The functions and elasticities of the cell are largely related to the structures of the cytoskeletons underlying the lipid bi-layer. Among various cell types, the Red Blood Cell (RBC) possesses a relatively simple cytoskeletal structure. Underneath the membrane, the RBC cytoskeleton takes the form of a two dimensional triangular network, consisting of nodes of actins (and other proteins) and edges of spectrins. Recent experiments focusing on the malaria infected RBCs (iRBCs) showed that there is a correlation between the elongation of spectrins in the cytoskeletal network and the stiffening of the iRBCs. Here we rationalize the correlation between these two observations by combining the worm-like chain (WLC) model for single spectrins and the Effective Medium Theory (EMT) for the network elasticity. We specifically focus on how the disorders in the cytoskeletal network affect its macroscopic elasticity. Analytical and numerical solutions from our model reveal that the stiffness of the membrane increases with increasing end-to-end distances of spectrins, but has a non-monotonic dependence on the variance of the end-to-end distance distributions. These predictions are verified quantitively by our AFM and micropipette aspiration measurements of iRBCs. The model may, from a molecular level, provide guidelines for future identification of new treatment methods for RBC related diseases, such as malaria infection.