Context : The long and almost continuous observations by Kepler show clear evidence of a granulation background signal in a large sample of stars , which is interpreted as the surface manifestation of convection . It has been shown that its characteristic timescale and rms intensity fluctuation scale with the peak frequency ( \nu _ { \mathrm { max } } ) of the solar-like oscillations . Various attempts have been made to quantify the observed signal , to determine scaling relations for its characteristic parameters , and to compare them to theoretical predictions . Aims : We aim to study different approaches to quantifying the signature of stellar granulation and to search for a unified model that reproduces the observed signal best in a wide variety of stars . We then aim to define empirical scaling relations between the granulation properties and \nu _ { \mathrm { max } } and various other stellar parameters . Methods : We use a probabilistic method to compare different approaches to extracting the granulation signal . We fit the power density spectra of a large set of Kepler targets , determine the granulation and global oscillation parameter , and quantify scaling relations between them . Results : We establish that a depression in power at about \nu _ { \mathrm { max } } /2 , known from the Sun and a few other main-sequence stars , is also statistically significant in red giants and that a super-Lorentzian function with two components is best suited to reproducing the granulation signal in the broader vicinity of the pulsation power excess . We also establish that the specific choice of the background model can affect the determination of \nu _ { \mathrm { max } } , introducing systematic uncertainties that can significantly exceed the random uncertainties . We find the characteristic frequency and amplitude of both background components to tightly scale with \nu _ { \mathrm { max } } for a wide variety of stars , and quantify a mass dependency of the latter . To enable comparison with theoretical predictions , we computed effective timescales and bolometric intensity fluctuations and found them to approximately scale as \tau _ { \mathrm { eff } } \propto g ^ { -0.85 } T ^ { -0.4 } and A _ { \mathrm { gran } } \propto ( g ^ { 2 } M ) ^ { -1 / 4 } ( or more conveniently R / M ^ { 3 / 4 } ) , respectively . Similarly , the bolometric pulsation amplitude scales approximately as A _ { \mathrm { puls } } \propto ( g ^ { 2 } M ) ^ { -1 / 3 } ( or R ^ { 4 / 3 } / M ) , which implicitly verifies a separate mass and luminosity dependence of A _ { \mathrm { puls } } . Conclusions : We provide a thorough analysis of the granulation background signal in a large sample of stars , from which we establish a unified model that allows us to accurately extract the granulation and global oscillation parameter .