## 5 Empirical Model  
### The Statistical Reliability of the Results  
Each measurement in laboratory conditions is burdened with a certain error, mainly because of apparatus and researchers' imperfections. Furthermore, in the case of soil laboratory tests, there is also a significant heterogeneity of the material, which further reduces the quality of the results. This phenomenon is clearly visible in damping tests in a resonant column apparatus. Despite the same torsional frequency excitation, the same shear strain and the same effective stress obtained damping ratio values may differ by a few tenths. To confirm the reliability of the obtained damping ratio values, the standard statistical operations were carried out. Every damping test was repeated ten times at each shear strain so the database of over 12,000 damping ratio values was gathered. For ten repetitions at each shear strain, authors determined a minimum, maximum, and average value. The average value of the damping ratio was the reference result on the basis of which statistical parameters like standard deviation, interval, median, variation, standard error of the mean value, and absolute and relative measurement error were determined. Afterward, the calculated parameters were averaged over the entire consolidation stage and, in the subsequent move, overall consolidation stages for one sample. The results of these calculations for individual samples and for all conducted research are summarized in Table 4. It is worth noting that the standard deviations for all presented data are rather low, not exceeding the value of 0.2, except for the _K1_ and _K4_ samples. Hence, variations for these specimens are also the highest. Moreover, the same soils have the highest value of standard error of the mean value. However, this does not result in the biggest relative measurement error. This parameter is the highest for sample _K2_ and _K8_. Nevertheless, calculated average statistical volatility parameters and errors for the whole study are still very low, which shows high accuracy and correctness of the damping ratio recorded during the laboratory tests. Experimental Model Describing the Damping Properties of Soil in the Range of Small and Medium Strains. One of the main aims of the following study was to find the factors affecting the dynamic properties of selected clayey soils from Warsaw. These properties are represented in the article by damping ratio, i.e., minimal damping ratio (\(D_{MIN}\)) in the elastic range of shear strain and damping ratio (\(D\)) in the elastoplastic and plastic range of shear strain. In order to find the relationship between the damping ratio and the factors described in Section 4, the correlation analysis was carried out. This method, utilizing a linear correlation model, checks the relationship between two variables. That allows the expected outcome of the variable based on another variable that is correlated with the first variable to be calculated. The main parameter describing strength of the correlation is the Pearson correlation coefficient, otherwise known as the correlation coefficient (\(R\)). It takes values from -1 to 1. The closer to 0 the correlation coefficient is, the weaker the correlation is. The \(R\) equals -1 or 1 indicates a perfect linear relationship. Moreover, the sign at \(R\) value indicates the direction of a correlation.