Square of the standard deviation

Variance Another common measure of spread is the square of the standard deviation, or the variance. The standard deviation, rather than the variance, is usually reported because the units for standard deviation are the same as that for the mean value. 

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. 

Equation 3.2 is essentially the root of the sum of the squares of the standard deviations of the toleranees in the staek, whieh equals the standard deviation of the assembly toleranee, lienee its other name, Root Sum Square or RSS model. This ean be represented by ... 

The peak width at the points of inflexion of the elution curve is twice the standard deviation of the Poisson or Gaussian curve and thus, from equation (8), the variance (the square of the standard deviation) will be equal to (n), the total number of plates in the column. 

The second moment is taken about the mean and is referred to as the variance or square of the standard deviation defined by... 

The square of the standard deviation is called the variance. A further measure of precision, known as the Relative Standard Deviation (R.S.D.), is given by ... 

Variance The mean square of deviations, or errors, of a set of observations the sum of square deviations, or errors, of individual observations with respect to their arithmetic mean divided by the number of observations less one (degree of freedom) the square of the standard deviation, or standard error. 

A central concept of statistical analysis is variance,105 which is simply the average squared difference of deviations from the mean, or the square of the standard deviation. Since the analyst can only take a limited number n of samples, the variance is estimated as the squared difference of deviations from the mean, divided by n - 1. Analysis of variance asks the question whether groups of samples are drawn from the same overall population or from different populations.105 The simplest example of analysis of variance is the F-test (and the closely related t-test) in which one takes the ratio of two variances and compares the result with tabular values to decide whether it is probable that the two samples came from the same population. Linear regression is also a form of analysis of variance, since one is asking the question whether the variance around the mean is equivalent to the variance around the least squares fit. 

Or equivalently, we calculate the variance of AT, which is the square of the standard deviation ... 

Equation 59-11 indicates that the correlation coefficient is represented by the square root of one minus the ratio comprised of the square of the standard error of performance, to the square of the standard deviation of all X],... 

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. 

The variance is the square of the standard deviation(s) i.e., s2. However, the former is fundamentally more important in statistics than the latter, whereas the latter is employed more frequently in the treatment of chemical data. 

As noted earlier, the variance V is the square of the standard deviation 5 thus, if each error is independent of the other, the overall variance for the above expression can be written as ... 

A statistical measure of the dispersion of the distribution of the random variable, typically obtained by taking the expected value of the square of the difference between the random variable and its mean. The variance is the square of the standard deviation. See Statistics (A Primer)... 

Note that Equation (9) implies that the square of the standard deviation a2 is the second moment of d relative to the mean d. Higher order moments can be used to represent additional information about the shape of a distribution. For example, the third moment is a measure of the skewness or lopsidedness of a distribution. It equals zero for symmetrical distributions and is positive or negative, depending on whether a distribution contains a higher proportion of particles larger or smaller, respectively, than the mean. The fourth moment (called kurtosis) purportedly measures peakedness, but this quantity is of questionable value. 

The F test tells us whether two standard deviations are "significantly different from each other. F is the quotient of the squares of the standard deviations ... 

Excel now goes to work and prints results in cells El to G13 of Figure 4-8. Mean values are in cells F3 and G3. Cells F4 and G4 give variance, which is the square of the standard deviation. Cell F6 gives pooled variance computed with Equation 4-9. That equation was painful to use by hand. Cell F8 shows degrees of freedom (df= 13) and tca cu illed = 20.2 from Equation 4-8 appears in cell F9. 

Squaring of the standard deviation term gives the variance, a term which is used in rate theory description for the gas chromatographic process (see Section 2.2.2). 

We know from statistical treatment that standard deviations are not additive. However, variances, the square of the standard deviation, are additive. In terms of the chromatographic process three diffusive process variables contribute to zone spreading. Thus, we can sum these variables in terms of variances to give... 

As an example, consider the data on day-to-day and interindividual variability of fruit growers respiratory and dermal exposure to captan shown in Table 7.3 (de Cock et al., 1998a). The ratio of the 97.5th percentile to the 2.5th percentile of the exposure distribution R95) is usually larger for the intraindividual or day-to-day variability, when compared to the interindividual variability. The variance ratio, k, can be calculated from the Rgs values, since the standard deviation of each exposure distribution is equal to In R95/3.92, and the square of the standard deviation gives the variance. For the respiratory exposure, this results in a variance ratio k of 32.8, whereas for dermal exposure of the wrist the variance ratio is considerably lower, approximately 3.0. What are the implications of these variance ratios for the number of measurements per study subject For a bias of less than 10 % (or /P > 0.90), the number of repeated measurements per subject... 

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