The estimators proportional to (N-A(xy)N-A(x,z)N-A(y,z))(1/2) and (N-L(x)N-L(y)N-L(z)) obtained by profile and intercept counts are recommended by ASTM Standards E 112 for 3D grain intensity lambda of anisotropic grain structures (N-A(center dot) are estimated in three mutually perpendicular planes suitably oriented with respect to the prevailing anisotropy direction and N-L(center dot) are intercept intensities along three suitably selected mutually perpendicular directions). However, X is also related to the profile and intercept intensities lambda', lambda", namely lambda = c'('lambda')(3/2) = c''(lambda")(3). The induced intensities are estimated by arithmetic means N-A = (N-A(x,y)+N-A(x,z)+N-A(y,z))/3, N-L = (N-L(x)+N-L(y)+N-L(z))/3. A simple model is developed, in which the role of geometric and arithmetic means of profile and intercept counts in the grain size estimation is elucidated. Its results are compared with computer simulated random tessellations with prevailing rod-like and plate-like cells and applied to anisotropic grain structures of pure Al produced by ECAP and to platinum based composite Pt - 0.5 Y2O3.