Cosmological observations provide more accurate values both for the background evolution of the Universe and for the structure formation. These values are given by the so-called dark energy equation of state, ω, and the growth index parameter, γ. From these observed parameters, one can reconstruct the model functions in scalar-tensor gravity theories. However, there is a long-standing debate about the (in)equality between conformally transformed frames in scalar-tensor gravity models and some cosmological observables have been shown to be frame independent and others frame dependent. We focus on the cosmological evolutions of the frame-dependent observables, which are functions of the frame-independent parameter, the redshift. Thus, if one measures the cosmological parameters ω and γ from the frame-dependent cosmological observables in one specific frame, then all of the other observables should also be interpreted in that frame with the same cosmological parameters obtained from that measurement. If one analyzes the same observables in another frame, then the obtained cosmological parameters should be different from those of the original frame. This explicitly shows the frame dependence of cosmological observables. Also, our method provides a model-independent analysis of scalar-tensor theories of gravity models about various observables in both frames.