# Probabilistic Analysis of Fatigue Behavior of Single Lap Riveted Joints
## 4 Statistical Analysis: Anderson-Darling Test  
In order to use the test data for setting up probabilistic methods, the experimental results should be fitted by a distribution function. Many distribution types, such as normal, log-normal, Weibull, Extreme Value, Exponential, etc., are available in statistics and it is very important to determine the distribution that better fits the test data. The Anderson-Darling test is very suitable [24] and used for application types herein presented [25; 26; 27]. It allows understanding if there are probabilities that a certain distribution could represent the analyzed data set. For datasets related to fatigue life and critical crack size, the following distribution types were tested: normal, log-normal, extreme value type and Weibull. The statistical test was performed by means of MATLAB(r) R2019a code. Anderson-Darling test [28] is characterized by the statistic value \(A^{2}\), also called goodness of fit statistic, evaluated by Equation (1):  
\[A_{n}^{2}=n\int_{-\infty}^{+\infty}\frac{\left[F_{n}(x)-F_{0}(x)\right]^{2}}{ F_{0}(x)[1-F_{0}(x)]}dF_{0} \tag{1}\]  
\(A^{2}\) is used to test the hypothesis that a random sample \(X_{1}\),..., \(X_{n}\), with empirical distribution \(F_{n}(x)\) comes from a continuous population with distribution function \(F(x)\), where \(F(x)=F_{0}(x)\) for some completely specified distribution function \(F_{0}(x)\). \(F_{n}(x)\) is defined as the proportion of the sample \(X_{1}\),..., \(X_{n}\) which is not greater than \(x\). The value \(A^{2}\) must be compared to its critical value \(A^{2}_{crit}\), which depends on the significance level (\(\alpha\)) and on the type of distribution. When the \(A^{2}\) value is lower than \(A^{2}_{crit}\), the supposed distribution type describes adequately the dataset. Vice versa, the hypothesis is rejected. Table 5 reports the values \(A^{2}\) related to fatigue life and critical crack size datasets. The table also reports the values \(A^{2}_{crit}\) at the significance level of 5% (\(\alpha\) = 0.05) for each type of distribution [29]. Immediately, it is possible to observe that for the fatigue life dataset at \(\sigma_{max}\) = 90 MPa there are no \(A^{2}\) values lower than \(A^{2}_{crit}\), so the test rejects the supposed distribution types. For crack length at 90 MPa, there is one value of \(A^{2}\) that satisfies the Anderson-Darling requirements. For the other datasets, at \(\sigma_{max}\) = 120 MPa and \(\sigma_{max}\) = 160 MPa, and for both fatigue life and crack size, there are at least two conditions that satisfy the hypothesis of the Anderson-Darling test, highlighted in red in Table 5.  
Usually, the test uses probability plots, in which ordered observations are plotted against their observed cumulative frequency, to further verify the best fit. The scale of the cumulative frequency was arranged so that a good fit (low values of \(A^{2}\)) results in a straight line and a bad fit (high values of \(A^{2}\)) in a significant deviation of the plotted points from a straight line. Probability plots of datasets are shown in the following figures: Figures 8-10 show the probability plots for fatigue life under the fatigue loads with a stress level of 90 MPa, 120 MPa and 160 MPa, respectively, considering normal, log-normal, extreme value and Weibull distributions. Similarly, Figures 11-13 show the probability plot for the critical crack size. It is possible to point out that the fatigue life dataset at a stress level of 90 MPa, as already mentioned, does not fit any supposed distribution (Figure 8). Anyway, splitting the data set into two sub-datasets, with separation at 781,700 cycles due to higher dispersion of subsequent data (Figure 3), which represents the 20% of the whole dataset, both fit normal distribution with \(A^{2}\) equals to 0.5789