Descriptive statistics correlation coefficient

The following description and corresponding MathCad Worksheet allows the user to test if two correlation coefficients are significantly different based on the number of sample pairs (N) used to compute each correlation. For the Worksheet, the user enters the confidence level for the test (e.g., 0.95), two comparative correlation coefficients, r, and r2, and the respective number of paired (X, Y) samples as N and N2. The desired confidence level is entered and the corresponding z statistic and hypothesis test is performed. A Test result of 0 indicates a significant difference between the correlation coefficients a Test result of 1 indicates no significant difference in the correlation coefficients at the selected confidence level. 

The description of large data tables by the usual univariate statistics (mean, standard deviation, range,. ..) and by histograms is still used in recent literature. Comparison between categories is made by the use of category means and ran s. Sometimes, the correlation coefficients are considered. The discussion of the extracted information can be wide-ranging and difficult to understand immediately. 

The accuracy of this system is dependent on the correlation coefficient of a retention description obtained from studies of QSRR, therefore, the selection of descriptors is the most basic and important task to construct RPS. This selection could be done with statistical framework, even if such description is not clearly derived from theories. The retention description obtained from QSRR studies is more effective for a rapid and accurate prediction of retention than that derived from theoretical models, because the former is simple and does not require introduction of a number of physicochemical parameters (they are often not clearly known and are very difficult and time-consuming to determine) for the latter case. By contrast, the consideration of physical meanings of descriptors derived from QSRR studies gave the overview of retention mechanisms in reversed-phase LC (7-10). That is to say, hydrophobicity, size and shape of alkyl-benzenes and PAHs are dominate factors controlling their retention. 

Starting from sixteen century onwards, the probability theory, calculus and mathematical formulations took over in the description of the natural real world system with uncertainty. It was assumed to follow the characteristics of random uncertainty, where the input and output variables of a system had numerical set of values with uncertain occurrences and magnitudes. This implied that the connection system of inputs to outputs was also random in behavior, i.e., the outcomes of such a system are strictly a matter of chance, and therefore, a sequence of event predictions is impossible. Not all uncertainty is random, and hence, cannot be modeled by the probability theory. At this junction, another uncertainty methodology, statistics comes into view, because a random process can be described precisely by the statistics of the long run averages, standard deviations, correlation coefficients, etc. Only numerical randomness can be described by the probability theory and statistics. 

There are several attractive features of such a mesoscopic description. Because the dynamics is simple, it is both easy and efficient to simulate. The equations of motion are easily written and the techniques of nonequilibriun statistical mechanics can be used to derive macroscopic laws and correlation function expressions for the transport properties. Accurate analytical expressions for the transport coefficient can be derived. The mesoscopic description can be combined with full molecular dynamics in order to describe the properties of solute species, such as polymers or colloids, in solution. Because all of the conservation laws are satisfied, hydrodynamic interactions, which play an important role in the dynamical properties of such systems, are automatically taken into account. 

This is for univariate data what happens in the case of multivariate (multiwavelength) spectroscopic analysis. The same thing, only worse. To calculate the effects rigorously and quantitatively is an extremely difficult exercise for the multivariate case, because not only are the errors themselves are involved, but in addition the correlation stmcture of the data exacerbates the effects. Qualitatively we can note that, just as in the univariate case, the presence of error in the absorbance data will bias the coefficient(s) toward zero , to use the formal statistical description. In the multivariate case, however, each coefficient will be biased by different amounts, reflecting the different amounts of noise (or error, more generally) affecting the data at different wavelengths. As mentioned above, these... 

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