Variance square root

The diagonal elements of this matrix approximate the variances of the corresponding parameters. The square roots of these variances are estimates of the standard errors in the parameters and, in effect, are a measure of the uncertainties of those parameters. 

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 overall standard deviation, S, is the square root of the average variance for the samples used to establish the control plot. 

To express the measure of dispersion in the original scale of measurement, it is usual to take the square root of the variance to give the standard deviation ... 

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

The standard deviation of the sample is defined as the square root of the variance. For the example,... 

Traditionally, column efficiency or plate counts in column chromatography were used to quantify how well a column was performing. This does not tell the entire story for GPC, however, because the ability of a column set to separate peaks is dependent on the molecular weight of the molecules one is trying to separate. We, therefore, chose both column efficiency and a parameter that we simply refer to as D a, where Di is the slope of the relationship between the log of the molecular weight of the narrow molecular weight polystyrene standards and the elution volume, and tris simply the band-broadening parameter (4), i.e., the square root of the peak variance. 

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... 

The standard deviation is the square root of the variance and is denoted by c (population) or S (sample). 

The Standard Error of Prediction (SEP) is supposed to refer uniquely to those situations when a calibration is generated with one data set and evaluated for its predictive performance with an independent data set. Unfortunately, there are times when the term SEP is wrongly applied to the errors in predicting y variables of the same data set which was used to generate the calibration. Thus, when we encounter the term SEP, it is important to examine the context in order to verify that the term is being used correctly. SEP is simply the square root of the Variance of Prediction, s2. The RMSEP (see below) is sometimes wrongly called the SEP. Fortunately, the difference between the two is usually negligible. 

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. 

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... 

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 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. 

Column-standardization is the most widely used transformation. It is performed by division of each element of a column-centered table by its corresponding column-standard deviation (i.e. the square root of the column-variance) ... 

Gaussian-shaped depth profiles of P with three parameters of maximum concentration (Cmax), projected range (Rp) and range straggling (ARp). The energy loss (dE/dx) and energy straggling ( 2 square root of the variance) of the a beam in the Si layer were taken into account ... 

Dispersion parameter for the distribution of measured values, sy, or analytical results, sx, for a given sample or the population, oy and ox. The SD is the square root of the variance. 

As a result of the linear regression, the values of fo (r) are now known for a sequence of discrete r. From Eq. (8.27) it is clear that fo (r) itself represents a weighted relative variance of the lattice distortions. If it is found to be almost constant, its square root directly describes the total amount of relative lattice distortion ( in percent ). 

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 ... 

It is precisely this variance of (g) that we are after, because its square root gives us the angular dependent linewidth. A general expression in matrix notation can be derived for the variance (Hagen et al. 1985c) ... 

Finally, reconverting variance back to SD by taking square roots on both sides of equation 41-18 ... 

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 ... 

Equation 52-149 presents a minor difficulty one that is easily resolved, however, so let us do so the difficulty actually arises in the step between equation 52-148 and 52-149, the taking of the square root of the variance to obtain the standard deviation conventionally we ordinarily take the positive square root. However, T takes values from zero to unity that is, it is always less than unity, the logarithm of a number less than unity is negative, hence under these circumstances the denominator of equation 52-149 would be negative, which would lead to a negative value of the standard deviation. But a standard deviation must always be positive clearly then, in this case we must use the negative square root of the variance to compute the standard deviation of the relative absorbance noise. 

We also have noted before that adding or subtracting noisy data causes the variance to increase as the number of data points added together [2], The noise of the first derivative, therefore, will be larger than that of the underlying absorbance band by a factor of the square root of two. 

For this reason, the reader will find another very interesting exercise to compute the sums of the squares of the coefficients for several of the sets of coefficients, to extend these results to both higher order derivatives and higher degree polynomials, to ascertain their effect on the variance of the computed derivative for extended versions of these tables. Hopkins [8] has performed some of these computations, and has also coined the term RSSK/Norm for the 2((coefl7Normalization factor)2) in the S-G tables. Since here we pre-divide the coefficients by the normalization factors, and we are not taking the square roots, we use the simpler term SSK (sum squared coefficients) for our equivalent quantity. Hopkins in the same paper has also demonstrated how the two-point... 

Let us discuss some of the terms in equation 70-20. The simplest way to think about the covariance is to compare the third term of equation 70-20 with the numerator of the expression for the correlation coefficient. In fact, if we divide the last term on the RHS of equation 70-20 by the standard deviations (the square root of the variances) of X and Y in order to scale the cross-product by the magnitudes of the X and Y variables and make the result dimensionless, we obtain... 

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