The "Central Limit Theorem" is, in my opinion, one of the most remarkable theorems in mathematics. It says that if a number of samples are taken from a population with any probability distribution, as long as the mean is $\displaystyle \mu$ and the standard deviation is $\displaystyle \sigma$, then the average of all the samples will be, at least approximately, normally distributed with mean $\displaystyle mu$ and $\displaystyle \sigma$. The sum of n such samples will be, at least approximately, approximately normal with mean $\displaystyle n\mu$ and standard deviation $\displaystyle \sigma\sqrt{n}$.
Any time we have a measurement of a physical quantity we can think of that as combining many "subquantities" and so expect that it will have, at least approximately, the normal distribution.
("At least approximately": if the population from which the samples are drawn is normal, both sum and average will have, exactly, the normal distribution. If the population is not normally distributed, the distribution of average and sum will approach normal as n goes to infinity.)
