*3.2. Sample Mean: Probability Distribution and Final Considerations*

The assumption of normality was verified for sample sizes *D* ≥ 30 independently on the distribution of the original sample variable *CPN*. However, in order to highlight the possible effect of a finite population, the normality assumption was checked for each Cluster and each value *N* < *Nmax* by means of the Kolmogorov-Smirnov (KS) test [48]. For *N* = *Nmax* no probability distribution can be defined since the variance is null.

The KS test was run under two different assumptions for the distribution of the sample mean:


Figure 8 shows the results of the Kolmogorov-Smirnov test for the two considered assumptions, in terms of percentage of samples passing/not passing the KS test for the four Clusters. It can be observed that under assumptions 1 and 2 the KS test is passed for all the Clusters for all the tested *N* values. This proves that the sample means are rigorously distributed by means of a normal model with the mean and variance correctly estimated by Equations (6) and (9), respectively. This also confirms that the estimation of the probability distribution of the sample mean is not affected by any finite population effect.

**Figure 8.** Percentage of samples passing/not passing the Kolmogorov-Smirnov test for the four tested Clusters under (**a**) assumption 1 and (**b**) assumption 2 for the underlying normal distribution of sample means.

Finally, for Cluster 2, Figure 9 shows a comparison between the empirical frequency and the normal probability models under the two assumptions, for the values *N* = 60 and *N* = 500; for these values, the KS test is passed under both assumptions. It can be noted that the CDF curves representing assumptions 1 and 2 are overlapped, highlighting the accuracy of the theoretical estimators adopted for the expected value and the standard deviation. Those results are shown for Cluster 2 but can be extended to all the Clusters.

**Figure 9.** Empirical vs. theoretical probability distributions of sample means for Cluster 2 under two different assumptions for the underlying normal model: (**a**) *N* = 60 and (**b**) *N* = 500.
