Fluctuations

The law of errors is the first example of a theory of fluctuations. It deals with sums of a large numberN of values σ1, . . . , σN occurring randomly with probability p(σ) equal for each pair of opposite values (i.e. p(σ) = p(−σ)), hence with 0 average. If the possible values of each σ are finitely many (and at least two) their sum can be of an order of magnitude as large as the number N , however such a large value is very improbable for large N and deviations from the average of the order of the square root of N follows the errors law, also called normal law, of Gauss