## 7 Practical Application of the CNM Precision Validity Score
With more than 100 measured CNM values, however, the result significantly improves here. We therefore point out the uncertainty of the validity score for small sample sizes and recommend its calculation for larger sample sizes. Furthermore, based on the estimation of the parameters of the population, a certain small difference might always occur. In practice, CNMs with a very small number of measurements are rarely carried out, so the score should be suitable for practical use. To illustrate the defined measure, Figure 2 shows the number of measurements, precision in terms of the confidence interval, and validity scores for \(\delta^{*}=100\) kbit/s for selected departments and islands of France. The subfigures are arranged according to the number of available measurement results in the specified area. Each point in the upper map of each subfigure represents a CNM measurement result. Below are the key figures that correspond to the figures defined in the paper in terms of precision. The validity score is given according to the Standard Normal Interval, Studentized \(t\) Interval, and various bootstrapping methods to show the differences. In the top row, departments with sufficient measurements are shown. All regions have a validity score of \(\geq\)100% and precision according to the confidence interval of at most \(\overline{y}_{S}\pm 50\) kbits/s for the average throughput. In the bottom row, islands and departments are listed for which too few measurements are available to determine the average downlink value with a precision of at least 100 kbit/s. For Ile-de-France, for example, 8.5 million measurement results are available, which is sufficient to ensure that the real average throughput is within a confidence interval of \(\overline{y}_{S}\pm 13.3\) kbit/s. The mean value of the measurements is 23.78 Mbit/s. More precisely, according to the statistical analysis based on the 8.5 million samples and their distribution, the real mean of the population is actually between 23.76 Mbit/s and 23.79 Mbit/s (confidence level \(\alpha=0.05\)). This range is less than 100 kbit/s, which results in a validity score of '\(\geq\)100%'. For Brittany, 421k measurements are available, which corresponds to a confidence interval of \(\overline{y}_{S}\pm 52.1\) kbit/s. Here, the mean value \(\overline{y}_{S}\) is 21.61 Mbit/s but it can only be limited to 21.556 Mbit/s to 21.660 Mbit/s according to the Studentized \(t\) Interval. This corresponds to a range of 104.20 kbit/s, which is above the desired precision. For this reason, the subfigure has a yellow background. The validity score shows what percentage of the measurement results were obtained to maintain the precision. In this case, about 7-10% of the measurements are missing for Brittany, depending on the calculation method and estimate of the standard deviation from the sample. In the case of the bottom row, we recommend taking more measurements to maintain the desired precision and to ensure comparability between the different regions in terms of the average throughput. The validity score indicates the relative number of measurements required to achieve the desired precision. The description with the key figures below each map also shows the differences between the individual calculation methods of the validity score. The score varies from 89.96% for Bootstrapping Basic to 92.14% for Studentized \(t\) Intervals for Brittany, for example. In general, it makes the most sense to trust the bootstrapping intervals, as they provide a good estimate for the underlying variance of the distribution of the population. However, bootstrapping methods are slower to compute. Based on our experience and the small differences in practice, we recommend using the method with Standard Normal Interval to calculate the validity score. In the following, we present another practical example. We show that the calculation of the average downlink throughput for France is (almost) possible for regions, but not at the departmental level, based on our given data set.