The definition of the average error of numerical methods (by example of a quadrature formula \(\tilde S(\)f)\(=\sum^{n}_{i=1}c_ if(a_ i)\) to approximate \(S(f)=\int fd\mu\) on a function class F) is difficult, because on many important sets F there is no natural probability measure. We define the average a posteriori error of an approximation \(\tilde S\) by an averaging process over the set of possible information, which is used by \(\tilde S \)(in the example of a quadrature formula, \(N(F)=\{(f(a_ 1),...,f(a_ n))| \quad f\in F\}\) is the set of possible information). As an application of the theory we consider the numerical integration of functions of the class \(F=\{f: [0,1]\to {\mathbb{R}}| \quad | f(x)-f(y)| \leq | x-y| \}.\) For arbitrary knots \(a_ i\) we determine the optimal coefficients \(c_ i\) for the approximation \(\tilde S(\)f)\(=\sum^{n}_{i=1}c_ if(a_ i)\) and compute the resulting average error. The latter is minimal for the knots \(a_ i=(3i-2)/(3n-1)\) \((i=1,...,n)\).