The statistical dynamics of partially incoherent ultrafast lasers are complex and chaotic, which is significant for fundamental research and practical applications. We experimentally and theoretically reveal the statistical dynamics of the spectral evolutions and correlations in an incoherent noise-like rectangle pulse laser (NLRPL). Based on statistical histogram analysis, the probability distribution asymmetry of the spectral intensity fluctuation is decayed with the wavelength far away from the spectral peak due to the detection noise. The full-spectral correlation values indicate that the spectral similarity between two round trips is exponentially weakened as the round-trip offset increases. By studying the correlation map of spectral components, we find that the area of the high-correlation region is relevant to the pump power, which is reduced by increasing the pump power. The mutual information of the spectra demonstrates that two spectral components with symmetry about the spectral peak have a statistical dependence. Experimental observations and statistical properties can coincide well with theoretical numerical simulations. We reveal the pump-dependent spectral correlation of the NLRPL and provide multiple statistical methods for the characterizations of chaotic dynamics in incoherent light sources.