The article deals with the family of expanding circle maps \(f_{k,\alpha,\varepsilon}\!: {\mathbb S}^1 \to {\mathbb S}^1\) which when written mod 1 are of the form \[ f_{k,\alpha,\varepsilon}: \;x \to kx + \alpha + \varepsilon\sin(2\pi x), \] where the parameter \(\alpha\) ranges in \({\mathbb S}^1\) and \(k \geq 2\). Let \(h_{k,\alpha,\varepsilon}\) be the measure theoretic entropy of \(f_{k,\alpha,\varepsilon}\) with respect to its absolutely continuous invariant measure, \(I_{k,\varepsilon}\) be the average of \(h_{k,\alpha,\varepsilon}\) over \(\alpha\), and \(J_{k,\varepsilon} = \int_0^1 \log|Df_{k,\alpha,\varepsilon}| \, dx\). The main result is the following inequalities: \[ I_{k,\varepsilon}< J_{k,\varepsilon} < \max_{\alpha} \;h_{k,\alpha,\varepsilon}, \eqno(1) \] provided \(\varepsilon\) is small, where the difference in the left inequality is of order \(\varepsilon^{2k+2}\) and the difference in the right inequality is of order \(\varepsilon^{k+1}\). The proofs are based on Fourier series and Taylor expansions by powers of \(\varepsilon\) for the densities of the invariant measures. The dominant coefficients of the expansions are calculated explicitly in terms of the Bessel functions. The authors compare this result with known entropy estimates for the families of expanding Blaschke products (for Blaschke products, \(I_{k,\varepsilon}\geq J_{k,\varepsilon}\) in contrast to \((1)\)). Estimates \((1)\) were at first obtained via numerical simulations. These simulations are represented and discussed in detail as well.