An expression for the probability density of any distribution of observed values (given background values of known accuracy) is derived from the properties of multivariate normal distributions. This is used in the quality control of observation-'good' and 'bad' observations are assumed to have errors from a normal distribution and from a distribution giving no useful information respectively. Three methods of quality control are presented and compared, two of these are based on the probability density derived above, and the third is based on a related maximum probability analysis, They differ in the optimality principal used: Individual Quality Control finds the most likely quality (i.e. good or bad) for each observation, given information from all the others; Simultaneous Quality Control finds the most likely combination of qualities; while Variational Quality Control is based on a variational analysis which finds the most likely true values. The quality control should be considered as part of the 'analysis' process of using the observations; these approaches to quality control are considered as approximations to a system giving the 'best' analysis, based on minimizing a Bayesian loss function. Approximations are also necessary in their practical implementations; the effect of these on various operational schemes is discussed. The multi-observation framework used includes the 'background' check as a special case, and it is extended to deal with observations with common sources of gross error. Applications to multi-level checks, bias checks and checks for known error patterns are sketched. As a by-product the standard statistical interpolation formulae are derived from the properties of normal distributions, thus demonstrating the implicit dependence of statistical interpolation on the normal distribution.
