Dispersions or mean square deviations

We restrict this discussion to dispersions or mean square deviations (MSD) of the Gibbs excess masses of component i = 1...N of a... 

The uncertainty of measurement represented by its dispersion or mean square deviation (MSD) of the Gibbs excess mass (o Qg) can be calculated from eq. (3.18) which results from eq. (3.14) or (3.34) by applying the Gauss... 

As already outlined in Chap. 2, Sects. 2.3 and 4.3 uncertainties or errors of measurements are important quantities for any kind of experimental work and should be provided for any quantity measured. Consequently we here denote the dispersions or mean square deviations (MSD), i = 1, 2 of the... 

We here present a formula allowing one to calculate numerically the dispersions or mean square deviation (MSD) of Gibbs excess masses of adsorbates which have been measured oscillometrically and... 

As a measure of variability or dispersion, the sum of squares considers how far the Xt s deviate from the mean. The mean sum of squares is called the variance (or mean squared deviation), and it is denoted by a2 for a population ... 

The size and composition dispersions of droplets can be estimated by using the maximum-term method, without performing detailed distribution calculations. This is accomplished by adapting the approach used for micellar systems,17-18 as described in Appendix C. The aggregate corresponding to the maximum a Xgo or Xgy/ is considered to provide the number-average size and composition of the equilibrium droplets. For each component k (=S, A, O, or W) present in the microemulsion, the mean-square deviation o 2(k) from the number-average number of molecules g (k) can be shown to be... 

In statistics, the square root of the dispersion variance is called the standard, or root mean square, deviation. In holds that [50]... 

A calculation of this sort ensures that the measure of dispersion is positive (squaring the deviations ensures that) and dividing by (n - 1) results in a quantity that represents an average of sorts. The sample variance is the "typical" or "average" squared deviation of observations from the sample mean. The use of the (n - 1) in the denominator may seem confusing, but the reason why this is done is that calculating the sample variance in this manner yields an unbiased estimator of the population variance, which is represented by the symbol o-. (The exact mathematical... 

Several ways may be used to characterize the spread or dispersion in the originai data. The range is the difference between the iargest vaiue and the smaiiest vaiue in a set of observations. However, aimost aiways the most efficient quantity for characterizing variabiiity is the standard deviation (aiso caiied the root mean square). 

Each individual measurement of any physical quantity yields a value A. But, independently of any possible observation errors associated with imperfect experimental measurements, the outcomes of identical measurements in identically prepared microsystems are not necessarily the same. The results fluctuate around a central value. It is this collection or Spectrum of values that characterizes the observable A for the ensemble. The fraction of the total number of microsystems leading to a given A value yields the probability of another identical measurement producing that result. Two parameters can be defined the mean value (later to be called the expected value ) and the indeterminacy (also called uncertainty by some authors). The mean value A) is the weighted average of the different results considering the frequency of their occurrence. The indeterminacy AA is the standard deviation of the observable, which is defined as the square root of the dispersion. In turn, the dispersion of the results is the mean value of the squared deviations with respect to the mean (A). Thus,... 

The standard deviation (or root-mean-square, rms, amplitude) always has the same dimensions as the original variable. It can be interpreted as a measure of the magnitude of the spread or dispersion of the original data from its mean. For this reason, it is used as a measure of the intensity of turbulence. 

Mean value and ran e are the simplest expressions and the most common. They are generally understood, but do not suffice at all in die characterization of a population (complete set) of values, or of a sample out of such a population. For variations in measurable quantities, such as delay time, the most useful measure of dispersion of a frequency distribution of these variations is the standard deviation, with which the reader must familiarize himself. The term standard deviation (represented by the Greek letter ff) Is the root-mean-square... 

Variance n One of the most used descriptive measures of a probability distribution, population or sample. It is equal to the square of the standard deviation. It is a measure of deviation from the expectation value or mean. It is a measure of statistical dispersion which quantifies the deviations from the expectation value as the mean of the squares of the distance from the mean. The variance of a random variable is defined as the second central moment of the random variable and is often denoted by cr and sometimes by Var(X), defined on a probability space, S, is given by ... 

Another parameter is called the standard deviation, which is designated as O. The square of the standard deviation is used frequently and is called the popular variance, O". Basically, the standard deviation is a quantity which measures the spread or dispersion of the distribution from its mean [L. If the spread is broad, then the standard deviation will be larger than if it were more constrained. 

Tlie expected value of a random variable X is also called "the mean of X" and is often designated by p. Tlie expected value of (X-p) is called die variance of X. The positive square root of the variance is called die standard deviation. Tlie terms and a (sigma squared and sigma) represent variance and standard deviadon, respectively. Variance is a measure of the spread or dispersion of die values of the random variable about its mean value. Tlie standard deviation is also a measure of spread or dispersion. The standard deviation is expressed in die same miits as X, wliile die variance is expressed in the square of these units. 

In this case the summation is the sum of the squares of all the differences between the individual values and the mean. The standard deviation is the square root of this sum divided by n — 1 (although some definitions of standard deviation divide by n, n — 1 is preferred for small sample numbers as it gives a less biased estimate). The standard deviation is a property of the normal distribution, and is an expression of the dispersion (spread) of this distribution. Mathematically, (roughly) 65% of the area beneath the normal distribution curve lies within 1 standard deviation of the mean. An area of 95% is encompassed by 2 standard deviations. This means that there is a 65% probability (or about a two in three chance) that the true value will lie within x Is, and a 95% chance (19 out of 20) that it will lie within x 2s. It follows that the standard deviation of a set of observations is a good measure of the likely error associated with the mean value. A quoted error of 2s around the mean is likely to capture the true value on 19 out of 20 occasions. 

A pdf or cdf is determined only approximately by any location index. For practical purposes it is sufficient to know the value of one location index—e.g., the mean—together with a measure indicating how the probability density is distributed around the chosen location index. There are several such measures called dispersion indexes. The dispersion index most commonly used and the only one to be discussed here is the variance V(x) and its square root, which is called the standard deviation a. 

The mean, p, characterizes the location of the data on the variable axis. The standard deviation, cr, and its square, the variance describe the dispersion of data around their mean (cf. Figure 2.2). The Greek letters are used by the statistician for true parameters of a population. Since only a limited number of measurements is available, the location and variance parameters must be estimated. The estimates are labeled by Latin letters or by a hat, for example, y. For estimations of the parameters of a Gaussian distribution, we obtain... 

One important function is the Gaussian distribution, often called the normal distribution or bell curve which is defined by two parameters. The mean, //, describes its average value. The variance denoted describes whether the distribution is narrow or dispersed. The square root of the variance is called the standard deviation and is denoted o. The Gaussian distribution is shown in Figure 15.1 and defined by the equation... 

The spread or dispersion of this set of values is measured by the sample variance, V, which is the sum of the squares of the deviation of each value from the mean, divided by one less than the number, n, of values in the set of samples If, instead of a limited set of samples, the entire set (population) of samples is available, the population variance, V, defined as the sum of the squares of the deviations divided by n, may be used. 

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