Central Limit Theorem

The central limit theorem states that given a distribution with a mean m and variance s2, the sampling distribution of the mean appraches a normal distribution with a mean and variance/N as N, the sample size, increases.

The central limit theorem explains why many distributions tend to be close to the normal distribution.

Here's a great learning example website: http://www.math.csusb.edu/faculty/stanton/m262/central_limit_theorem/clt.html

Addend:
If you are are averaging your measurements of a particular observable, your average's distribution may seem to tend toward a normal distribution. If the random variable that you are measuring is decomposable into a combination of several random variables your measurements may also seem to be normally distributed.