see a 1-to-1 mapping between the value of $\Delta A$ and the corresponding frequency. More importantly, the cumulative distribution for a given $\Delta A$ quantifies the fraction of the randomized realizations whose differential area is smaller than this value.

In the framework of a specific cosmological model, we can calculate a normalized absolute area difference between the GP reconstructed function of the actual data and its theoretical counterpart $$\Delta A= \int_{z_{\rm min}}^{z_{\rm
max}}dz_s\bigg(\frac{\big|\mathcal{D}_{\rm mock}(\langle z_l\rangle,
z_s)-\mathcal{D}_{\rm th}(\langle z_l\rangle,
z_s)\big|}{\sigma_{\mathcal{D}}}\bigg)\;.$$ Therefore, based on the assumption that a curve with a smaller $\Delta A$ is a better match to $\mathcal{D}_{\rm obs}$, the cumulative distribution can be directly used to estimate the probability (i.e., the p-value) that a cosmological model is well consistent with the observations. More specifically, in order to decide which cosmology is favored by the observational data, we perform model comparison statistics by calculating its $\Delta A$ and apply the 1-to-1 mapping to determine the probability that it is inconsistent with the SGL sample. The results for different cosmological scenarios on the reconstructed $\mathcal{D}$ observations are listed in Table 1 and discussed as follows. We stress here that the observational distance ratio $\mathcal{D}$ has the advantage that the Hubble constant 