\begin{table}%t3
\caption{\label{tab3}Averaged diagnostic flux ratios of HzRGs.}
%\centerline
 {\begin{tabular}{l l c c c c c c}
\hline\hline\noalign{\smallskip}
\multicolumn{1}{c}{Sample}&&
\multicolumn{1}{c}{Number}&
\multicolumn{1}{c}{$\langle z \rangle$}&
\multicolumn{1}{c}{$\langle\log L(\ion{He}{ii})\rangle$}&
\multicolumn{1}{c}{$Z_{\rm NLR}/Z_\odot^a$}&
\multicolumn{1}{c}{\ion{C}{iv}/\ion{He}{ii}}&
\multicolumn{1}{c}{\ion{C}{iii}]/\ion{C}{iv}}\\
\hline\noalign{\smallskip}
$1.2 < z < 2.0$&$41.5 < \log L($\ion{He}{ii}$)^b < 42.5$&8&1.70&41.96&$0.80^{\rm +0.38}_{-0.50}$&$1.82^{\rm +0.47}_{-0.37}$&$0.77^{\rm +0.19}_{-0.16}$\\
&$42.5 < \log L($\ion{He}{ii}$)^b < 43.0$&7&1.66&42.70&$1.27^{\rm +0.34}_{-0.33}$&$1.72^{\rm +0.41}_{-0.33}$&$0.49^{\rm +0.11}_{-0.09}$\\
&$43.0 < \log L($\ion{He}{ii}$)^b < 45.0$&2&1.81&43.32&$1.36^{\rm +0.67}_{-0.60}$&$0.98^{\rm +0.38}_{-0.27}$&$0.99^{\rm +0.40}_{-0.28}$\\
$2.0 < z < 2.5$&$41.5 < \log L($\ion{He}{ii}$)^b < 42.5$&4&2.28&42.25&$0.77^{\rm +0.61}_{-0.77}$&$1.34^{\rm +0.57}_{-0.40}$&$1.14^{\rm +0.43}_{-0.31}$\\
&$42.5 < \log L($\ion{He}{ii}$)^b < 43.0$&10&2.28&42.78&$1.54^{\rm +0.34}_{-0.31}$&$1.21^{\rm +0.22}_{-0.19}$&$0.65^{\rm +0.13}_{-0.11}$\\
&$43.0 < \log L($\ion{He}{ii}$)^b < 45.0$&6&2.24&43.22&$1.53^{\rm +0.28}_{-0.26}$&$1.33^{\rm +0.22}_{-0.19}$&$0.56^{\rm +0.10}_{-0.08}$\\
$2.5 < z < 3.8$&$41.5 < \log L($\ion{He}{ii}$)^b < 42.5$&4&2.78&42.39&$0.88^{\rm +0.25}_{-0.26}$&$1.47^{\rm +0.24}_{-0.21}$&$0.94^{\rm +0.14}_{-0.12}$\\
&$42.5 < \log L($\ion{He}{ii}$)^b < 43.0$&9&3.06&42.73&$1.32^{\rm +0.25}_{-0.22}$&$1.46^{\rm +0.21}_{-0.19}$&$0.61^{\rm +0.10}_{-0.08}$\\
&$43.0 < \log L($\ion{He}{ii}$)^b < 45.0$&7&2.67&43.40&$1.64^{\rm +0.23}_{-0.24}$&$1.30^{\rm +0.16}_{-0.15}$&$0.52^{\rm +0.09}_{-0.08}$\\
\hline
\end{tabular}}
\medskip
$^a$~Assuming the hydrogen density of $n_{\rm H}=10^4 {\rm cm}^{-3}$; $^b$~line luminosity of \ion{He}{ii}$\lambda$1640 in units of ergs~${\rm s}^{-1}$.
\end{table}