Summary statistics, such as the two-point correlation function, have also been used to characterize spatial structure of young stellar clusters. However, these methods have difficulty distinguishing between first moment (gradients in surface density) and second moment (clustering) phenomena in point processes (Bartlett 1974). The Q𝑄Qitalic_Q statistic (Cartwright & Whitworth 2004) is cleverly designed to address this problem: it uses the mean edge length of the minimum spanning tree graph normalized by the average distance between points. The Q𝑄Qitalic_Q parameter is calibrated through simulations so that fractal-like distributions of stars have Q0.8𝑄0.8Q>0.8italic_Q > 0.8. For MYStIX clusters the values of Q𝑄Qitalic_Q, in order of increasing central concentration, are: 0.61 for NGC 2264, 0.62 for NGC 6357, 0.71 for NGC 6334, 0.81 for Trifid, 0.82 for Tr 16, 0.83 for Tr 14, 0.89 for Orion, 0.9 for Tr 15, 0.91 for Eagle, and 0.92 for W 40. This agrees with the expectation that single clusters, like W 40, would have high central concentration, while a cluster with high sub-structure, like NGC 2264 would be more fractal.

This problem may also be addressed using the inhomogeneous two-point correlation function (Baddeley et al. 2000). In order to determine if the ellipsoid models are sufficient for describing the clustering of stars, the two-point correlation function is re-weighted by the subcluster mixture model, so that it will show clustering in the residuals that are not modeled. Values above the “null hypothesis” line indicate additional clustering, and statistical significance may be evaluated using the 99% confidence envelope. For the two cases above, the results indicate that the centrally concentrated W 40 cluster is adequately described by the ellipsoid model, while the model for NGC 2264 does not capture additional clustering at small separations.