Unity standard deviation

Table 10-7 Hypothetical data under two different scenarios for the experiment examining the effect of solvent and catalyst on yield, random variations (from Table 10-6) have zero mean and unity standard deviation...

In addition to uniformly distributed random numbers, the Gaussian distribution is also of classical and practical importance. The function gnormalQ (also in the prob.lua file) can be used to generate such random numbers by setting rand = gnormal as on line 5 of Listing 8.2. This function returns a Gaussian distribution with zero mean and a unity standard deviation. For other values of these parameters, the equation ... 

This result shows that the square root of the amount by which the ratio M /M exceeds unity equals the standard deviation of the distribution relative to the number average molecular weight. Thus if a distribution is characterized by M = 10,000 and a = 3000, then M /M = 1.09. Alternatively, if M / n then the standard deviation is 71% of the value of M. This shows that reporting the mean and standard deviation of a distribution or the values of and Mw/Mn gives equivalent information about the distribution. We shall see in a moment that the second alternative is more easily accomplished for samples of polymers. First, however, consider the following example in which we apply some of the equations of this section to some numerical data. 

The mean value of each of the distributions is obtained from these high, modal, and low values by the use of Eq. (9-101). If the distribution is skewed, the mean and the mode will not coincide. However, the mean values may be summed to give the mean value of the (NPV) as 161,266. The standard deviation of each of the distributions is calculated by the use of Eq. (9-75). The fact that the (NPV) of the mean or the mode is the sum of the individual mean or modal values implies that Eq. (9-81) is appropriate with all the A s equal to unity. Hence, by Eq. (9-81) the standard deviation of the (NPV) is the root mean square of the individual standard deviations. In the present case s° = 166,840 for the (NPV). 

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. 

The uncertainties given are calculated standard deviations. Analysis of the interatomic distances yields a selfconsistent interpretation in which Zni is assumed to be quinquevalent and Znn quadrivalent, while Na may have a valence of unity or one as high as lj, the excess over unity being suggested by the interatomic distances and being, if real, presumably a consequence of electron transfer. A valence electron number of approximately 432 per unit cell is obtained, which is in good agreement with the value 428-48 predicted on the basis of a filled Brillouin polyhedron defined by the forms 444, 640, and 800. ... 

The values in Table 10-6 were selected randomly, and have a mean of zero and a standard deviation of unity. When these error values are superimposed on the data, we arrive at the Table 10-7. 

Table 10-8 Part A - DATA ANOVA for the hypothetical data containing error with mean equal 0 and standard deviation (S) equal to unity...

Figure 44-10a Transmittance noise as a function of transmittance, for different values of reference energy S/N ratio (recall that, since the standard deviation of the noise equal unity, the set value of the reference energy equals the S/N ratio), (see Color Plate 11)...

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. 

When the noise is small the multiplication factor approaches unity, as we would expect. As we have seen for the previous two types of noise we considered, the nonlinearity in the computation of transmittance causes the expected value of the computed transmittance to increase as the energy approaches zero, and then decrease again. For the type of noise we are currently considering, however, the situation is complicated by the truncation of the distribution, as we have discussed, so that when only the tail of the distribution is available (i.e., when the distribution is cut off at +3 standard deviations), the character changes from that seen when most of the distribution is used. 

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. 

Since the standard deviations are unity and the variables are independent (zero covariance), the covariance-matrix of X is the identity matrix / and the contours of constant probability in the space 9T are given by... 

Each stated uncertainty in this and other tables represents one estimated standard error, propagated to parameters from uncertainties of measurements of wave numbers the uncertainties of the latter measurements were provided by authors of papers [91,93] reporting those data, and the weight of each datum in the non-linear regression was taken as the reciprocal square of those uncertainties. As the reduced standard deviation of the fit was 0.92, so less than unity, the authors... 

According to apphcation of Dunham s formalism to analysis of molecular spectra, as for GaH and H2, these radial coefficients of seven types represent many Dunham coefficients Ym and their auxiliary coefficients Zki of various types that collectively allow wave numbers of observed transitions to be reproduced almost within their uncertainty of measurement through formula 54. Mostly because of inconsistency between reported values of frequencies of pure rotational transitions [118,119], the reduced standard deviation of the fit reported in table 3 is 1.25, slightly greater than unity that would be applicable with consistent data for which uncertainty of each measurement were carefully assigned. 

In addition to the temporal correlation coefficient, the spatial correlation coefficient was calculated approximately for fixed values of time. Except for one of the mathematical models, all techniques showed a better temporal correlation than spatial correlation. The two correlation coefficients are cross plotted in Figure 5-6. Nappo stressed that correlation coefficients express fidelity in predicting tends, rather than accuracy in absolute concentration predictions. Another measure is used for assessing accuracy in predicting concentrations the ratio of predicted to observed concentration. Nappo averaged this ratio over space and over time and extracted the standard deviation of the data sample for each. The standard deviation expresses consistency of accuracy for each model. For example, a model might have a predicted observed ratio near unity,... 

Autoscaling is a technique for coding data so that the mean is zero and the standard deviation is unity. What should Cy, and Jy, be to autoscale the nine responses in Section 3.1 (Hint see Figure 3.3.)... 

Now if we have data of C/Cj. as a function of pipe radius, r, we can use standard least-squares techniques to estimate Ko, Ki, K, . In addition, we can find the standard deviations of the estimates of Ki by the least-squares procedure, which gives an indication of the precision of the data. The first constant, Ko, should be unity if we have a perfect mass balance, and the deviation from this value gives an estimate of the reliability of the data. Knowing the injection tube size, we can find the Ni/q from the least squares K from Eq. (59). 

Therefore the square root of the amount by which the molecular weight ratio exceeds unity measures the standard deviation of the distribution relative to the number average molecular weight. 

The preexponential factor accomplishes the normalization of the function that is, the integral of the function over all possible values of x (— oo to oo) equals unity. In a broad distribution a is large, and the exponential does not drop off as rapidly as in a narrow distribution (recall that all deviations are measured relative to the standard deviation). 

Since the area under the curve is always unity, a narrow distribution will show larger values of f x) at the maximum, whereas a broader distribution will have a smaller value for the function at the maximum. This is why the standard deviation appears in the denominator of the preexponential normalization factor. 

Gaussian distribution Theoretical bell-shaped distribution of measurements when all error is random. The center of the curve is the mean, p, and the width is characterized by the standard deviation, a. A nortnalized Gaussian distribution, also called the normal error curve, has an area of unity and is given by... 

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