## 4 Defining Statistical Validity for CNMs  
The problem of validity of measurements was generally extensively studied in various research in different domains [4; 22; 23; 24]. _Validity_ is, in addition to reliability and objectivity, a quality criterion for models, measurement, or test procedures [23; 25]. _Validity_: a measurement is valid if it actually measures what it is intended to measure, and thus, delivers credible results. _Reliability_: reliability relates to whether your research produces reliable results when done repeatedly. _Objectivity_: research is objective if there are no unwanted influences from people involved. Validity is fulfilled if the measurement method measures the characteristic with sufficient _accuracy_ that it is supposed to measure or that it pretends to measure [4; 25]. In empirical terms, validity denotes the agreement of the content of an empirical measurement with the logical measurement concept in reality. In general, this is the degree of accuracy with which the feature that is to be measured is actually measured. Definitions can be found in [23; 25]. Fundamental general work on sampling and sample theory is given, for example, in [26]. The accuracy of a measurement is further given by the _precision_ and _trueness_ of a measurement [4, 27]. The International Standard ISO 5725 [4] defines them as follows. \"The general term accuracy is used in ISO 5725 to refer to both trueness and precision. (...) Trueness refers to the closeness of agreement between the arithmetic mean of a large number of test results and the true or accepted reference value. Precision refers to the closeness of agreement between test results.\"  
The precision describes the spread of the results. The trueness ensures that the results also correspond to the correct or true value and are not distorted by the measurement concept, i.e., the representativeness must be ensured in such a way that _no bias_ or _systematic errors_ occur due to the measurement concept, even if the results are already precise. In the literature, the concept of validity is commonly further divided into several empirical and theoretical validity aspects for measurements [23]. These include construct validity, convergent validity, discriminant validity, or content validity [28]. In the following, we assume that the measured values were selected in the sense of the characteristic to be recorded (construct validity). Furthermore, in this work we only deal with questions about the degree of precision. In psychology [23] and medicine [24], studies on medication or treatment programs are regularly carried out. Generalized statements are drawn there from a finite number of observations, and in this case a sample. The studies are commonly performed (i) as representative as possible and (ii) until the desired precision prevails. In addition, the systematic error in election polls is kept low in electoral research [29] by a representative selection of the surveyed citizens to satisfy the validity [22]. Given a CNM \(S\) with scope \(m\), i.e., a measurement can be seen as a sample with \(m\) observations. Let \(S\subseteq U\) be the CNM with \(U\) as the finite underlying population \(U=\{1,\ldots,n\}\) with \(n\in\mathbb{N}\). For each element \(i\in U\) the value of a variable \(y\) can be measured. The vector of these values \(y_{i}\) is denoted by \(y_{U}\). The aim of the measurement is now to estimate a characteristic \(\Theta(y_{U})\) of \(U\) with the help of a sample \(S\). The characteristic to be estimated is often the population mean \(\mu=\hat{y}_{U}=\sum_{i\in U}\frac{y_{i}}{N}\) or the absolute sum with \(y_{U+}=\sum_{i\in U}y_{i}\). The measurement plan \(p(S)\) on \(S\) of the possible samples \(S\subseteq U\) assigns a measurement probability to each sample: \(p:S\rightarrow[0,1]\). CNMs result in uncontrolled observations without statistical certainty. The values observed in the measurement \((y_{i_{1}},\ldots,y_{i_{n}})\) are denoted by \(y_{S}\).