Due to uncertainty in manufacturing processes, applied probability and statistics have been widely applied for quality and productivity improvement. In spite of significant achievements made in causality modeling for control of process variations, there exists a lack of understanding on error equivalence phenomenon, which concerns the mechanism that different error sources result in identical variation patterns on part features. This so called error equivalence phenomenon could have dual effects on dimensional control: significantly increasing the complexity of root cause identification, and providing an opportunity to use one error source to counteract or compensate the others. Most of previous research has focused on analyses of individual errors, process modeling of variation propagation, process diagnosis, reduction of sensing noise, and error compensation for machine tool. This dissertation presents a mathematical formulation of the error equivalence to achieve a better, insightful understanding, and control of manufacturing process. The first issue to be studied is mathematical modeling of the error equivalence phenomenon in manufacturing to predict product variation. Using kinematic analysis and analytical geometry, the research derives an error equivalence model that can transform different types of errors to the equivalent amount of one base error. A causal process model is then developed to predict the joint impact of multiple process errors on product features. Second, error equivalence analysis is conducted for root cause identification. Based on the error equivalence modeling, this study proposes a sequential root cause identification procedure to detect and pinpoint the error sources. Comparing with the conventional measurement strategy, the proposed sequential procedure identifies the potential error sources more effectively. Finally, an error-canceling-error compensation strategy with integration of statistical quality control is proposed. A novel error compensation approach has been proposed to compensate for process errors by controlling the base error. The adjustment process and product quality will be monitored by quality control charts. Based on the monitoring results, an updating scheme is developed to enhance the stability and sensitivity of the compensation algorithm. These aspects constitute the "Error Equivalence Theory". The research will lead to new analytical tools and algorithms for continuous variation reduction and quality improvement in manufacturing.