Mean arithmetic

the arithmetic mean is simply referred to as mean and is expressed by 

Assume that the inspection department of an organization involved in the manufacture of systems for use in the area of transportation inspected ten identical systems and discovered 5, 4, 6, 7, 9, 10, 3, 2, 8, and 1 defects in each system. Calculate the average number of defects per system (i.e., arithmetic mean). 

By inserting the specified data values into Equation (2.1), we obtain 

the average number of defects per system is 5.5. In other words, the arithmetic mean of the given data set is 5.5. 

Assume that the following values represent the number of monthly patient safety-related incidents in a health care facility during a 9-month period 12,10, 8,6,4,15,9, 7, and 20. Calculate the average or mean number of patient safety-related incidents per month. 

The distribution of molecular weights in a polymer sample is commonly expressed as the proportions of the sample with particular molecular weights. The various molecular weight averages used for polymers can be shown to be simply arithmetic means of molecular weight distributions. 

Let us assume that unit volume of a polymer sample contains a total of A molecules consisting of a molecules with molecular weight M, U2 molecules 

The ratio a,/A represents the proportion of molecules with molecular weight Mf. Denoting this proportion by fu the arithmetic mean molecular weight will be given by 

Equation (4.2) gives the arithmetic mean of the distribution of molecular weights. Almost all molecular weight averages can be related to this equation. 

The distribution of molecular weights in a polymer sample is commonly expressed as the proportions of the sample with particular molecular weights. The mass of data contained in the distribution can be analyzed more easily by condensing the information into parameters that present a concise picture of the distribution and describe its various aspects. One such summarizing parameter is the arithmetic mean that is often used with synthetic polymers. The various molecular weight averages used for polymers can be shown to be simply arithmetic means of molecular weight distributions. 

Number-Average Molecular Weight Mn) stnd Degree of Polymerization (DPn) 

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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. 

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. 

As Eq. (29) shows, corresponds to the arithmetic mean and to the geometric mean of the homodimeric parameters AA and BB. 

This is similar in spirit to the arithmetic-mean rule but with each individual r,) being weighted according to the square of its value. The well depth in this function starts with a formula proposed by Slater and Kirkwood for the Cg coefficient of the dispersion series expansion ... 

When we report the result of a measurement a , there are two things a person reading the report wants to know the magnitude (size) of the measurement and the reliability of the measurement (its scatter ). If measuring errors are random, as they very frequently are, the magnitude is best expressed as the arithmetic mean p of N repeated tr ials xi... 

If very many measurements are made of the same variable a , they will not all give the same result indeed, if the measuring device is sufficiently sensitive, the surprising fact emerges that no two measurements are exactly the same. Many measurements of the same variable give a distribution of results Xi clustered about their arithmetic mean p. In practical work, the assumption is almost always made that the distribution is random and that the distribution is Gaussian (see below). 

In so doing, we obtain the condition of maximum probability (or, more properly, minimum probable prediction error) for the entire distribution of events, that is, the most probable distribution. The minimization condition [condition (3-4)] requires that the sum of squares of the differences between p and all of the values xi be simultaneously as small as possible. We cannot change the xi, which are experimental measurements, so the problem becomes one of selecting the value of p that best satisfies condition (3-4). It is reasonable to suppose that p, subject to the minimization condition, will be the arithmetic mean, x = )/ > provided that... 

This method, because it involves minimizing the sum of squares of the deviations xi — p, is called the method of least squares. We have encountered the principle before in our discussion of the most probable velocity of an individual particle (atom or molecule), given a Gaussian distr ibution of particle velocities. It is ver y powerful, and we shall use it in a number of different settings to obtain the best approximation to a data set of scalars (arithmetic mean), the best approximation to a straight line, and the best approximation to parabolic and higher-order data sets of two or more dimensions. 

Clearly, proposing arbitrary candidates for p and selecting the one with the smallest value of xi — p) to find x is not very efficient, nor can it be readily generalized. This is especially so because, even with a data set of integral numbers, the arithmetic mean does not have to be an integer. 

The sum of squares of differences between points on the regression line yi at Xi and the arithmetic mean y is called SSR... 

Look up the experimental values of the first ionization potential for these atoms and calculate the average difference between experiment and the computed values. Depending on the source of your experimental data, the arithmetic mean difference should be within 0.010 hartrees. Serious departrues from this level of agreement may indicate that you have one or more of your spin multiplicities wrong. 

Raw data are collected observations that have not been organized numerically. An average is a value that is typical or representative of a set of data. Several averages can be defined, the most common being the arithmetic mean (or briefly, the mean), the median, the mode, and the geometric mean. 

The median of a set of numbers arranged in order of magnitude is the middle value or the arithmetic mean of the two middle values. The median allows inclusion of all data in a set without undue influence from outlying values it is preferable to the mean for small sets of data. 

Another measure of dispersion is the coefficient of variation, which is merely the standard deviation expressed as a fraction of the arithmetic mean, viz., s/x. It is useful mainly to show whether the relative or the absolute spread of values is constant as the values are changed. 

The sound absorption of materials is frequency dependent most materials absorb more or less sound at some frequencies than at others. Sound absorption is usually measured in laboratories in 18 one-third octave frequency bands with center frequencies ranging from 100 to 5000 H2, but it is common practice to pubflsh only the data for the six octave band center frequencies from 125 to 4000 H2. SuppHers of acoustical products frequently report the noise reduction coefficient (NRC) for their materials. The NRC is the arithmetic mean of the absorption coefficients in the 250, 500, 1000, and 2000 H2 bands, rounded to the nearest multiple of 0.05. 

The U.S. ambient air standard has been estabHshed as 1.5 as a quarterly arithmetic mean. Some state standards are more restrictive, eg,... 

It has been found (7,9) that Z) > 0, ie, the maximum of an asymmetrical dye, X, is shifted to the short-wavelength region with respect to the arithmetical mean of the parent dye maxima. The phenomenon has been named a deviation (7). The positive deviations in PMDs are explained by the bond order alternation within the polymethine chain caused by different contributions of both end groups to the dye energetic stabiUty (7,9,10). The deviation reaches its maximum at Tqi > 45° and Tqi < 45° if the end groups have > 45° and 4>q2 > 45°, or < 45° and 4>q2 < 45°, then the... 

The simplest calculation of the mean, referred to as arithmetic mean (count mean diameter) for data grouped in intervals, consists of the summation of all diameters forming a population, divided by the total number of particles. It can be expressed mathematically by equation 1 ... 

The mean volume (mass diameter) is the arithmetic mean diameter of all the particle volumes or masses forming the entire population and, for spherical particles, can be expressed as in equation 2 ... 

The diametei of average mass and surface area are quantities that involve the size raised to a power, sometimes referred to as the moment, which is descriptive of the fact that the surface area is proportional to the square of the diameter, and the mass or volume of a particle is proportional to the cube of its diameter. These averages represent means as calculated from the different powers of the diameter and mathematically converted back to units of diameter by taking the root of the moment. It is not unusual for a polydispersed particle population to exhibit a diameter of average mass as being one or two orders of magnitude larger than the arithmetic mean of the diameters. In any size distribution, the relation ia equation 4 always holds. 

Thep and q denote the integral exponents of D in the respective summations, and thereby expHcitiy define the diameter that is being used. and are the number and representative diameter of sampled drops in each size class i For example, the arithmetic mean diameter, is a simple average based on the diameters of all the individual droplets in the spray sample. The volume mean diameter, D q, is the diameter of a droplet whose volume, if multiphed by the total number of droplets, equals the total volume of the sample. The Sauter mean diameter, is the diameter of a droplet whose ratio of volume-to-surface area is equal to that of the entire sample. This diameter is frequendy used because it permits quick estimation of the total Hquid surface area available for a particular industrial process or combustion system. Typical values of pressure swid atomizers range from 50 to 100 p.m. 

A quantitative expression of these observations is shown in equation 1, where is the observed absorption maximum for the unsymmetrical carbocyanine and Xj is the arithmetic mean (isoenergetic wavelength) for the absorption maxima of the related symmetrical dyes. 

Thus the average cost per share for John is the arithmetic mean of pi, po,. . . , pn, whereas that for Mary is the harmonic mean of these n numbers. Since the harmonic mean is less than or equal to the arithmetic mean for any set of positive numbers and the two means are equal only i pi=po = =pn, we conclude that the average cost per share for Mary is less than that for John if two of the prices Pi are distinct. One can also give a proof based on the Gaiichy-Schwarz inequality, To this end, define the vectors... 

The distribution function/(x) can be taken as constant for example, I/Hq. We choose variables Xi, X9,. . . , Xs randomly from/(x) and form the arithmetic mean... 

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