## 1 Introducing the Statistical Organization of Spray Data  
A spray is a two-phase flow of droplets interacting with a gaseous continuous phase. The physical process of liquid atomization depends on the atomizer type and breakup process, and once completed, the droplets formed have multiple sizes and velocities, and statistical histograms are the most common way of organizing the large amount of data on their characteristics. In the sense of data organization, the histograms organizing the sizes and velocities of droplets by classes do not represent a probability of occurrence, as in conventional statistical analysis, but a probability of _presence_ of droplets in a spray, since the atomization mechanisms already occurred. This small language shift allows considering each probability value as representing the degree of relevance of a certain class in the spray--a notion which will be essential to understand _drop size diversity_. In practice, after sorting data by classes, the way histograms represent the probability of presence is dividing the counts in each class (\(n_{k}\)) by the total sample size (\(N\)), \(p_{k}=n_{k}/N\). However, if there is a need to increase the detail of the distribution, one can build the discrete distribution in terms of density of probability of presence by considering the bin width (\(\delta D_{k}\)) in the probability value as \(pd_{k}=p_{k}/\delta D_{k}\). The bin width is constant if size classes are regularly spaced within the spectrum, or can vary the size if irregularly spaced. In the case of discrete probability distributions, the sum of all probability values associated with each class \(k\) is equal to one, \(\sum p_{k}=1\), which means that each probability value \(p_{k}\) corresponds to the weight a number of drops within a characteristic class \(k\) has in the entire spray. It is why one designates this way of presenting spray data as a _number-weighted_ probability distribution. There are other ways as shown later. The interpretation of the probability as a number-weighted value of class \(k\), for example, applied to drop sizes \(d_{k}\), allows for calculating the moments of the size distribution, such as the average size of droplets, \(d_{10}=\sum_{k}d_{k}p_{k}=\sum_{k}d_{k}pd_{k}\partial D_{k}\), whether using a probability discrete distribution or a probability discrete density distribution, respectively. Although this is basic statistical knowledge, in several research works, it is unclear which is the distribution reported, considering that each approach (probability or probability density) reacts differently when we increase the detail of a distribution by changing the bin width \(\delta D_{k}\), as illustrated in the example of Figure 1.  
While a smaller bin size implies a higher number of classes, in probability distributions, it leads, ultimately, to a uniform probability distribution with one class per sample (Figure 1a). However, in probability density distributions, it increases the detail to allow identifying eventual multimodalities does to different drop cluster with similar characteristics, or increase the noise in \(pd_{k}\) values, as illustrated in the examples depicted in Figure 1b. The purpose of remarking the difference between probability and probability density distribution is particularly relevant when comparing experimental with simulated drop characteristics. In the case of using a probability distribution (\(p_{k}\)), the number of classes must be the same, or the representative values of each class (\(d_{k}\) for drop size and \(u_{k}\) for one velocity component). Otherwise, if the presentation of spray data opts for the probability density distribution, which is dimensional (\(pd_{k}\) [\(\mu\)m\({}^{-1}\)]), it is not necessary to use the same number of classes. Ultimately, to avoid erroneous comparisons between experimental and simulated data, it is essential to be clear about the approach followed when presenting spray data in statistical format.