## 6 Precision
### Standard Error and Confidence Intervals in the Context of CNMs  
The standard error (SE) is the standard deviation for a measured characteristic \(\Theta\) on the sampling distribution, i.e., it is a measure of how much an observed parameter in a sample deviates on average from the true parameter of the population. Speaking for CNMs, this corresponds to the variability of the measurement results of the users evaluating the same characteristic \(\Theta\) with estimator \(T(y_{S})\). The variability of the characteristic is firstly given by the spread of the values in the population \(U\) itself, i.e., the variance of \(y_{U}\) with \(Var(y_{U})=\mathbb{E}\Big{[}(y_{U}-\overline{y}_{U})^{2}\Big{]}\) and, secondly due to the nonexhaustive measurement methodology with sample observations \(S\subseteq U\) on the population \(U\). Thus, the standard error decreases as the population variance decreases. Furthermore, it decreases the more individual values are measured. SE is defined as standard deviation \(\sigma\) for the measured characteristic \(\Theta\) with \(\sigma(\Theta)=\sqrt{Var(\Theta)}\). Please note that we use the symbol \(\sigma\) in our work for the standard deviation of a specific evaluation or characteristic \(\Theta\) of CNM data. Other standard deviations are indicated by lowercase letters, for example \(s\), to distinguish between the two standard deviations with different data. If the characteristic to be measured is the mean value (\(\Theta=\overline{y}_{U}\)), \(\sigma\) is called standard error of the mean (SEM). The standard deviation of the population being sampled is seldom known. Thus, SEM on the sampling distribution \(S\) is estimated by  
\[\sigma(\overline{y}_{U})\approx\frac{s}{\sqrt{m}}\, \tag{1}\]  
where \(s\) is the standard deviation calculated by an estimator on sample \(S\), and \(m=|S|\) is the size of the sample. \(m\) is inversely included in the SEM, which means that the SEM decreases with increasing sample size. The estimator, i.e., the sample standard deviation \(s\) of the observations \(y_{i}\), is defined as  
\[s=\sqrt{\frac{1}{m-1}\sum_{i\in S}(y_{i}-\overline{y}_{S})^{2}}\, \tag{2}\]  
where \(y_{i}\) are the measured values, \(\overline{y}_{S}\) is the sample mean, and \(m\) is the size of the sample. \(\frac{1}{m-1}\) ensures that \(s\) is an unbiased estimator. Using \(s\), SEM \(\sigma(\overline{y}_{U})\) can be estimated as \(\frac{s}{\sqrt{m}}\), resulting in an absolute value for the degree of dispersion for a characteristic \(\Theta\) when sampling. Using SEM, confidence intervals (CIs) propose a range of plausible values for an unknown parameter of the real population (e.g., the mean \(\overline{y}_{U}\)). The interval has an associated significance level that the exact parameter \(\overline{y}_{U}\) is in the proposed range \(CI_{\alpha}\). The confidence interval for the mean is defined as  
\[CI_{\alpha}=[\overline{y}_{S}-z_{\frac{\alpha}{2}}\frac{s}{\sqrt{m}},\ \overline{y}_{S}+z_{\frac{\alpha}{2}}\frac{s}{\sqrt{m}}]\, \tag{3}\]  
with \(\overline{y}_{S}\) as sample mean, \(z_{\frac{\alpha}{2}}\) as quantile at \(\frac{\alpha}{2}\) for a given distribution, and \(\alpha\) is the chosen significance level. For crowdsourced measurements, this gives the possibility to quantify how precise a characteristic \(\Theta\) can generally be determined in terms of the number of measurements and a given significance level [30]. We use this in the following to define the minimal number of crowdsourced measurements (i.e., CNM observations) needed to achieve a certain precision of the data with respect to the pure number of measurements at a given confidence level. To maintain a precision given by the maximum absolute difference \(\delta^{*}=|\overline{y}_{S}-\overline{y}_{U}|\)[30] between the estimated mean value \(\overline{y}_{S}\) of the CNM \(S\) and the exact one \(\overline{y}_{U}\) of the underlying population, the minimum number of measurements \(n_{abs.