Since we are considering a spectral range limited to $1.5<\Gamma<2.3$ , the sum in Eq. (3) is slightly less than 1 (say by a factor f) if the spectral dispersion $\sigma_{\Gamma}\ga 0.2$ . In this case we simply scale the p_i weights up by f to make Eq. (3) hold.

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Although the data samples used by Ueda et al. (2003) and La Franca et al. (2005) are not completely independent from a statistical point of view, we note that the output XLFs have been instead obtained with two completely independent analysis methods. Therefore, we safely apply the $\chi^2$ test to the combined Ueda et al. (2003) and La Franca et al. (2005) XLF datapoints. At any rate, when considering only the sample by La Franca et al. (2005), which is the larger among the two, we obtained $R_{\rm S}$ and $R_{\rm Q}$ values very similar to those from the combined fit.

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