Arithmetic mean 67, normal

The resulting similarity measures are overlap-like Sa b = J Pxi ) / B(r) Coulomblike, etc. The Carbo similarity coefficient is obtained after geometric-mean normalization Sa,b/ /Sa,a Sb,b (cosine), while the Hodgkin-Richards similarity coefficient uses arithmetic-mean normalization Sa,b/0-5 (Saa+ b b) (Dice). The Cioslowski [18] similarity measure NOEL - Number of Overlapping Electrons (Eq. (10)) - uses reduced first-order density matrices (one-matrices) rather than density functions to characterize A and B. No normalization is necessary, since NOEL has a direct interpretation, at the Hartree-Fodt level of theory. 

The idea behind measures of location and central tendency is contained within the notion of the average. There are predominantly three summary statistics that are commonly used for describing this aspect of a set of data the arithmetic mean - normally shortened to the mean, the mode and the median. 

For examples of different types of similarity measures, see Table 6-2. The Tanimoto similarity measure is monotonic with that of Dice (alias Sorensen, Czekanowski), which uses an arithmetic-mean normaJizer, and gives double weight to the present matches. Russell/Rao (Table 6-2) add the matching absences to the nor-malizer in Tanimoto the cosine similarity measure [19] (alias Ochiai) uses a geometric mean normalizer. 

It is shown that the energy of a normal covalent bond A-B between unlike atoms is probably represented more closely by the geometric mean of the bond energies for A-A and B-B than by their arithmetic mean. 

It may be mentioned here that the mode which represents the most commonly occurring size in a given distribution is not of much use in mineral processing since it does not describe fully the characteristics of a group of particles. The arithmetic mean diameter suffers from the same limitation except when the distribution is a normal one. The harmonic mean diameter is related to the specific surface area. It is, therefore, useful in such mineral processing operations where surface area is an important parameter. 

In case of unsymmetric distributions both geometric mean and median are smaller than the arithmetic mean. In the same way as the distribution converges towards a normal one, geometric mean and median turn into the arithmetic mean. 

As a rule, the average blank is estimated from repetition measurements of a - not too small - number of blank samples as arithmetic mean yBL. If there is information that another than normal distribution applies, then the mean of this other distribution should be estimated (see textbook of applied statistics see Arnold [1990] Davies and Goldsmith [1984] Graf et al. [1987] Huber [1981] Sachs [1992]). 

Fig. 2 Normal, or Gaussian, size-frequency distribution curve. Percentage of particles lying within 1 and 2 standard deviations about the arithmetic mean diameter are indicated.

Nitzberg R. Analysis of the Arithmetic Mean CFAR Normalizer for Fluctuating Targets, IEEE Transactions on AES, 1, pp. 44-47, 1978. 

The classical and most used estimator for a central value is the arithmetic mean x (x mean in R-notation). Throughout this book the term mean will be used for the arithmetic mean. For a normal or approximately normal distribution the mean is the best (most precise) central value. 

Note that z can be larger than the number of objects, n, if for instance repeated CV or bootstrap has been applied. The bias is the arithmetic mean of the prediction errors and should be near zero however, a systematic error (a nonzero bias) may appear if, for instance, a calibration model is applied to data that have been produced by another instrument. In the case of a normal distribution, about 95% of the prediction errors are within the tolerance interval 2 SEP. The measure SEP and the tolerance interval are given in the units of v, and are therefore most useful for model applications. 

Arithmetic mean Distribution functions Normal distribution Rectangular distribution... 

For many purposes in analytical chemistry the normal distribution is a very useful model. The area imder the normal curve can be divided in various sections characterized by the arithmetic mean and a multiple of the standard deviation. These areas can then be interpreted as proportions of observations falling within ranges defined by specific probabilities. 

If a consensus value is used as the assigned value there are different possibilities to calculate it. If the arithmetic mean is used, an outlier test is required. But in many eases these tests are not very satisfaetoiy, espeeially if several outliers are present. If the tests are strietly used, they ean only be apphed to normally distributed data, whieh is usually not the case in trace analysis. 

Most readers will be familiar with the bell-shaped normal distribution plotted in Fig. 9.12. When applied to the size distribution of particles, for example, such a distribution is fully characterized by the arithmetic mean D and the standard deviation a, where a is defined such that 68% of the particles have sizes in the range D a In the log-normal distribution, the logarithm of the diameter D is assumed to have a normal distribution. (Either logarithms to the base 10 or loga-... 

Estimates of these parameters, based upon small samples of data, are designated x and s respectively. The corresponding parameters and their estimates of the normal distribution are the arithmetic mean, and x, and the standard deviation, a and s. 

Normal Random Variable. The probability density function of a normally distributed random variable, y, is completely characterized by its arithmetic mean, y, and its standard deviation, a. This is abbreviated as N (y,cr2) and written as ... 

The statistical tests previously described assume that the data follow a normal distribution. However, the results obtained by several analytical methods follow different distributions. These distributions are either asymmetric or symmetric but not normally distributed. In some approaches, these distributions are considered to be aberrant values superimposed on the normal distribution. In the following approach, the arithmetic mean is replaced by the median (cf. 21.1) and the standard deviation is replaced by the mean deviation, MD. 

The Formulation of the Electronegativity Scale.—In Section 3-4 it was pointed out that the values of the difference between the energy D A—B) of the bond between two atoms A and B and the energy expected for a normal covalent bond, assumed to be the arithmetic mean or the geometric mean of the bond energies Z>(A—A) and Z>(B—B), increase as the two atoms A and B become more and more unlike with respect to the qualitative property that the chemist calls electronegativity, the power of an atom in a molecule to attract electrons to itself. Thus both A, the deviation from the arithmetic mean, and A, the deviation from the geometric mean, increase rapidly in the sequence HI,... 

Only a few surveys have been done to measure toxaphene residues in agricultural soils. Carey et al. [27,28] surveyed cropland in the United States in 1971, collecting 1473 soil samples in 37 states that covered the eastern, southern and mid-western portion of the country and four western states California, Oregon, Washington and Idaho. Results have been summarized by Li et al. [26]. Only 33 of the sites reported toxaphene use and 92 soil samples were positive. The maximum and arithmetic mean of positive samples were 36 330 and 281 ngg-1 dry weight, respectively. Soil surveys have recently been done at farms in Alabama in 1996 and 1999-2000 [29-31], Louisiana and Texas in 1999 [29,30] and Georgia and South Carolina in 1999 [32]. Total toxaphene concentrations in the latter studies ranged from < 3-6500 ngg-1 (Table 2), and residues appeared to be log-normally distributed [29]. 

Empirical normal distributions often exhibit left (positive) or right (negative) skewness, in other words they are not symmetrical around the arithmetic mean. The skewness in-... 

Here, Joi and parameters defining the log-normal distribution. Joi is the median diameter, and cumulative-distribution curve has the value of 0.841 to the median diameter. In Joi and arithmetic mean and the standard deviation of In d, respectively, for the log-normal distribution (Problem 1.3). Note that, for the log-normal distribution, the particle number fraction in a size range of b to b + db is expressed by /N(b) db alternatively, the particle number fraction in a parametric range of Info to Info + d(lnb) is expressed by /N(lnb)d(lnb). 

The sizes of a powder sample are found to follow a log-normal distribution with arithmetic mean diameter, (b) the surface mean diameter, (c) the volume mean diameter, (d) the Sautermean diameter, and (e) the DeBroucker mean diameter. 

U.S. coal and equipped with ESPs for the northeastern U.S.) are selected, the variability of the emitted particle composition is reduced substantially. As an example of component constructed using these principles and data in the library, we give in Table IV the fine-particle component for eastern U.S. plants with ESPs. Here we have used arithmetic means and standard deviations, as that is the form in which components are normally entered into CMB calculations (13). A better representation of the particles as received at ambient sampling sites could, in principle, be obtained... 

With normally distributed data the arithmetic and geometric means would be equal, but with positively skewed data the arithmetic mean is always greater than the geometric mean. 

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