It is important for detecting the anomaly in power systems before it expands and causes serious faults such as power failures or system blackout. With the deployments of phasor measurement units (PMUs), massive amounts of synchrophasor measurements are collected, which makes it possible for the real-time situation awareness of the entire system. In this paper, based on random matrix theory (RMT), a data-driven approach is proposed for anomaly detection in power systems. First, spatio-temporal data set is formulated by arranging high-dimensional synchrophasor measurements in chronological order. Based on the Ring Law in RMT for the empirical spectral analysis of `signal+noise' matrix, the mean spectral radius (MSR) is introduced to indicate the system states from the macroscopic perspective. In order to realize anomaly declare automatically, an anomaly indicator based on the MSR is designed and the corresponding confidence level $1-α$ is calculated. The proposed approach is capable of detecting the anomaly in an early phase and robust against random fluctuations and measuring errors. Cases on the synthetic data generated from IEEE 300-bus, 118-bus and 57-bus test systems validate the effectiveness and advantages of the approach.