In this paper , we constrain the dimensionless Compton wavelength parameter B _ { 0 } of f ( R ) gravity as well as the mass of sterile neutrino by using the cosmic microwave background observations , the baryon acoustic oscillation surveys , and the linear growth rate measurements . Since both the f ( R ) model and the sterile neutrino generally predict scale-dependent growth rates , we utilize the growth rate data measured in different wavenumber bins with the theoretical growth rate approximatively scale-independent in each bin . The employed growth rate data come from the peculiar velocity measurements at z = 0 in five wavenumber bins , and the redshift space distortions measurements at z = 0.25 and z = 0.37 in one wavenumber bin . By constraining the f ( R ) model alone , we get a tight 95 % upper limit of \log _ { 10 } B _ { 0 } < -4.1 . This result is slightly weakened to \log _ { 10 } B _ { 0 } < -3.8 ( at 2 \sigma level ) once we simultaneously constrain the f ( R ) model and the sterile neutrino mass , due to the degeneracy between the parameters of the two . For the massive sterile neutrino parameters , we get the effective sterile neutrino mass m _ { \nu, { sterile } } ^ { eff } < 0.62 eV ( 2 \sigma ) and the effective number of relativistic species N _ { eff } < 3.90 ( 2 \sigma ) in the f ( R ) model . As a comparison , we also obtain m _ { \nu, { sterile } } ^ { eff } < 0.56 eV ( 2 \sigma ) and N _ { eff } < 3.92 ( 2 \sigma ) in the standard \Lambda CDM model .