## 4 Results and Discussions
### Assumption Testing for Statistical Analysis  
Based on the above analysis, we hypothesize that the probability distribution of the microstructure of NS-modified silty clay can be described by the four distribution functions. In this study, the hypothesis is assessed by the chi-square test as follows:  
**Hypothesis H1:**_The observations obey a probability distribution \(F\) and the following statistical methods can be used ([PERSON] et al., 2008) ([PERSON] et al., 2008) ([PERSON] et al., 2008)._  
\[\chi^{2}=\sum_{i=1}^{n}\frac{f_{i}^{2}}{np_{i}}-n \tag{4}\]  
where \(\chi^{2}\) is the random variable and follows a \(\chi^{2}(k-r-1)\) distribution; we divide the possible values under \(H_{0}\) into disjoint subsets \(A_{1},A_{2},\cdots,A_{k}\); \(f_{i}\) (\(i=1,2,\cdots,k\)) is the number of the observation data \(x_{1},x_{2},\cdots,x_{n}\) that fall into \(A_{i}\); based on the hypothetical distribution \(X\) in \(H_{0}\), we can calculate the probability \(p_{i}\) of event \(A_{i}\); \(k\) is the amount of observation categories; and \(r\) is the distribution parameters amount of the distribution \(F\)  
Figure 11: The distribution laws of potential probability distributions and measured \(V_{D<0.1~{}\upmu\text{m}}\).  
respectively. If the following inequality is true, we accept the hypothesis; in other words, these observations agree well with the distribution \(F\).  
\[\chi^{2}<\chi_{\alpha}^{2}(k-r-1) \tag{5}\]  
where \(\alpha\) is the statistical significance level, and it is 5%, generally. In addition, we determine the acceptable level of hypothetical use with a chi-square test effect evaluation index (CEI).  
\[\text{CEI}=\frac{\chi^{2}}{\chi_{\alpha}^{2}(k-r-1)} \tag{6}\]  
Obviously, if the statistical effect is better, the value of CEI will be smaller. By the chi-square test, Weibull distribution of microstructure parameters with \(\text{CEI}>1\). However, the others are less than 1, which satisfies Equation (5). Hence, the hypothesis of Weibull distribution should not be accepted. The CEI values of the probability distributions of median pore diameters are shown in Figure 12. The CEI values of the normal, lognormal, and Gamma distributions are almost the same when the NS is added at 1%.  
However, at other NS levels, the CEI results of lognormal and Gamma distributions are bigger than those of the normal distribution, meaning that the normal distribution has a better effect to describing the statistical characteristic of the median pore diameter of NS-modified Qinghai-Tibet silty clay.  
Especially, for the NS addition of 0.25%, the CEI results of lognormal and Gamma distributions are far more than that of normal distribution. Therefore, the normal distribution is most consistent with the median pore radius probability distribution of the NS-modified Qinghai-Tibet silty clay.  
The CEI results of the normal, lognormal, and Gamma distributions of \(V_{D<0.1~{}\upmu\text{m}}\) are denoted in Figure 13. The CEI points of Gamma and lognormal distributions are bigger than those of normal distribution when 0.25% and 1% NS is added. In general, the normal distribution has the smallest CEI values among the three distributions. Consequently, it can be modeled well by the normal distribution that the measured results of the \(V_{D<0.1~{}\upmu\text{m}}\)  
Figure 12: CEI results for the normal, lognormal, and Gamma distributions of the median pore diameters.