## 5 Results and Discussion
### Statistical Evaluation  
Several methods were used to evaluate the proposed \(\mathrm{V_{s}}\)-SPT-N values empirical relationships for all, sandy, and clayey soil types (Table 3). From the regression analysis, evaluated regression lines have an \(\mathrm{\bar{R}^{2}}\) that ranges from 0.73 to 0.80, suggesting a moderate to strong fit of the proposed empirical function to the obtained dataset. Evaluation using the coefficient of correlation (r) and the Spearman rank coefficient (r\({}_{\mathrm{s}}\)) suggested that there is moderate to strong correlation between the \(\mathrm{V_{s}}\) and SPT-N values from the dataset.  
An alternative method applied to evaluate the validity of the proposed empirical relations was through an analysis of the residuals. Graphical methods for analyzing the residuals applied in this research included the development of residual histograms and probability-probability (p-p) plots (Figure 7). Histogram plots of the residuals determined whether the variance of the datasets was normally distributed. An ideal histogram of the residuals, which follows a bell-shaped distribution and which is distributed around zero, suggests that the normality assumption is likely true. In this analysis, histograms were produced from standardized residual values, which is solely the raw residual divided by its standard error. The method of standardizing residuals is applied to transform data, such that its mean centers at zero and the standard deviation equals one. This technique also identified potential outliers, such that residuals with standard deviations less than \(-3\) or greater than \(3\), could be identified. In this analysis, histograms from the datasets suggested that the residuals generally followed a normal distribution, and outliers with standard deviations greater than \(13\) were not detected.  
Figure 6: (**A**). Uncorrected SPT-N and \(\mathrm{V_{s}}\) pairs for clayey soils and the determined empirical correlation. (**B**). Comparison of the proposed empirical correlation for clayey soil types and other published works [14,22,23,28,30,35,60,61,66].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Soil Type** & \begin{tabular}{c} **Proposed** \\ **Correlation** \\ \end{tabular} & \begin{tabular}{c} **Coefficient of** \\ **Regression (R\({}^{2}\))** \\ \end{tabular} & \
\begin{tabular}{c} **Coefficient of** \\ **Correlation(r)** \\ \end{tabular} &
\begin{tabular}{c} **Spearman** \\ **Rank Correlation (r\({}_{\mathrm{s}}\))** \\ \end{tabular} \\ \hline All soils & \(\mathrm{V_{s}}=56.82\mathrm{N^{0.4861}}\) & 0.7515 & 0.8767 & 0.8424 \\ Sandy soil & \(\mathrm{V_{s}}=45.07\mathrm{N^{0.5534}}\) & 0.7922 & 0.9050 & 0.8192 \\ Clayey soil & \(\mathrm{V_{s}}=70.26\mathrm{N^{0.4220}}\) & 0.7334 & 0.8453 & 0.8493 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary of the proposed correlations from this study.  
The second graphical technique applied was a p-p plot. This type of plot compares the empirical cumulative distribution function with the theoretical cumulative distribution function. Moreover, this plot can also show whether the residuals or errors follow the assumed normality. Points on the p-p plot must follow the 1:1 diagonal line for normally distributed data.  
Finally, the measured V\({}_{\text{s}}\) and estimated V\({}_{\text{s}}\) from each empirical correlation were compared, in order to assess the performance of the regression models (Figure 8). Data pairs mostly lay in the region between the lines 1:1.50 and 1:0.65.  
Figure 7: Histogram of the standardized residuals and the probability–probability (p-p) plots for (**A**) all soils, (**B**) sandy soils, and (**C**) clayey soils.