Errors, quadratic addition

Let a measurement of the physical quantity yield the random variable X, and the deviation of X from be due to N independent (uncorrelated) errors. Hypothetical measurements with only one of these errors present would yield the deviations . If all these differences can be described by distributions with zero means and variances then the difference

follows a distribution of zero mean and variance

( Convolution). Expressed in errors rather than variances, one has the rule of quadratic addition of errors:

which can also be written

For errors of normal distribution, the total error will also have a normal distribution. For large N, the total error will have normal distribution for any distribution of the ( central limit theorem).