PSI - Issue 84

Michele D’Amato et al. / Procedia Structural Integrity 84 (2026) 1175–1182

1180

committing a type I error, i.e., rejecting a true null hypothesis. The decision criterion is based on the p-value associated with each statistical test. If p- value ≥ α , the null hypothesis cannot be rejected, and the result is classified as NSS. On the contrary if p-value < α , the null hypothesis is rejected, and the result is considered SS. Drawing on material strength data from the database, in this study goodness-of-fit tests using values sourced from both acceptance certificates and in situ testing have been performed. Specifically, the Shapiro–Wilk (Shapiro et al., 1965; Shapiro et al., 1968) and D’Agostino–Pearson (D’Agostino et al., 1973; D’Agostino et al., 1990) tests are employed to assess distributive normality. These tests have been limited to only variables with a statistically significant number of data points. The Shapiro–Wilk test is widely regarded as one of the most robust tools for assessing normality, applicable to datasets with more than three observations. A key characteristic of this test is the assumption of data uniqueness. It presumes a continuous distribution of ordered values without identical data (ties), which is a crucial consideration when dataset preparation. The test verifies distribution normality by dividing the square of an appropriate sample linear combination by the usual symmetric estimate of the variance. The D’Agostino–Pearson test is a robust and versatile normality test, particularly effective for sample sizes exceeding eight observations. Unlike the Shapiro–Wilk test, it can accommodate datasets containing identical data. The method employs an Omnibus test to identify non- normality by analyzing both skewness (√b 1 ) and kurtosis (b 2 ). These two measures are transformed into standardized normal deviates (Z) to determine how closely the sample distribution aligns with a normal distribution. Shapiro–Wilk ( W ) and D’Agostino–Pearson ( K 2 ) statistics are are determined using the equations reported in Table 1. Goodness-of-fit tests are conducted in order to determine how well specific probability density functions fit the measured data of material strength. The analyzed variables are categorized based on the material type (reinforcing steel, concrete, and prestressing steel), data sources (acceptance certificates and in situ tests), structural elements, and declared strength classes. For each variable, descriptive statistical indicators were first computed, including measures of central tendency (mean and median), dispersion (coefficient of variation), and shape (skewness and kurtosis). Then, the distribution normality test is carried out by removing any outliers. Normality is assessed using primarily the Shapiro– Wilk test at a significance level α = 1%, and statistical testing was limited to variables with a sufficiently large number of observations. Due to the limited dataset for concrete, goodness-of-fit testing was feasible only for pier concrete, for which log-normality is evaluated using logarithmically transformed values. The Shapiro–Wilk results indicate that the normal distribution assumption cannot be rejected for reinforcing steel (for all variables examined) and prestressing steel (only for no. 6 variables examined). For concrete piers, the log normal distribution provides an adequate representation of the data. Moreover, three prestressing steel variables (Bars 1050, Braids 1860, and Strands 1860) initially showed statistically significant under the Shapiro–Wilk test; however, these datasets contain many identical data. Therefore, the D’Agostino–Pearson test, which is more robust in the presence of ties, was applied. The results confirmed that, when repetitions are properly accounted for, the normality assumption cannot be rejected for these variables. The results of Shapiro–Wilk ( W ) and D’Agostino-Pearson ( K 2 ) tests are summarized in Table 2. Table 1. Equations for the Shapiro-Wilk (W) and D’Agostino-Pearson statistics. Shapiro–Wilk statistic D’Agostino–Pearson statistic ⋅ = 1 ) 2 ∑ ( − ) 2 = 1 2 = 2 (√ 1 )+ 2 ( 2 ) = (∑ 4.1. Goodness-of-fit test results