## 1 Introducing the Statistical Organization of Spray Data
The question is why one should change or tune the number of classes when describing the spray characteristics. The underlying idea of increasing the number of classes is to obtain a greater detail of the probability distributions and detect eventual multimodality associated with clusters of data with dissimilar characteristics. In the case of drop sizes, an example generating such multimodality would be the presence of multiple atomization mechanisms (e.g., aerodynamic and/or hydrodynamic). In addition, in the case of the velocity, different two-phase flow events like the impact of droplets on solid surfaces contain information about the axial velocity component with positive values from those impinging on the surface, but the secondary droplets resulting after impact have a negative velocity component. Therefore, what is the criterion for choosing a given number of classes, \(k\)? In addition, should the spacing of these classes be regular or irregular? Considering regularly spaced classes, one of the basic principles introduced by [PERSON] [1] stated that the number of classes \(k>\log_{2}(N)\), with \(N\) as the total sample size. [PERSON] [2] further elaborated on [PERSON]'s rule, but the problem is the over-smoothing of histograms produced and its applicability  
Figure 1: Example of the effect of changing the bin size \(\delta\) in discrete probability (**a**) and probability density (**b**) size distributions. The simulated data follow a lognormal distribution function. limited to a number of data samples below 200, as analyzed by [PERSON] [3]. The alternatives for the [PERSON]'s and [PERSON]'s rules are the:  
* [PERSON]'s rule for the _bin width_ as \(\delta D=3.49 sN^{-1/3}\), where \(s\) is the standard deviation [4];
* [PERSON] rule, also for the _bin width_ as \(\delta D=2\left(IQR\right)N^{-1/3}\), with IQR as the _interquartile range_[5];
* [PERSON]'s rule for the number of classes is \(k=2N^{1/3}\)[6];
* and a rule based on J = 6 interlaced Fibonacci series with a number of classes defined as \(k=J\ln(N)/\ln(1.618)\)[7]. Considering a simulated example of two clusters of droplets, each following a lognormal distribution function  
\[f_{LN}(d_{g},\gamma)=\frac{1}{d\gamma\sqrt{2\pi}}\exp\left(-\frac{\ln(d/d_{g} )^{2}}{2\gamma^{2}}\right) \tag{1}\]  
with \(d_{g}\) as the geometric diameter and \(\gamma\) as the geometric standard deviation, the final distribution function is a mixture between the two as \(f(d)=w_{1}f_{LN}(40,0.5/\sqrt{6})+(1-w_{1})f_{LN}(70,0.5/\sqrt{6})\) with \(w_{1}=0.3\). Figure 2 shows the effect of using different criteria to organize drop size data with (a) \(N=10^{4}\) and (b) \(N=10^{5}\) measurements in the form of probability density distributions of drop size. For the two sample sizes tested, the Sturges' rule clearly over-smooths the distribution's bimodality. The Scott's rule generates less classes but is enough to capture the multimodality of the drop size distribution while producing the minimum number of empty classes. The Freedman and [PERSON] proposal, and the interlaced Fibonacci series, can provide greater detail, but the results with the lower number of samples (Figure 2a) prove the cost of increasing the number of classes by a larger noise  
Figure 2: Example of the effect of changing the number of classes in details obtained on the probability density distribution of a mixture between two clusters described by distinct lognormal distributions, considering different sample sizes of (**a**) \(N=10^{4}\) and (**b**) \(N=10^{5}\) droplets. observed in probability density values. It is noteworthy that the best approach to represent drop size distributions for comparison purposes is a probability density (\(pdf_{k}\) [\(\mu\)m\({}^{-1}\)]), since the amplitude of distributions with a different number of classes does not change as the amplitude of probability distributions (\(p_{k}\)). The second approach to define classes in drop size statistics (less so in velocity) is the use of irregular bin widths.