Implied volatility and realized volatility are two different ways to describe the variations in the price returns of financial data. Their relationship, however, remains unclear in the literature. In this research, we establish a theoretical connection between these two types of volatility. We find that depending on the smoothness level of the cumulative implied volatility, the implied­realized association can be represented differently. Moreover, this relationship can be tested empirically. Using a non­parametric approach and martingale decomposition, on the one hand we are able to obtain estimators for the two variations in price returns. We examine the statistical properties of the procedure, in particular, how to set confidence intervals, and we investigate the impact of the estimation scheme on trading error. On the other hand, we can combine our theoretical findings with the ideas of ANOVA in analyzing stock­return variation in incomplete financial markets. This latter method decomposes the variation content implied from an option into two parts, the variation ''observed'' from historical price returns, and the residual variation (< Z, Z >) which may contain the variation from one or several extra instruments. A main device in the theoretical analysis is finding the small interval asymptotic distribution for the estimation error of < Z, Z >. Finally we discuss the implications of residual variation on the volatility model, and carry out numerical experiments and a data analysis with S&P500 data.