always go bell. But always check: real data might have heavier tails than the math wishes.