Denominate numbers

Numerator Number of animals that died. Denominator Number of animals exposed, b Discontinuous exposure for 6 hours daily, 5 days per week. 

Number of measures (denominator) Number of measures (numerator of the fraction F) ... 

Align level of detail with the quality of the data. For example, segment-specific accident rates should not be applied to rail or highways unless good data exist for both numerators (number of accidents) and denominators (number of miles). Also, ensure that there have been a sufficient number of miles traveled across a segment so sound and justifiable accident rates can be developed. 

There are two kinds of numbers absolute numbers and denominate numbers. 

The factor of 2 in the denominator of the H2 molecule s rotational partition function is the "symmetry number" that must be inserted because of the identity of the two H nuclei. 

The F statistic describes the distribution of the ratios of variances of two sets of samples. It requires three table labels the probability level and the two degrees of freedom. Since the F distribution requires a three-dimensional table which is effectively unknown, the F tables are presented as large sets of two-dimensional tables. The F distribution in Table 2.29 has the different numbers of degrees of freedom for the denominator variance placed along the vertical axis, while in each table the two horizontal axes represent the numerator degrees of freedom and the probability level. Only two probability levels are given in Table 2.29 the upper 5% points (F0 95) and the upper 1% points (Fq 99). More extensive tables of statistics will list additional probability levels, and they should be consulted when needed. 

Defining the sample s variance with a denominator of n, as in the case of the population s variance leads to a biased estimation of O. The denominators of the variance equations 4.8 and 4.12 are commonly called the degrees of freedom for the population and the sample, respectively. In the case of a population, the degrees of freedom is always equal to the total number of members, n, in the population. For the sample s variance, however, substituting X for p, removes a degree of freedom from the calculation. That is, if there are n members in the sample, the value of the member can always be deduced from the remaining - 1 members andX For example, if we have a sample with five members, and we know that four of the members are 1, 2, 3, and 4, and that the mean is 3, then the fifth member of the sample must be... 

Since the total numbers of dyads and triads always occur as ratios in Eqs. (7.73) and (7.74), both the numerators and denominators of these ratios can be divided by the total number of dyads or triads to convert these total numbers into fractions That is, p-Jp = Pi/Ps- Thus the fractions in Table 7.9... 

The concept of a mass-transfer unit was developed many years ago to represent more rigorously what happens in a differential contactor rather than a stagewise contactor. For a straight operating line and a straight equilibrium line with an intercept of zero, the equation for calculating the number of mass-transfer units based on the overall raffinate phase N r is identical to the Kremser equation except for the denominator when the extraction factor is not equal to 1.0 [Eq. (15-23)]. 

Fxample 5 Number of Transfer Units Let us calculate the numher of transfer units required to achieve the separation in Example 3. The solution to the problem is the same as in Example 3 except that the denominator is changed in the final equation [Eq. (15-25)] ... 

In the numerator we have the speed and the flow. If we were comparing similar pumps into an application, these multiplied numbers would mostly be a constant. In the denominator we have the NPSHr of the pump (or competing pumps under comparison for an application). As the NPSHr of the pump goes down, the Nss value rises. As the Nss value increases, the operating window of the pump narrows. 

The result is a modified Euler number. You can prove to yourself that the pressure drop over the particle can be obtained by accounting for the projected area of the particle through particle size, S, in the denominator. Thus, by application of dimensional analysis to the force balance expression, a relationship between the dimensionless complexes of the Euler and Reynolds numbers, we obtain ... 

Practical probability is the limit of two ratios (Section 2.2). The numerator is the number of cases of failure of the type of interest (N) the denominator, the nonnalizing term is the time duration over which the failures occurred or the total number of challenges to the system. The former has the units of per time and may be larger than 1, hence it cannot be probability which must be less than 1. The latter is a dimensionless number that must be less than 1 and can be treated as probability. 

The numerator is a random normally distributed variable whose precision may be estimated as V(N) the percent of its error is f (N)/N = f (N). For example, if a certain type of component has had 100 failures, there is a 10% error in the estimated failure rate if there is no uncertainty in the denominator. Estimating the error bounds by this method has two weaknesses 1) the approximate mathematics, and the case of no failures, for which the estimated probability is zero which is absurd. A better way is to use the chi-squared estimator (equation 2,5.3.1) for failure per time or the F-number estimator (equation 2.5.3.2) for failure per demand. (See Lambda Chapter 12 ),... 

Failure rates are computed by dividing the total number of failures for the equipment population under study by the equipment s total exposure hours (for time-related rates) or by the total demands upon the equipment (for demand-related rates). In plant operations, there are a large number of unmeasured and varying influences on both numerator and denominator throughout the study period or during data processing. Accordingly, a statistical approach is necessary to develop failure rates that represent the true values. 

Number of demands The number of actual or estimated challenges placed upon a component to perform its function within the data window the demand-related failure probability denominator. 

Once it is determined that data exist, the next step is to begin the collection process. If sufficient thought and training is provided in the development and operation of the maintenance and operating reporting systems, much of the collection process can be automated. Automation assumes that a well-thought-out taxonomy is in place. If this is not the case, then an analyst must collect and review the records manually. In either case, the analyst must collect data from the plant sources previously discussed in order to determine the numerator (number of failures within a unique plant equipment population), and denominator (the operating time or number of demands for the equipment) of the equation to calculate failure rates. 

The numbers iVj and N- are only equal if there are no degeneracies. The sum in the denominator runs over all available molecular energy levels and it is called the molecular partition function. It is a quantity of immense importance in statistical thermodynamics, and it is given the special symbol q (sometimes z). We have... 

Bnich-kupfer, n. scrap copper, -last, /. breaking load, -metall, n. broken metal, scrap metal, -modul, m. modulus of rupture, -probe, /. breaking test, breakdown test, -punkt, m. breaking point, -riss, m. (Meial.) failure crack, -silber, n. broken silver, scrap silver, -spaonung,/. breaking stress tensile strength, -stein, m. quarry stone broken stone, -stelle,/. broken place, place of fracture. -strich, m. (Math.) fraction stroke (between numerator and denominator), -stiick, n. fragment shred, -stiicke, pi. debris scrap, -teil, m. fraction, -zahl, /. fractional number. 

For a system of multiple components of valves, pipe, and fittings. Equation 2-25 can be used to establish a component size to which each separate resistance can be expressed as a common denominator, or common pipe size. Under these conditions, the corrected Rvalues are additive and can be used as one number in Equation 2-27. These types of corrections should be made to improve and more accurately represent the pressure drop calculations. 

Scope, 52 Basis, 52 Compressible Flow Vapors and Gases, 54 Factors of Safety for Design Basis, 56 Pipe, Fittings, and Valves, 56 Pipe, 56 Usual Industry Pipe Sizes and Classes Practice, 59 Total Line Pressure Drop, 64 Background Information, 64 Reynolds Number, R,. (Sometimes used Nr ), 67 Friction Factor, f, 68 Pipe—Relative Roughness, 68 Pressure Drop in Fittings, Valves, Connections Incompressible Fluid, 71 Common Denominator for Use of K Factors in a System of Varying Sizes of Internal Dimensions, 72 Validity of K Values,... 

Let us now consider the same charged sphere immersed in various liquids with widely different values of n. By diluting water with dioxane at constant temperature, we can reduce n from 3.3 X 1022 toward zero. Clearly when n, the number of dipoles per unit volume, approaches zero, the total entropy lost per unit volume must approach zero. From this point of view the expression (170) is seen to have a somewhat paradoxical appearance, since e, which, according to Table 32, is roughly proportional to n, occurs in the denominator. This means that, as the number of dipoles per cubic centimeter decreases, the total amount of entropy lost progressively increases. The reason for this is that, when... 

Count the number of significant figures in the numerator and in the denominator the smaller of these two numbers is the number of significant figures in the quotient... 

In this generalized equation, (75), we see that again the numerator is the product of the equilibrium concentrations of the substances formed, each raised to the power equal to the number of moles of that substance in the chemical equation. The denominator is again the product of the equilibrium concentrations of the reacting substances, each raised to a power equal to the number of moles of the substance in the chemical equation. The quotient of these two remains constant. The constant K is called the equilibrium constant. This generalization is one of the most useful in all of chemistry. From the equation for any chemical reaction one can immediately write an expression, in terms of the concentrations of reactants and products, that will be constant at any given temperature. If this constant is measured (by measuring all of the concentrations in a particular equilibrium solution), then it can be used in calculations for any other equilibrium solution at that same temperature. 

In this equation the denominator is (n — 1) rather than n when the number of values is small. 

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