## 3 Results and Discussion
### Hardness Analysis  
The procedure previously described was employed once more in order to analyze the distribution of the values of hardness. The results of the Vickers hardness measurement are collected in Table 6.  
Figure 12 shows a Weibull graph, where \(y=\ln(\ln(1-P_{\mathrm{f}})^{-1})\) represents the double logarithmic transformation of the cumulative probability of failure and \(x=\ln(\mathrm{HV}_{10})\) describes the natural logarithm of the failure of Vicker hardness. The black dots correspond to the experimental data points. The linear regression model, fitted to the data points, yielded the parameters describing the Weibull distribution listed in Table 7. \begin{table}
\begin{tabular}{c c c} \hline
**Experiment No.** & \(HV_{10}\)**, **GPa** & **In(In(\(1-P_{\mathrm{f}}\))\({}^{-1}\))** \\ \hline
1 & 17.5 \(\pm\) 0.5 & \(-\)3.218 \\
2 & 18.2 \(\pm\) 0.5 & \(-\)2.355 \\
3 & 18.65 \(\pm\) 0.5 & \(-\)2.068 \\
4 & 19.2 \(\pm\) 0.5 & \(-\)1.349 \\
5 & 19.32 \(\pm\) 0.5 & \(-\)1.061 \\
6 & 20.2 \(\pm\) 0.5 & \(-\)0.630 \\
7 & 20.35 \(\pm\) 0.5 & \(-\)0.486 \\
8 & 20.65 \(\pm\) 0.5 & \(-\)0.342 \\
9 & 21.55 \(\pm\) 0.5 & \(-\)0.198 \\
10 & 21.8 \(\pm\) 0.5 & \(-\)0.054 \\
11 & 22.2 \(\pm\) 0.5 & 0.377 \\
12 & 22.5 \(\pm\) 0.5 & 0.521 \\
13 & 22.55 \(\pm\) 0.5 & 0.665 \\
14 & 23.2 \(\pm\) 0.5 & 0.809 \\
15 & 23.25 \(\pm\) 0.5 & 0.953 \\ \hline \end{tabular}
\end{table} 
Table 6: Vickers hardness results and parameters of the Weibull distribution for the sintered samples of ZrO\({}_{2}\)–5 wt.% CeO\({}_{2}\). Figure 12: Weibull distribution of the hardness for ZrO\({}_{2}\)–5 wt.% CeO\({}_{2}\) sintered sample. The relatively high Weibull modulus, \(m\), of 13.159 suggests a narrow range of the hardness values, indicating a constant material hardness. The scale parameter \(\sigma_{0}\), that does not exceed 21.535 GPa, is the characteristic hardness value reached or exceeded by as much as 63.2% of the population. The value of the Weibull modulus of 13.159 GPa is consistent with results obtained from the bending strength analysis presented in Section 3.3 and with those published in other reports [44; 45; 46]. The expected value to failure (mean value to failure) was calculated based on Equation (8). The average durability of the sample in terms of hardness measured by the expected value to failure is equal to 20.079 GPa, Table 8.  
The median and the mean are almost identical, suggesting a moderately symmetrical distribution. The reliability function (the probability that the sample will keep its properties without failure) as well as failure function (the probability of failure) are shown in Figure 13. (Both were plotted based on the parameters from the linear fit shown in Figure 12). The failure function, widely known as the hazard rate, and the probability density of failure are shown in Figure 14. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Fit Parameters**} & \multicolumn{4}{c}{**Statistics of the Fit**} & \multicolumn{2}{c}{**Weibull Distribution Parameters**} \\ \hline
**a\({}_{0}\)** & **a\({}_{1}\)** & **r** & **s** & **Reduced \(\chi^{2}\)** & **m** & **HV\({}_{10}\), GPa** \\ \hline
13.159 \(\pm\) 3.43 \(\times\) 10\({}^{-6}\) & \(-\)40.394 \(\pm\) 2.092 & 0.999 & 0.0052 & 1.14 \(\times\) 10\({}^{-14}\) & 13.159 \(\pm\) 3.43 \(\times\) 10\({}^{-6}\) & 21.535 \(\pm\) 4.21 \(\times\) 10\({}^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The parameters of the Weibull distribution of the hardness for ZrO\({}_{2}\)–5 wt.% CeO\({}_{2}\) sintered sample. Figure 13: The reliability and failure functions drawn based on the parameters from the linear fit shown in Table 7 for the experimental hardness for ZrO\({}_{2}\)–5 wt.% CeO\({}_{2}\) sintered sample. The black points on the graph represent experimental data, whilst the solid lines represent the Weibull model. A thorough analysis of the data from the Weibull model depicted in Figure 12 reveals the presence of three distinct groups in the distribution.