In Internet of Things (IoT) systems in which a large number of IoT devices are connected to each other and to third-party servers, it is crucial to verify whether each device operates appropriately. Although anomaly detection can help with this verification, individual devices cannot afford this process because of resource constraints. Therefore, it is reasonable to outsource anomaly detection to servers; however, sharing device state information with outside servers may raise privacy concerns. In this paper, we propose a method to compute the $L_p$ distance privately for even $p>2$ using inner product functional encryption and we use this method to compute an advanced metric, namely p-powered error, for anomaly detection in a privacy-preserving manner. We demonstrate implementations on both a desktop computer and Raspberry Pi device to confirm the feasibility of our method. The experimental results demonstrate that the proposed method is sufficiently efficient for use in real-world IoT devices. Finally, we suggest two possible applications of the proposed computation method for $L_p$ distance for privacy-preserving anomaly detection, namely smart building management and remote device diagnosis.