The nonlinear propagation of finite energy Airy beam (FEAB) in Kerr media was investigated numerically. The characteristics of whole beam self-focusing (WBSF) and small-scale self-focusing (SSFS) of the FEAB were thoroughly studied during its nonlinear propagation. It was found that the SSFS is more likely to take place than the WBSF, and the growth rate of noise modulation decreases with increasing the truncation factor. By comparing the gain spectra of SSFS for different truncaed factors, it can been seen that the truncation factor has a large impact on the gain spectrum of the FEAB indicating the possibility of the breakup of the FEAB induced by noise is inversely proportional to the truncated factor. In addition, the gain spectrum of FEAB is more similar to that of the Gaussian beam fitted from the main lobe of FEAB in the case of larger truncation factor. These results are very important for the potential application of the FEAB.