## 4 Results
### Confidence Considerations  
To check the confidence of our above assumption, possible sources for errors were investigated. Individual measurements at onshore locations were recorded at different hours and days, so they can \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Boundary Layer State**} \\ \hline  & **Unstable** & **Neutral** & **Stable** & **Total** \\ \hline
**All** & 84 & 23 & 12 & 119 \\
**Offshore** & 65 & 0 & 0 & 65 \\
**Onshore** & 19 & 23 & 12 & 54 \\ \hline
**All \(s_{p}^{2}\
eq\mathbf{0}\)** & 7 & 4 & 0 & 11 \\
**Probability (\%)** & 8.3 & 17.4 & 0.0 & 9.2 \\ \hline
**Offshore \(s_{p}^{2}\
eq\mathbf{0}\)** & 4 & 0 & 0 & 4 \\
**Probability (\%)** & 6.2 & -/- & -/- & -/- \\ \hline
**Onshore \(s_{p}^{2}\
eq\mathbf{0}\)** & 3 & 4 & 0 & 7 \\
**Probability (\%)** & 15.8 & 17.4 & 0.0 & -/- \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Correlation of the occurrence probability of a high shape factor to boundary layer state for all 119 measurements. be assumed to be independently and randomly sampled. The probability of occurrence of a high \(s_{p}^{2}\) event under combined unstable or neutral conditions was estimated to be \(\Phi_{0}=0.167\). The statistical test was based on a binomial distribution with the states _event occurred_ and _no event occurred_. To check the occurrence probability of an event in the onshore stable boundary layer states, for the null-hypothesis, an occurrence probability was assumed to be the same as for unstable/neutral states, and finding no event was just bad luck. The alternative hypothesis postulates a much lower probability of \(\Phi_{1}=0.062\) for high-\(s_{p}^{2}\) events under stable conditions (see Table 4). The number of expected events for \(H_{0}\) in onshore stable conditions is given in Table 4, and \(E_{0}=2\) and \(H_{0}\) were accepted if at least one event was found. However, no event of a non-Gaussian distribution was found under onshore stable conditions, and \(H_{0}\) was rejected. The type I error rate was about 11.2%. This is also the level of significance (\(\alpha\)). To improve the statistical significance, the number of samples in stable conditions had to be raised up to 17 for \(\alpha<5\%\) (or 26 for \(\alpha<1\%\)), but has not been done so far. Since the offshore measurements were often consecutive within the same conditions, and no observations in neutral or stable conditions were acquired, the statistical confidence could not be estimated for the offshore turbulence. However, the calculated occurrence probability gave a good approximation to allow the quality of the type II error for the onshore unstable and neutral boundary layers to be checked. The setup for this test is presented in Table 5 and the errors were 11.4% for type I (\(H_{0}\) was rejected despite it being true) and 14.9% for type II (\(H_{0}\) was accepted when it was really false). \begin{table}
\begin{tabular}{l l} \hline \hline
**Hypotheses:** & \\ \hline \(H_{0}\) & Occurrence probability is \(\Phi_{0}=0.167\) in stable conditions \\ \(H_{1}\) & Occurrence probability is \(\Phi_{1}<\Phi_{0}\) in stable conditions \\ \hline \multicolumn{3}{l}{**Error Type I**} \\ \hline Number of samples & 12 \\ Occurrence probability & \(\Phi_{0}=0.167\) \\ Expected occurrences & \(E_{0}=2.00\) \\ Variance & \(V_{0}=1.67\) \\ \hline Acceptance region for \(H_{0}\) & \{1...12\} \\ Rejection region for \(H_{0}\) & \{0\} \\ \hline Probability of error & 11.2\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Error estimation for the occurrence of non-Gaussian turbulence for onshore stable conditions.