A derivation of the single particle (SP) model is made from a Doyle-Fuller-Newman (DFN) model for electrodes composed of uniformly sized spherical electrode particles of one chemistry. The derivation uses a formal asymptotic method based on the disparity between the size of the thermal voltage and that of the characteristic change in overpotential that occurs during (de)lithiation. Comparison is made between the SP model and the DFN model for electrodes made from: lithium nickel manganese cobalt oxide (NMC), graphite and lithium iron phosphate (LFP). These are used to identify regimes where the SP model is accurate. For most chemistries, even at moderate rates, there are discrepancies between the DFN model and the SP model due to spatial non-uniformities in the electrolyte. Motivated by these discrepancies a correction term to the SP model is derived. Incorporating this into the SP model gives the corrected SP (cSP) model whose accuracy is very significantly improved over the SP model. Generalised versions of the cSP model for graded electrodes containing multiple electrode particle sizes (or chemistries) in different regions of the electrode, are also derived. The results of this generalisation to the cSP model compare favourably to the full DFN model, even at relatively high discharge rates, where the active electrode material is either graphite or a particular NMC variant.