The present invention relates to positive electrode active substance particles comprising a compound having a spinel type structure comprising at least Li, Ni and Mn, and having an Li content which is controlled such that a molar ratio of Li(Ni+Mn) therein is 0.3 to 0.65, an Ni content of 5 to 25% by weight, an Na content of 0.05 to 1.9% by weight and an S content of 0.0005 to 0.16% by weight, a sum of the Na content and the S content being 0.09 to 1.9005% by weight. The positive electrode active substance particles according to the present invention can be suitably used as positive electrode active substance particles for non-aqueous electrolyte secondary batteries which can exhibit a high discharge voltage and an excellent discharge capacity.