We report the results from a study of the full sample of ∼6.031 × 10 7 K ± → π ± π 0 π 0 decays recorded by the NA48/2 experiment at the CERN SPS. As first observed in this experiment, the π 0 π 0 invariant mass (M 00 ) distribution shows a cusp-like anomaly in the region around M 00 = 2m + , where m + is the charged pion mass. This anomaly has been interpreted as an effect due mainly to the final state charge exchange scattering process π + π − → π 0 π 0 in K ± → π ± π + π − decay. Fits to the M 00 distribution using two different theoretical formulations provide the presently most precise determination of a 0 − a 2 , the difference between the ππ S-wave scattering lengths in the isospin I = 0 and I = 2 states. Higher-order ππ rescattering terms, included in the two formulations, allow also an independent, though less precise, determination of a 2 .