Relationship between the low-frequency spectral index (${\alpha }_{54}^{144}$) and redshift. Left: Distributions of optical-selected (blue) and radio-selected (red) quasars. Right: Distributions for extended sources (yellow), optical compact sources (blue), and radio-compact sources (red). Error bars indicate the standard error of the mean ($\sqrt{\overline{x}/N}$) within each redshift bin. Blue, red, and orange lines represent optical-selected quasars, radio-selected quasars, and extended sources, respectively. Vertical lines mark the median values of each parameter in the corresponding histograms. Redshift bins were determined using Knuth’s binning method to optimize bin width (Knuth 2006). No significant trend is observed in the low-frequency spectral index (${\alpha }_{54}^{144}$) with redshift for either sample.