Reliability of battery diagnosis depends on accurate estimation of the state of charge (SOC) and battery characterizing parameters including maximum capacity, internal impedance, polarization coefficients, and their probabilistic characterizations. This paper develops a framework that employs real-time operating data to estimate jointly the SOC and parameters, performs statistical analysis to derive quantitative diagnostic procedures with error analysis. Convergence of the algorithms, asymptotic distributions, and diagnosis reliability analysis are performed rigorously by using stochastic differential equations, central limit theorems, and large deviations principles. Simulated case studies and experimental data are used to illustrate the diagnosis algorithms and their capabilities. Experimental studies are conducted to verify the results.