Statistical definitions

The numerical value of the exponent k determines which moment we are defining, and we speak of these as moments about the value chosen for M. Thus the mean is the first moment of the distribution about the origin (M = 0) and is the second moment about the mean (M = M). The statistical definition of moment is analogous to the definition of this quantity in physics. When Mj = 0, Eq. (1.11) defines the average value of M this result was already used in writing Eq. (1.6) with k = 2. 

By combining random flight statistics from Chap. 1 with the statistical definition of entropy from the last section, we shall be able to develop a molecular model for the stress-strain relationship in a cross-linked network. It turns out to be more convenient to work with the ratio of stretched to unstretched lengths L/Lq than with y itself. Note the relationship between these variables ... 

Planck Bed. Ber., 1908, 633) has, from the fundamental statistical definition of entropy, deduced the equation ... 

EWLS regression is by statistical definition the optimum method, and has the added benefits that (a) uncertainties of the slope and intercept of the regression are easily estimated, and (b) the probability that the data support assumption (1) above can be readily calculated. 

In general, all observed intemuclear distances are vibrationally averaged parameters. Due to anharmonicity, the average values will change from one vibrational state to the next and, in a molecular ensemble distributed over several states, they are temperature dependent. All these aspects dictate the need to make statistical definitions of various conceivable, different averages, or structure types. In addition, since the two main tools for quantitative structure determination in the vapor phase—gas electron diffraction and microwave spectroscopy—interact with molecular ensembles in different ways, certain operational definitions are also needed for a precise understanding of experimental structures. 

Operational definitions of molecular structure are needed to clarify experimental significance. In addition, some statistical notation is needed to clarify physical meaning. All statistical definitions hinge on the minimum of potential energy in a bound electronic state, which defines the equilibrium geometry or r,-intemuclear distance type. 

Facts include statistics, definitions, recorded statements, and observations. For example, a writer is drafting an essay assessing the flat tax from her outline (Lesson 6). Here is her thesis ... 

Sample The term sample refers to a single chemical sample as is understood by a chemist. This is in contrast to the statistical definition of a sample, which is a collection of observations from a population. 

One approach that QA would use to assure itself that a given process (step) is under control is the effort associated with the concept of process capability. Ekvall and Juran [15] defined the concept as the measured inherent reproducibility of the product turned out by the process. The statistical definition of process capability is that all the measured values fall within a 6-sigma range (i.e., range of the minimum to maximum limits). The information is used to show that the process is under control over a period of time as well as determine whether there is any drifting or abnormal behavior from time to time. Process validation is a QA tool in this case because its data will be used as the origin for the data curve developed for the process capability concept. 

Entropy is a measure of the degree of randomness in a system. The change in entropy occurring with a phase transition is defined as the change in the system s enthalpy divided by its temperature. This thermodynamic definition, however, does not correlate entropy with molecular structure. For an interpretation of entropy at the molecular level, a statistical definition is useful. Boltzmann (1896) defined entropy in terms of the number of mechanical states that the atoms (or molecules) in a system can achieve. He combined the thermodynamic expression for a change in entropy with the expression for the distribution of energies in a system (i.e., the Boltzman distribution function). The result for one mole is ... 

Considering calibrated gels or randomly crosslinked chains, the effect of linkage of monomeric units on NMR is well detected in spite of the purely statistical definition of the segmental spacing between crosslinks in vulcanised chains. 

The methods of quantitative analysis presented in this chapter are not unique to chromatography, but their principles will be discussed as they relate to chromatographic measurements. They will be preceded by a short section dealing with conventional statistical definitions and terms. 

For Landau free energy (m = 0, - the order parameter), Zt is a partition function of an extended system with additional variable . The integral denoted in square brackets in Eq. (3-12) is simply the configurational partition function of the system with a fixed value of -Z. The statistical definition of the Gibbs free energy function combined with Eq (12) results in the following expression ... 

It is obvious that not all residues that are folded are protected from exchange, since they can be exposed to the solvent in the native state or become exposed because ad]acent or complementary surfaces become unfolded. The statistical definition of the protection factors has the same form as that of the stability constants (equation 1) and can be expressed in terms of the folding probabilities... 

Fig. 23. Statistical definition of the limit of determination by the confidence interval in the analytical result98)...

---