Outlier detection often relies on measuring how far values deviate from a "typical" center. While variance and standard deviation can be used in simple z-score based methods, they arenot robust: a few extreme outliers can inflate the mean and standard deviation, masking the very outliers you want to find. A widely taught robust alternative is themedian absolute deviation (MAD), which is based on the median rather than the mean and therefore resists distortion by extreme values.
MAD is computed by first taking the median of the data, then computing the absolute deviation of each point from that median, and finally taking the median of those deviations. Because medians are stable under extreme values, MAD provides a strong baseline for identifying unusually distant points. Many textbooks and data analysis references present MAD as a robust scale estimator for outlier detection, often combined with a threshold rule such as flagging points whose deviation exceeds a constant multiple of MAD (with a scaling factor sometimes used to make it comparable to standard deviation under normality assumptions).
In NumPy, you can implement MAD using np.median() and np.abs(). Mode is generally not useful for continuous numeric outlier detection, and variance/standard deviation are more sensitive to outliers than MAD. Thus, among the given options, the best statistical measure for detecting outliers robustly is the median absolute deviation.