which phenomena follow normal distribution?

Question

In nature so many phenomena following normal distribution such as human length, grass length etc.

My question is what what natural phenomena must follow normal distribution? Is there any criterion that can describe them all and exclude the others? What do all of them have in common?

Answer 1

Owing to the central limit theorem, as long as your data points are taken from independent random variables, and you have sufficiently many data points, the data will be approximately normally distributed, no matter what distribution the random variable has. A remarkable result, the central limit theorem applies to any natural phenomenon for which the observed random variables are independent.

Answer 2

Some natural phenomenon that follow the normal distribution exactly are

1. The velocities of molecules in the ideal gas. More generally, velocities of the particles in any system in thermodynamic equilibrium will have normal distribution, due to the maximum entropy principle.
2. The probability density function of a ground state in a quantum harmonic oscillator.
3. The position of a particle that experiences diffusion.

Many approximately normal things occur because of the central limit theorem.