# Multivariate Analysis for Prediction of Splitting Tensile Strength in Concrete Paving Blocks
## 3 Results  
The database is obtained from measurements on 300 sample paving blocks; the tensile splitting strength is the response variable, and the models consider two groups of predictor variables, the first group of five variables (length, width, thickness, mass of the fresh paving block, and water absorption percentage) and the second group of two variables (density of the fresh paving block and percentage of absorption). As the first point in the data processing, an analysis of missing data is carried out, resulting in the database being complete; therefore, it does not require any imputation method. Outliers are determined by Mahalanobis distances, which represent the distance between each data point and its center of mass. For the first group of predictor variables, the square of the distance is calculated following a chi-square distribution with six degrees of freedom. The data belonging to the area under the curve of 99.9% of the distribution are preserved, evidencing seven records outside that represent 2.3% of the total database, which must be excluded for the analysis, resulting in a database of analysis that is made up of 293 records that enter the multivariateregression analysis of the first group of predictors. Similarly, for the multivariate regression analysis for the second group of predictors with three degrees of freedom, four records outside 99.9% of the distribution were detected, which are excluded and represent 1.3% of the total database, so the analysis database contains 296 records for the second group of predictor variables. A total of 80% of the database was separated for training records and 20% for validation of the models. The Table 1 shows the descriptive statistics in the sample of each variable, the measures of central tendency such as the mean and median, and the measures of dispersion such as the standard deviation, the range, and the coefficient of variation. Additionally, a histogram is included, where the values of the variable are grouped into intervals, and each rectangular bar indicates the frequency of the data in each interval, the red line indicates the density function for the behavior of the data in the sample, and at the bottom is the boxplot diagram and the points that indicate the data with its dispersion. \begin{table}
\begin{tabular}{l l l l} \hline \hline \multicolumn{1}{c}{**Variable**} & \multicolumn{1}{c}{**Measures**} & \multicolumn{1}{c}{
\begin{tabular}{c} **Histogram, Density Function,** \\ **Boxplot (Below), and Dots of Data** \\ \end{tabular} } \\ \hline \multirow{7}{*}{Tensile splitting strength, MPa} & Max: & 5.47 & MPa \\  & Min: & 1.31 & MPa \\  & Mean: & 3.82 & MPa \\  & Median: & 3.89 & MPa \\  & Standard deviation: & 0.74 & MPa \\  & Variance: & 0.55 & MPa\({}^{2}\) \\  & Range: & 4.16 & MPa \\  & Coefficient of variation: & 19.42 & \% \\ \hline \multirow{7}{*}{Thickness, mm} & Max: & 62.75 & mm \\  & Min: & 57.19 & mm \\  & Mean: & 60.08 & mm \\  & Median: & 60.08 & mm \\  & Standard deviation: & 0.85 & mm \\  & Variance: & 0.72 & mm\({}^{2}\) \\  & Range: & 5.56 & mm \\  & Coefficient of variation: & 1.41 & \% \\ \hline \multirow{7}{*}{Width, mm} & Max: & 102.03 & mm \\  & Min: & 98.77 & mm \\  & Mean: & 100.13 & mm \\  & Median: & 100.14 & mm \\  & Standard deviation: & 0.63 & mm \\  & Variance: & 0.39 & mm\({}^{2}\) \\  & Range: & 3.26 & mm \\  & Coefficient of variation: & 0.63 & \% \\ \hline \multirow{7}{*}{Length, mm} & Max: & 201.00 & mm \\  & Min: & 198.00 & mm \\  & Mean: & 199.68 & mm \\  & Median: & 200.00 & mm \\  & Standard deviation: & 0.60 & mm \\  & Variance: & 0.35 & mm\({}^{2}\) \\  & Range: & 3.00 & mm \\  & Coefficient of variation: & 0.30 & \% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Descriptive statistics of the variables.