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We study the rate of growth of entire functions that are distributionallyirregular for the differentiation operator D. More specifically, given $p \in[1,\infty ]$ and $b \in (0,a)$, where $a = \frac{1}{2 max\{2,p\}}$, we provethat there exists a distributionally irregular entire function $f$ for theoperator D such that its p-integral mean function $M_p(f,r)$ grows not morerapidly than $e^r r^{-b}$. This completes related known results about thepossible rates of growth of such means for D-hypercyclic entire functions. Itis also obtained the existence of dense linear submanifolds of H(C) all whosenonzero vectors are D-distributionally irregular and present the same kind ofgrowth.

Order of growth of distributional irregular entire functions for the differentiation operator