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Anomaly detection in hyperspectral images is of great interest in the target detection domain since it requires no prior information and makes full use of the spectral differences revealed in hyperspectral images. The current anomaly detection methods are susceptible to anomalies in the processing window range or the image scope. In addition, for the local anomaly detection methods themselves, it is difficult to determine the window size suitable for processing background statistics. This paper proposes an anomaly detection method based on the random selection of background pixels, the random-selection-based anomaly detector (RSAD). Pixels are randomly selected from the image scene to represent the background statistics; the random selections are performed a sufficient number of times; blocked adaptive computationally efficient outlier nominators are used to detect anomalies each time after a proper subset of background pixels is selected; finally, a fusion procedure is employed to avoid contamination of the background statistics by anomaly pixels. In addition, the real-time implementation of the RSAD is also developed by random selection from updating data and QR decomposition. Several hyperspectral data sets are used in the experiments, and the RSAD shows a better performance than the current hyperspectral anomaly detection algorithms. The real-time version also outperforms its real-time counterparts.