In general, the more a measurement is like the sum of independent variables with equal influence on the result, the more normality it exhibits.

The formula is regarded as "a stepping stone in the theory of sums of independent random variables".

In general we have for the sum of random variables:

The expression above can be extended to a weighted sum of multiple variables:

This formula may be derived from what we know about the variance of a sum of independent random variables.

Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean.

Another reason can be seen by looking at the formula for the kurtosis of the sum of random variables.

To this end, assume that is a sum of random variables such that the and variance .

If is a sum of random variables, the spreading of mass takes place in an dimensional space.

It is the distribution of the sum of independent exponential variables with mean .