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Entropy and complexity of the electroencephalogram (EEG) have recently been proposed as measures of depth of anesthesia and sedation. Using surrogate data of predefined spectrum and probability distribution we show that the various algorithms used for the calculation of entropy and complexity actually measure different properties of the signal. The tested methods, Shannon entropy (ShEn), spectral entropy, approximate entropy (ApEn), Lempel-Ziv complexity (LZC), and Higuchi fractal dimension (HFD) are then applied to the EEG signal recorded during sedation in the intensive care unit (ICU). It is shown that the applied measures behave in a different manner when compared to clinical depth of sedation score the Ramsay score. ShEn tends to increase while the other tested measures decrease with deepening sedation. ApEn, LZC, and HFD are highly sensitive to the presence of high-frequency components in the EEG signal.