## 2 Statistical Analysis of the Maximum Localized Imperfection (\(\omega\))  
Although there is a considerable uncertainty when characterizing localized imperfection (\(\omega\)) in cold-formed hollow sections, experimental data on the maximum localized imperfection (\(\omega_{\text{max}}\)) are available for the particular case of RHS and SHS stainless steel specimens. A statistical analysis of experimental results of the \(\omega_{\text{max}}\) from the literature [25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50  
The probability distribution for \(\omega_{\max}\) among the samples collected in the literature was identified by statistical distribution tests (Anderson-Darling method), as well as from probability plots. Both distribution tests and probability plots were performed using the statistical software Minitab 18 [32].  
For the case of distribution tests, Anderson-Darling statistics (AD) and \(p\)-values measure how well the specified distributions fit the data. For a given sample data and distribution, the smaller the AD, the better the distribution fits the data. Higher \(p\)-values indicate a better fit, and \(p\)-values less than 0.05 typically indicate that the data do not follow the specified distribution. The indicator LRT P is for three-parameter distributions only. A lower LRT P indicates that the related two-parameter distribution can be significantly improved by a third parameter. Goodness of fit test results for 16 different distribution tests are shown in Figure 6a. The Box-Cox transformation and the Johnson transformation are disregarded as the target is to identify probability distribution rather than to perform any transformation. It is found that the log-normal distribution (AD = 1.016, \(p\)-value = 0.011) represents the best fit for the data of \(\omega_{\max}\).  
Probability plots are another efficient way to determine whether the specified distribution fits the sample data. The closer the data to the middle straight line, the better the distribution fits the data. The probability plot of the sample data is shown in Figure 6b. It is observed that the data points are in close agreement with the center straight line. It again demonstrates that the sample data follow the log-normal distribution. The histogram of \(\omega_{\max}\) is shown in Figure 6c. The log-normal distribution is fitted to the histogram. Comparison of the cumulative probability (CDF) curve against the log-normal  
Figure 5: Distribution of \(\omega\) in the longitudinal direction for two tubes reported in the work of [19]. (a) RHS \(100\times 40\times 2\); (b) SHS \(60\times 60\times 3\).  
distribution is shown in Figure 6d, in which CDF determines the probability that an observation will be less than or equal to a certain value.  
Some standards provide the maximum allowable value of localized imperfection in the transverse direction [33; 34]. The tolerance for convexity/concavity of a cross-section specified in EN-10219-2:2006 [34] is \(\min\{\omega_{\max}/\text{b},\,0.5\,\text{mm}\}\), where \(\omega_{\max}/\text{b}\leq 0.008\); b is the side (straight side of the cross-section) length; and \(\omega_{\max}\) represents the maximum deviation from the straight side.