## 4 Results and Discussion: Statistical Approach
### Multiple-Regression Analysis
#### 4.3.1 Standardization  
In multiple-regression modeling, a variable is expressed as a function of the values of a set of variables. In this case, the aim is to express the UPV readings as a function \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Mechanical Property**} & \multicolumn{3}{c}{**Most Accurate Model**} & \multicolumn{1}{c}{**Linear Model**} \\ \cline{2-5}  & \(a\) & \(b\) & **R\({}^{2}\) (\%)** & **R\({}^{2}\) (\%)** \\ \hline Compressive strength & 21.353 & 194.252 & 97.91 & 87.51 \\ Modulus of elasticity & 23.404 & 208.482 & 90.31 & 62.54 \\ Splitting tensile strength & 25.383 & 31.679 & 64.86 & 53.94 \\ Flexural strength & 25.398 & 41.214 & 73.90 & 62.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simple-regression analysis. that is dependent on a set of mechanical properties. If the real values of the mechanical properties and the UPV readings are used, then the function will not only be influenced by the statistical relation between the UPV readings and the mechanical properties, but also by their values, since all these properties are not expressed on comparable scales, so the values will also influence the estimates [43]. This aspect is reflected by the simple example of a flexural strength of 7 MPa, that is a high value, while a compressive strength of 7 MPa is a very low value, although the value of both mechanical properties is the same, which is because the scales of both mechanical properties are not comparable. This difference in the scales affects the estimation by multiple regression, and the importance of each mechanical property in the value of the UPV is not shown in the expression, otherwise the value of each one would influence the final result. Before developing the multiple-regression models, both the UPV and the mechanical properties were standardized to address the above-mentioned problem of non-comparable scales. Thus, units were eliminated, obtaining dimensionless values in the interval (\(-2\), \(2\)) for both UPV and mechanical properties, and all properties were therefore expressed in values comparable to each other [54,55]. Standardization was performed according to Equation (2) (\(X_{S}\) is the standardized property; \(X_{exp}\) is experimental value of the property to be standardized, in its corresponding units (MPa, GPa, or km/s); \(X_{m}\) is the arithmetic mean of the values of the property to be standardized for all SCC mixes, in its corresponding units; \(X_{max}\) is the upper limit of the 95% confidence interval of the values of the property to be standardized for all SCC mixes, in its corresponding units; and \(X_{min}\) is the lower limit of the 95% confidence interval of the values of the property to be standardized for all SCC mixes, in its corresponding units). These values for each property, obtained from the experimental results shown in Section 3, are detailed in Table 4.  
\[X_{S}=\frac{X_{exp}-X_{m}}{X_{max}-X_{min}} \tag{2}\]  
The standardization procedure, in addition to making all the properties comparable with each other and enabling the use of multiple-regression models, defines two aspects of the properties to be known simply on the basis of their values. On the one hand, the sign can be used to determine whether a property for any mix is higher (positive sign) or lower (negative sign) than the arithmetic mean of the values of this property for all the SCC mixes, which are shown in Table 4. On the other hand, its value informs us that if that mechanical property for any mixture is within the 95% confidence interval obtained for the values of that property for all the SCC mixes (Table 4), then the standardized value must be between \(-0.50\) and \(0.50\). In summary:  
* A standardized value between \(0.00\) and \(0.50\) indicates that the property is above the mean and within the confidence interval. * A value greater than \(0.50\) shows that the property is above the upper limit of the confidence interval.