The electroencephalogram (EEG) is a widely-used assay of neural function in research and medicine, whose advantages include high temporal resolution , portability, noninvasiveness, low cost, and ease of use. However, typical EEG recordings contain significant artifact from non-neural sources. These are often of high amplitude and at frequencies that overlap with neural signals. Most EEG analysis therefore involves removing artifacts based on their temporal or spatial features prior to quantitative analysis. While effective, these strategies can remove significant signal along with noise, and can be time-consuming and introduce bias. Here, I develop an alternative approach: tools that are tolerant of these outliers for a mainstay of EEG analysis; namely, spectral estimation. The basic strategy is to apply quantile statistics to multitaper spectral calculation. I then develop confidence intervals for these robust spectral estimates, as well as a novel spectral comparison test. Using simulated EEG data as well as healthy control human recordings, I show that the robust power spectral estimator is less sensitive to artifacts that affect the EEG power spectrum, compared to the standard method. Additionally, the robust approach to spectral comparison resulted in fewer false positives, false negatives, and errors of sign, compared with the standard two-group test approach. Lastly, I applied the robust method to spectrogram data from patients with disorders of consciousness who exhibit paradoxical activation in response to zolpidem treatment. The analysis shows that the robust method reduces movement and other artifacts even when all the data are used together, while reducing analysis time. I also corroborate findings from previous papers, including the restoration of posterior alpha rhythm seen in one patient after treatment. The main drawback of the robust approach, which is also illustrated by this application, is that it may discard spectral features that are intermittent, especially when they co-occur with artifacts. Thus, the new approach, while it reduces sensitivity to artifact, may be best used as an adjunct to artifact removal, rather than a replacement. In summary, I have created a toolkit that combines robust statistics with spectral methods and demonstrated its utility for EEG analysis.