  A quasi-static approximation (QSA) for modified gravity can be applied in a
number of ways. We consider three different analytical formulations based on
applying this approximation to: (1) the field equations; (2) the equations for
the two metric potentials; (3) the use of the attractor solution derived within
the Equation of State (EoS) approach. We assess the veracity of these
implementations on the effective gravitational constant ($\mu$) and the slip
parameter ($\eta$), within the framework of Horndeski models. In particular,
for a set of models we compare cosmological observables, i.e., the matter power
spectrum and the CMB temperature and lensing angular power spectra, computed
using the QSA, with exact numerical solutions. To do that, we use a newly
developed branch of the CLASS code: QSA_class. All three approaches agree
exactly on very small scales. Typically, we find that, except for $f(R)$ models
where all the three approaches lead to the same result, the quasi-static
approximations differ from the numerical calculations on large scales ($k
\lesssim 3 - 4 \times 10^{-3}\,h\,{\rm Mpc}^{-1}$). Cosmological observables
are reproduced to within 1% up to scales ${\rm K} = k/H_0$ of the order of a
few and $\ell>5$ for the approaches based on the field equations and on the
EoS, and we also do not find any appreciable difference if we use the
scale-dependent expressions for $\mu$ and $\eta$ with respect to the value on
small scales, showing that the formalism and the conclusions are reliable and
robust, fixing the range of applicability of the formalism. We discuss why the
expressions derived from the equations for the potentials have limited
applicability. Our results are in agreement with previous analytical estimates
and show that the QSA is a reliable tool and can be used for comparison with
current and future observations to constrain models beyond $\Lambda$CDM.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology