## 6 Precision
### CNM Precision Validity Score
Interval}}\)
11:endif\(\triangleright\) Quantile \(z_{\frac{\alpha}{2}}\) was calculated
12:\(\bar{y}_{S}\leftarrow\sum_{i\in S}\frac{y_{i}}{m}\)\(\triangleright\) Assumption: mean value \(\bar{y}_{S}\) and sample standard deviation \(s\) is stable
13:\(s\leftarrow\sqrt{\frac{1}{m-1}\sum_{i\in S}(y_{i}-\overline{y}_{S})^{2}}\)
14: Direct calculation of \(n^{min}_{abs}(\delta^{*})\) or \(n^{min}_{\text{\emph{rel. }}}(\gamma^{*})\) according to Equation (5), resp. Equation (7)
15: Calculate \(Val.\ Score_{\text{\emph{prec. }}}\) with \(\frac{m}{n^{min}_{eq}(t^{*})}\) with \(q\in\{abs.,rel.\}\) according to Equation (10)\(\triangleright\) Output \(Val.\ Score_{\text{\emph{prec.}}}\). ```  
**Algorithm 1**\(Val.\ Score_{\text{\emph{prec. }}}\) with Standard Normal Interval, Studentized \(t\) Interval (_approx. only_), or Bootstrapping. \begin{table}
\begin{tabular}{l l c c c c} \hline \hline
**Name** & **Confidence Interval Method** & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & **Remarks** \\ \hline \hline Val. Score (Norm.) & Standard Normal Interval & \(\bigcirc\) & \(-\) & \(\bigcirc\) & Version with the lowest computational complexity, \\  & & & & & Normal theory must be applicable \\  & & & & & Iterative calculation with correction of the standardized \\  & & & & & \(\bigcirc\) & \(\bigcirc\) \\ Val. Score (Stud-t) & Studentized \(t\) Interval & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) \\  & & & & & & \\ Val. Score (Boot Basic) & Basic Bootstrap Confidence Interval & \(\bigcirc\) & \(-\) & \(-\) & Basic bootstrapping procedure for taking into account \\  & & & & & any underlying distribution of the measured values \\ Val. Score (Boot Perc.) & Percentile Confidence Interval & \(\bigcirc\) & \(-\) & \(-\) & Bootstrapping where the bootstrap estimates are lined \\  & & & & & up from lowest to highest \\ Val. Score (BCa) & Bias Corrected and Accelerated Confi- & \(\bigcirc\) & \(-\) & \(-\) & Second-order accurate interval, adjusts for skewness \\  & & & & & in the bootstrap distribution \\ \hline \hline \end{tabular}  
* \(\bigcirc\) = Computational effort (\(\bigcirc\)/\(\bigcirc\)/\(\bigcirc\) = low/medium/high), \(\blacksquare\) = Applicable with few numbers of measurements (\(\bigcirc\)/\(\bigcirc\) = yes/no), \(\bigcirc\) = Normal theory must be applicable to estimate the distribution of the statistics (\(\bigcirc\)/\(\bigcirc\) = yes/no).  
\end{table} 
Table 2: Overview of the different versions of the CNM Precision Validity Score. The most relevant method for practice is highlighted in gray. ```
1:\(\hat{y}_{S}\leftarrow\sum_{i\in S}\frac{y_{i}}{m}\)\(\triangleright\) Assumption: mean value \(\hat{y}_{S}\) and sample standard deviation \(s\) is stable
2:\(s\leftarrow\sqrt{\frac{1}{m-1}\sum_{i\in S}(y_{i}-\overline{y}_{S})^{2}}\) (otherwise re-calculate it inside repeat...until with \(\hat{m}\) loop after incrementing \(\hat{m}\))
3: Temporary \(\hat{m}\) with \(\hat{m}\gets 0\) (or reasonable low start value \(m_{0}\))
4:repeat\(\triangleright\) Determine min. measurements required: \(n_{rel. }^{min}\)
5: Increment \(\hat{m}\) (since condition has not yet been reached)
6: Calculate \(z_{\frac{s}{2}}\) according to Student's \(t\) distribution with \(\hat{m}\) degrees of freedom
7:until\(z_{\frac{s}{2}}\cdot\frac{s}{\sqrt{m}}\leq\delta^{*}\), resp. \(\frac{z_{\frac{s}{2}}\cdot\delta^{-\sqrt{\hat{m}}}}{|\overline{y}_{S}|}\leq \frac{\gamma^{*}}{1+\gamma^{*}}\)\(\triangleright\) Depending on or absolute (\(\delta^{*}\)) or relative precision (\(\gamma^{*}\))
8:\(n_{q}^{min}\leftarrow\hat{m}\)
9: Calculate Val. Score\({}_{\text{prec. }}\) with \(\frac{m}{n_{q}^{min}(t^{*})}\) according to Equation (10)\(\triangleright\) Output \(Val.\ Score_{\text{prec. }}\)
```  
**Algorithm 2**\(Val.\ Score_{\text{prec. }}\) iterative approach with Studentized \(t\) Interval for CNM \(S\).