In the zoom lens of digital still cameras with the variation of the image plane generated by various symmetric error factors such as curvature, thickness and refractive index error of each lens surface about the optic axis, we induce a theoretical condition to fix constantly the image plane by translating the compensator group of the zoom lens by using the Gaussian bracket. We confirm the validity of this condition by using three examples of general zoom lens types with 3, 4, and 5 groups, respectively. When these error factors are randomly changed within the range of tolerance according to the Monte Carlo method, we verify that the distributions of the degree of moving of the compensator are normal distributions at three zoom lens types. From capability analysis using these results, we theoretically propose the method estimating the standard deviation, that is, sigma-level, as a function of the maximum movement of the compensator.