A robust method for determining the electron temperature and density of atmospheric-pressure non-equilibrium argon plasmas is reported. The methodology is based on the analysis of the continuum and line spectra of the plasma. Assuming that the electron energy distribution function (EEDF) is expressed as a two-temperature generalized EEDF (GEEDF), the gamma value of the GEEDF is determined through a grid search of the continuum spectrum analysis given by the bremsstrahlung process, which minimizes the mean-squared logarithmic error (MSLE). In addition, the relationship between the gamma value and the electron temperature and density is determined. Utilizing this relationship, the electron temperature and density are determined by minimizing the MSLE between the excited-state densities obtained from the line spectrum analysis and numerically calculated using the collisional-radiative model. This methodology yielded results that satisfied both continuum and line spectrum analyses. In addition, the same analysis was conducted either by continuum spectrum analysis or by line spectrum alone to compare the results.