We propose a numerical validation of a probabilistic approach applied to estimate the relative accuracy between two Lagrange finite elements P k and Pm, (k < m). In particular, we show practical cases where finite element P k gives more accurate results than finite element Pm. This illustrates the theoretical probabilistic framework we recently derived in order to evaluate the actual accuracy. This also highlights the importance of the extra caution required when comparing two numerical methods, since the classical results of error estimates concerns only the asymptotic convergence rate.