Variance mean square root

A brief digression. In the language of statistics, the results for each of the stepped distributions in Figure 10-1 constitute a sample1 of the population that is distributed according to the continuous curve for the universe. A sample thus contains a limited number of x s taken from the universe that contains all possible z s. All simple frequency distributions are characterized by a mean and a variance. (The square root of the variance is the standard deviation.) For the population, the mean is u and the variance is a2. For any sample, the mean is x and the (estimate of) variance is s2. Now, x and s2 for any sample can never be as reliable as p and a2 because no sample can contain the entire population ir and s2 are therefore only the experimental estimates of g and cr2. In all that follows, we shall be concerned only with these estimates for simplicity s sake, we shall call s2 the variance. We have already met s—for example, at the foot of Table 7-4. 

The replication mean square, 0 075 in this case, is a measure of error variance The square root of this number is the standard deviation of experimental f test eTror if the experiment actually was repeated twice of test error, only, if the two results represent two analyses on each experiment. 

The normal distribution is characterized by two parameters. The center of the distribution, the mean, is given the symbol p. The width of the distribution is governed by the variance, The square root of the variance, ct, is the standard deviation. 

In a more general case, we have to carry out the following calculations firstly the values of the tj variable using relation (5.58) where Pj is the j regression coefficient, and Sp, represents the corresponding Pj mean square root of variance Sp ... 

It is then important to show that, in case of generalization, the mean square root of the variances with respect to the mean Pj value as well as its variances have the quality to respect the law of the accumulation of errors [5.13, 5.16, 5.19]. As a result, the mean square root of the variances will have a theoretical expression, which is given by... 

Once a significant difference has been demonstrated by an analysis of variance, a modified version of the f-test, known as Fisher s least significant difference, can be used to determine which analyst or analysts are responsible for the difference. The test statistic for comparing the mean values Xj and X2 is the f-test described in Chapter 4, except that Spool is replaced by the square root of the within-sample variance obtained from an analysis of variance. 

The square root of the variance is the standard deviation. The mean absolute error MAE is... 

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. 

Cliebysliev s theorem provides an interpretation of the sample standard deviation, tlie positive square root of the sample variance, as a measure of the spread (dispersion) of sample observations about tlieir mean. Chebyshev s tlieorem states tliat at least (1 - 1/k ), k > 1, of tlie sample observations lie in tlie... 

To compute the variance, we first find the mean concentration for that component over all of the samples. We then subtract this mean value from the concentration value of this component for each sample and square this difference. We then sum all of these squares and divide by the degrees of freedom (number of samples minus 1). The square root of the variance is the standard deviation. We adjust the variance to unity by dividing the concentration value of this component for each sample by the standard deviation. Finally, if we do not wish mean-centered data, we add back the mean concentrations that were initially subtracted. Equations [Cl] and [C2] show this procedure algebraically for component, k, held in a column-wise data matrix. 

An alternative method of variance scaling is to scale each variable to a uniform variance that is not equal to unity. Instead we scale each data point by the root mean squared variance of all the variables in the data set This is, perhaps, the most commonly employed type of variance scaling because it is a bit simpler and faster to compute. A data set scaled in this way will have a total variance equal to the number of variables in the data set divided by the number of data points minus one. To use this method of variance scaling, we compute a scale factor, sr, over all of the variables in the data matrix, 8g,... 

Normalization is performed on a sample by sample basis. For example, to normalize a spectrum in a data set, we first sum the squares of all of the absorbance values for all of the wavelengths in that spectrum. Then, we divide the absorbance value at each wavelength in the spectrum by the square root of this sum of squares. Figure C7 shows the same data from Figure Cl after variance scaling Figure C8 shows the mean centered data from Figure C2 after variance... 

The standard deviation s is the square root of the variance graphically, it is the horizontal distance from the mean to the point of inflection of the distribution curve. The standard deviation is thus an experimental measure of precision the larger s is, the flatter the distribution curve, the greater the range of. replicate analytical results, and the Jess precise the method. In Figure 10-1, Method 1 is less precise but more nearly accurate than Method 2. In general, one hopes that a and. r will coincide, and that 5 will be small, but this happy state of affairs need not exist. 

The second order central moment is used so frequently that it is very often designated by a special symbol o2 or square root of the variance, o, is usually called the standard deviation of the distribution and taken as a measure of the extent to which the distribution is spread about the mean. Calculation of the variance can often be facilitated by use of the following formula, which relates the mean, the variance, and the second moment of a distribution... 

The variance about the mean, and hence, the confidence limits on the predicted values, is calculated from all previous values. The variance, at any time, is the variance at the most recent time plus the variance at the current time. But these are equal because the best estimate of the current time is the most recent time. Thus, the predicted value of period t+2 will have a confidence interval proportional to twice the variance about the mean and, in general, the confidence interval will increase with the square root of the time into the future. 

Just as in everyday life, in statistics a relation is a pair-wise interaction. Suppose we have two random variables, ga and gb (e.g., one can think of an axial S = 1/2 system with gN and g ). The g-value is a random variable and a function of two other random variables g = f(ga, gb). Each random variable is distributed according to its own, say, gaussian distribution with a mean and a standard deviation, for ga, for example, (g,) and oa. The standard deviation is a measure of how much a random variable can deviate from its mean, either in a positive or negative direction. The standard deviation itself is a positive number as it is defined as the square root of the variance ol. The extent to which two random variables are related, that is, how much their individual variation is intertwined, is then expressed in their covariance Cab ... 

Poisson-distributed noise, however, has an interesting characteristic for Poisson-distributed noise, the expected standard deviation of the data is equal to the square root of the expected mean of the data ([11], p. 714), and therefore the variance of the data is equal (and note, that is equal, not merely proportional) to the mean of the data. Therefore we can replace Var(A s) with Es in equation 47-17 and Var(A r) with Et ... 

A variety of statistical parameters have been reported in the QSAR literature to reflect the quality of the model. These measures give indications about how well the model fits existing data, i.e., they measure the explained variance of the target parameter y in the biological data. Some of the most common measures of regression are root mean squares error (rmse), standard error of estimates (s), and coefficient of determination (R2). 

The statistics of the normal distribution can now be applied to give more information about the statistics of random-walk diffusion. It is then found that the mean of the distribution is zero and the variance (the square of the standard deviation) is na2), equal to the mean-square displacement, . The standard deviation of the distribution is then the square root of the mean-square displacement, the root-mean-square displacement, + f . The area under the normal distribution curve represents a probability. In the present case, the probability that any particular atom will be found in the region between the starting point of the diffusion and a distance of J (the root-mean-square displacement) on either side of it, is approximately 68% (Fig. 5.6b). The probability that any particular atom has diffused further than this distance is given by the total area under the curve minus the shaded area, which is approximately 32%. The probability that the atoms have diffused further than 2f is equal to the total area under the curve minus the area under the curve up to 2f. This is found to be equal to about 5%. Some atoms will have gone further than this distance, but the probability that any one particular atom will have done so is very small. 

Both correlation and variance analysis results showed that the hypothesis on the linear correlation between inter-laboratory data and the homogeneity of the corresponding variances is true for all data sets, at the for 95% confidence level. Table 2 presents a typical example of such a comparison. Based on the detected property of homogeneous variances, root-mean-square standard deviation, S, for all melted snow samples was estimated S = 0.32 0.06 for 95% confidence level [3]. 

The standard deviation a is the square-root of the variance and has the same unit as the random variable. A random variable is standardized (or reduced) if its variance is unity and centered if its mean is zero. 

Variance k2 characterizes the sharpness of the profile, i.e., whether it changes abruptly or smoothly from 0 to 1 at the mean time. The smaller is its value, the more are the residence times centered about the mean, and the sharper is the profile. It is usual practice to report its square root, the standard deviation (STD), as this gives a measure on the same scale. 

Thus, when a property of the sample (which exists as a large volume of material) is to be measured, there usually will be differences between the analytical data derived from application of the test methods to a gross lot or gross consignment and the data from the sample lot. This difference (the sampling error) has a frequency distribution with a mean value and a variance. Variance is a statistical term defined as the mean square of errors the square root of the variance is more generally known as the standard deviation or the standard error of sampling. 

Like the median, neither of these ranges accounts for the numerical values of all the data only their relative magnitudes. The standard deviation, which is the square root of the variance, accounts for the individual magnitudes and is a measure of the average squared-deviation of individual values from the sample mean. If the individual values are denoted by j/j, j = 1,..., n and the sample mean by y, then the sample variance is... 

I. A measure, symbolized by square root of the variance. Hence, it is used to describe the distribution about a mean value. For a normal distribution curve centered on some mean value, fjt, multiples of the standard deviation provides information on what percentage of the values lie within na of that mean. Thus, 68.3% of the values lie within one standard deviation of the mean, 95.5% within 2 cr, and 99.7% within 3 cr. 2. The corresponding statistic, 5, used to estimate the true standard deviation cr = (2(Xi - x) )/(n - 1). See Statistics (A Primer)... 

Statistical Analysis. Analysis of variance (ANOVA) of toxicity data was conducted using SAS/STAT software (version 8.2 SAS Institute, Cary, NC). All toxicity data were transformed (square root, log, or rank) before ANOVA. Comparisons among multiple treatment means were made by Fisher s LSD procedure, and differences between individual treatments and controls were determined by one-tailed Dunnett s or Wilcoxon tests. Statements of statistical significance refer to a probability of type 1 error of 5% or less (p s 0.05). Median lethal concentrations (LCjq) were determined by the Trimmed Spearman-Karber method using TOXSTAT software (version 3.5 Lincoln Software Associates, Bisbee, AZ). 

It is noteworthy that from a modeling perspective, 0j is also a scaling factor, since the expectation operator and the variance are of different dimensions. If it is desirable to obtain a term that is dimensionally consistent with the expected value term, then the standard deviation of z0 may be considered, instead of the variance, as the risk measure (in which standard deviation is simply the square root of variance). Moreover, 0i represents the weight or weighting factor for the variance term in a multiobjective optimization setting that consists of the components mean and variance. 

Standard Deviation - the positive square root of the variance, which in turn is defined as the sum of squares of the deviations of the observations from their mean (x) divided by one less than the number of observations (n - 1). 

Recall our short discussion in Section 18.5 where we learned that turbulence is kind of an analytical trick introduced into the theory of fluid flow to separate the large-scale motion called advection from the small-scale fluctuations called turbulence. Since the turbulent velocities are deviations from the mean, their average size is zero, but not their kinetic energy. The kinetic energy is proportional to the mean value of the squared turbulent velocities, Mt2urb, that is, of the variance of the turbulent velocity (see Box 18.2). The square root of this quantity (the standard deviation of the turbulent velocities) has the dimension of a velocity. Thus, we can express the turbulent kinetic energy content of a fluid by a quantity with the dimension of a velocity. In the boundary layer theory, which is used to describe wind-induced turbulence, this quantity is called friction velocity and denoted by u. In contrast, in river hydraulics turbulence is mainly caused by the friction at the... 

The sample variance is essentially the sum of the squares of the deviation of the data points from the mean value divided by (n-1). A large value of variance indicates that the data are widely spread about the mean. In contrast, if all values for the data points were nearly the same, the sample variance would be very small. The standard deviation sx is defined as the square root of the variance. The standard deviation is expressed by the same units as random variable values. Both standard deviation and the average are expressed by the same units. This characteristic made it possible to mutually compare variability of different distributions by introducing the relative measure of variability, called the coefficient of variation ... 

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