<p>A statistical structure is a pair of a Riemannian metric and a torsion-free affine connection that satisfies the Codazzi equation, and a manifold endowed with a statistical structure is called a statistical manifold. The Tchebychev vector field is an object unique to statistical manifolds, which plays an important role in affine differential geometry. Under an appropriate setting, the Tchebychev vector field can be expressed as the tension field of the identity map between two different statistical manifolds. We observe the statistical biharmonicity of this identity map. Within this setting, we determine the class of statistical manifolds for which the identity map is a statistical biharmonic.</p>