Table 3: Best-fit results on the scaling relations corrected by the cosmological factor Ez(dashed and dotted lines in the plots; see Eq. (5)). The temperature, $T_{\rm gas, 6}$, is in unit of 6 keV; the luminosity, $L_{\rm bol, 44}$, in 1044  h70-2 erg s-1; the total mass, $M_{\rm tot, 14}$, in $10^{14}~ h_{70}^{-1}~ M_{\odot}$; the gas mass, $M_{\rm gas, 13}$, in $10^{13}~ h_{70}^{-5/2} ~M_{\odot}$. When the slope A is fixed, we estimate the error-weighted mean of $(\log Y -A \log X)$ and evaluate the error after resampling Y and Xby 1000 times according to their uncertainties. The scatter on Y is measured as $\left[ \sum_{j=1,N} \left(\log Y_j -\alpha -A \log X_j \right)^2 /N \right]^{1/2}$. Note that the scatter along the X-axis can be estimated as $\sigma_{\log X} = \sigma_{\log Y} / A$.
Relation (Y-X)$\alpha$A$\sigma_Y$
$E_z^{-1} L_{\rm bol, 44}-T_{\rm gas, 6}$ $0.79 (\pm0.07)$ $3.72 (\pm0.47)$0.35
  $1.00 (\pm0.02)$2.00 (fixed)0.41
$E_z M_{\rm tot, 14}-T_{\rm gas, 6}$ $0.75 (\pm0.03)$ $1.98 (\pm0.30)$0.15
  $0.74 (\pm0.02)$1.50 (fixed)0.15

$E_z M_{\rm gas, 13}-T_{\rm gas, 6}$		 $0.79 (\pm0.03)$ $2.37 (\pm0.24)$0.17
  $0.92 (\pm0.01)$1.50 (fixed)0.22

$E_z^{-1} L_{\rm bol, 44}- E_z M_{\rm tot, 14}$		 $-0.63 (\pm0.32)$ $1.88 (\pm0.42)$0.47
  $0.09 (\pm0.03)$1.33 (fixed)0.52


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