## 4 Statistical Analysis  
### Assessing Means of Two Samples Groups with ANOVA  
Comparing the mean values from two or more populations using Analysis of Variance (ANOVA) can provide valuable insights into the differences or similarities between the populations. ANOVA is a statistical method that allows us to determine whether there are statistically significant differences between the means of three or more groups. By conducting an ANOVA analysis, we can gain several key advantages and insights. Identifying Differences: One of the primary benefits of comparing mean values with ANOVA is the ability to determine whether there are significant differences between the populations. By analyzing the variance both within groups and between groups, ANOVA allows us to assess whether the observed differences in means are likely due to random chance or if there is a true effect. This analysis allows for partitioning the total variability in the data into different sources: the variability within groups and the variability between groups.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Group**} & \multicolumn{2}{c}{**Density of Specimens (kg/m3)**} \\ \cline{2-3}  & **Cube** & **Cored** \\ \hline
703-9 & 2217.7 & 2274.7 \\
433-9 & 2242.4 & 2272.0 \\
703-6 & 2177.6 & 2204.6 \\
433-6 & 2230.5 & 2259.4 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Density of specimen groups.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **703-6** & **703-9** & **433-6** & **433-9** \\ \hline Test results & 9.68 & 13.36 & 9.44 & 13.46 \\ Test results & 9.97 & 14.76 & 8.36 & 14.77 \\ Test results & 9.45 & 14.76 & 8.81 & 13.36 \\ Test results & & & & 12.61 \\ Test results & & & & 12.24 \\ Mean of tests & 9.70 & 14.29 & 8.87 & 13.29 \\ Standard dev. & 0.261 & 0.808 & 0.542 & 0.973 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Compressive strength of cube specimens in MPa for different CSRE mixtures tested after 120 days.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **703-6** & **703-9** & **433-6** & **433-9** \\ \hline Test results & 10.11 & 10.77 & 9.39 & 13.24 \\ Test results & 10.91 & 13.12 & 9.40 & 12.67 \\ Test results & 8.95 & 11.53 & 8.83 & 11.51 \\ Test results & 8.21 & 14.28 & & \\ Test results & 9.98 & 14.18 & & \\ Test results & 9.96 & & & \\ Mean of tests & 9.69 & 12.78 & 9.21 & 12.47 \\ Standard dev. & 0.955 & 1.575 & 0.326 & 0.882 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Compressive strength of cylinder specimens in MPa for different CSRE mixtures cored and tested after 120 days.  
This breakdown helps us understand how much of the overall variance is due to differences between populations and how much is due to random variation within groups. It is to emphasize that ANOVA results can often be generalized beyond the specific sample studied, providing insights into broader populations or settings. This generalizability is particularly valuable when making inferences or decisions based on the results of the analysis [49; 50; 51; 52]. Conducting an ANOVA requires the following four assumptions to be met:  
* independence of random variables for each population,
* measurability of variables,
* normality of the distribution of random variables in each population,
* variance homogeneity in all populations.  
Independence and measurability of the crush-test are provided. Each crush-test was made for a separate CSRE cube or cylinder. The results--the compressive strengths--were calculated based on the readouts of crushing force from the instrument separately for each sample. To confirm the normal distribution of the compressive strengths in each set of samples, the Shapiro-Wilk tests are done. Homogeneity of variances is verified throughout [PERSON]'s tests [49; 50; 51]. The sequence of calculations is presented in Figure 14.