The conformal anomaly number for new two-dimensional critical points obtained by adding a slightly relevant perturbation φ (renormalization group eigenvalue y ⪡ 1) to a given critical theory is obtained to lowest order in y to be c′ = c − y 3/ b 2 + …, where b is the operator product expansion coefficient in φφ ∼ (− b) φ. This problem is equivalent to determining the effect of a relevant perturbation on finite size scaling of the free energy on a strip. Corrections to scaling of the free energy are computed for small perturbations from the new fixed point. Conditions for the applicability of the Kac formula at the new critical point are derived. The results are used to explain the way the relevant operators change position in the conformal grid in the series of Z 2-multicritical points in the Andrews, Baxter and Forrester models. For the random and replicated q-state Potts models the conformal anomaly is obtained perturbatively in q − 2; it is shown that the scaling dimension of the random energy operator cannot be obtained from the Kac formula for general q near 2; modular invariance is shown to imply an infinite operator product algebra and negative multiplicities. The finite-size correction to the conformal anomaly and of the energy correlation length of the random Ising model in strips is obtained.