This paper deals with statistical submanifolds and a family of statistical connections on them. The geometric structures such as the second fundamental form, curvatures tensor, mean curvature, statistical Ricci curvature and the relations among them on a statistical submanifold of a statistical manifold equipped with F-statistical connections are examined. The equations of Gauss and Codazzi of F-statistical connections are obtained. Such structures when the statistical submanifolds are conjugate symmetric are discussed. We present a inequality for statistical submanifolds in real space forms with respect to F-statistical connections. Also, we obtain a basic inequality involving statistical Ricci curvature and the squared F-mean curvature of a statistical submanifold of statistical manifolds.
