It's hard to explain the precise way in which an understanding of the central limit theorem has changed my life. However, knowing how any random distribution sums up to a Gaussian has subtly changed how I perceive and comprehend the world around me. Over time, this has added to a significant number of choices and decisions that I've subconsciously made, informed only by rough estimates of a mean value and it's standard deviation.
Just a word of warning, the central limit theorem is a bit misleading.
Assuming the sum independent and identically distributed random variables converges to a distribution, that distribution is not necessarily the Gaussian, but a larger family called the Levy-Stable distributions [1].
Levy-Stable distributions are "heavy-tailed" in that far away from zero they behave like (inverse) power laws. This is probably why you see so many power laws in nature (gravity, "small-world" networks, income distribution, galaxy distribution density, etc.). This was one of the central themes of Mandelbrot's soapbox, that power laws were more fundamental than normal distributions. Mandelbrot gets remembered for the highly symmetric and pretty fractal pictures but those images (Koch curve, Dragon Curve, Sierpinski gasket, etc.) are like a focusing on a sine wave when talking about Fourier analysis.
The central limit theorem applies to sums of random variables with finite variance [2]. Once you relax the condition of finite variance, or finite mean for that matter, Levy-Stable distributions are the more likely result.
[2] https://en.wikipedia.org/wiki/Central_limit_theorem " ... Mathematically, if X ... is a random sample ... taken from a population with mean ... and finite variance ... the limiting form of the distribution ... is the standard normal distribution." (emphasis mine)
On the other hand, for many systems the energy is related (in some way) to the square of the value. For example, voltages, currents, photon counts, etc. So if energy is finite, then you get CLT behavior.
It's hard to explain the precise way in which an understanding of the central limit theorem has changed my life. However, knowing how any random distribution sums up to a Gaussian has subtly changed how I perceive and comprehend the world around me. Over time, this has added to a significant number of choices and decisions that I've subconsciously made, informed only by rough estimates of a mean value and it's standard deviation.