4 Individual populations

Figure 4: Color-color diagrams of the average populations

Figure 5: CPF and histograms for the color indexes for all the classes of object.

  
4.1 Average and Variances

For each class of MBOSS, we compute the average color indexes. Table 3 lists the average colors of the various classes of objects together with the square root of their variances (which will become equivalent to the standard deviation for large samples with a $\sim$normal distribution). These values are of practical interest, for instance when preparing observations of an object whose colors are not known.

Figure 4 displays the average populations' colors in a set of color-color diagrams.

Figure 5: Continued. CPF and histograms for the Gradient $\cal S$.

  
4.2 Histograms and cumulative probability functions

It is customary to visually compare distributions of objects using their histograms, i.e. the number of objects if given bins. Such histograms are displayed in Fig. 5. However, one has to be extremely careful in working with such plots: the size of the bin has a strong influence on the shape of the final histogram. Indeed, binning the data is equivalent to smoothing the data with a window equal to the bin size. Structures in the distribution that have a size similar to or smaller than the bin will be masked in the histogram, an effect that can create dangerously convincing - but wrong - artifacts.

A better way to represent a distribution is its Cumulative Probability Function (CPF). If one of the sample is x₁, x₂, ... , x_n(e.g. the V-R color indexes of n Centaurs), the corresponding CPF F(x) is the fraction of the sample whose value is smaller than x. The CPF always has a typical "S'' shape, with $F(-\infty)=0$, and F increases by step at each x_i till it reaches a value of 1 when x is larger than all the x_i. The advantage of the CPF is that no information is lost with respect to the original distribution. While its use is not as instinctive as that of the histogram, it is a worthwhile exercise to train ones eye to use them. The CPFs are also displayed in Fig. 5