Plasticity and robustness, which are two basic concepts in the evolution of developmental dynamics, are characterized in terms of the variance of phenotype distribution. Plasticity concerns the response of a phenotype against environmental and genetic changes, whereas robustness is the degree of insensitivity against such changes. Note that the sensitivity increases with the variance, and the inverse of the variance works as a measure of the robustness. First, it is found that the response ratio is proportional to the phenotype variance, as described by extending the fluctuation-response relationship in statistical physics. Next, it is shown that through the course of robust evolution, the phenotype variance caused by genetic change decreases in proportion to that by noise during the developmental process. This evolution, resulting in increased robustness, is achieved only when the noise in the developmental process is sufficiently large; in other words, robustness to noise leads to robustness to mutation. For a system that achieves robustness in the phenotype, it is also found that the proportionality between the two variances also holds across different phenotypic traits. These general relationships for plasticity and robustness in terms of fluctuations are demonstrated using macroscopic phenomenological theory, simulations of gene-expression dynamics models with regulation networks, and laboratory selection experiments. It is also shown that an optimal noise level compatibility between robustness and plasticity is achieved to cope with a fluctuating environment.