  We study sample-to-sample fluctuations in a critical two-dimensional Ising
model with quenched random ferromagnetic couplings. Using replica calculations
in the renormalization group framework we derive explicit expressions for the
probability distribution function of the critical internal energy and for the
specific heat fluctuations. It is shown that the disorder distribution of
internal energies is Gaussian, and the typical sample-to-sample fluctuations as
well as the average value scale with the system size $L$ like $\sim L
\ln\ln(L)$. In contrast, the specific heat is shown to be self-averaging with a
distribution function that tends to a $\delta$-peak in the thermodynamic limit
$L \to \infty$. While previously a lack of self-averaging was found for the
free energy, we here obtain results for quantities that are directly measurable
in simulations, and implications for measurements in the actual lattice system
are discussed.
