We investigate the mechanism of structural coloration by quasi-ordered nanostructures in bird feather barbs. Small-angle X-ray scattering (SAXS) data reveal the structures are isotropic and have short-range order on length scales comparable to optical wavelengths. We perform angle-resolved reflection and scattering spectrometry to fully characterize the colors under directional and omni-directional illumination of white light. Under directional lighting, the colors change with the angle between the directions of illumination and observation. The angular dispersion of the primary peaks in the scattering/reflection spectra can be well explained by constructive interference of light that is scattered only once in the quasi-ordered structures. Using the Fourier power spectra of structure from the SAXS data we calculate optical scattering spectra and explain why the light scattering peak is the highest in the backscattering direction. Under omni-directional lighting, colors from the quasi-ordered structures are invariant with the viewing angle. The non-iridescent coloration results from the isotropic nature of structures instead of strong backscattering.