Variance of a distribution

The last example brings out very clearly that knowledge of only the mean and variance of a distribution is often not sufficient to tell us much about the shape of the probability density function. In order to partially alleviate this difficulty, one sometimes tries to specify additional parameters or attributes of the distribution. One of the most important of these is the notion of the modality of the distribution, which is defined to be the number of distinct maxima of the probability density function. The usefulness of this concept is brought out by the observation that a unimodal distribution (such as the gaussian) will tend to have its area concentrated about the location of the maximum, thus guaranteeing that the mean and variance will be fairly reasdnable measures of the center and spread of the distribution. Conversely, if it is known that a distribution is multimodal (has more than one... 

The variance of a distribution is a measure of how much the individual data points vary from the distribution mean, i.e., how spread out the data is. The variance is the average of the squared deviations from the mean and is calculated using the formula presented below. Using the data for the number of hazards identified in various locations above, the variance provides an indication as to how much the number of accidents reported differs from the average. (When the variance is calculated for a sample, as is the case in this example, the sum of the squared deviation scores is divided by N - 1, where N is the number of observations. When the variance is calculated for an entire population, the sum of the squared deviation scores is divided by N.)... 

It is possible to represent the variance of a distribution diagrammatically on a cumulative frequency diagram. To do this, cumulative frequencies are calculated in the way illustrated in Table 12.4 for the three distributions of days lost used in Figure 12.7. The cumulative frequencies so calculated are then plotted on a diagram as shown in Figure 12.8 where it can be seen that when the distribution is even, as in distribution 3, the cumulative frequency shows as a straight line. 

Nor should the foregoing be misinterpreted. The mean and variance of a distribution may be quite repeatible even if the overall shape is not reliably reproduced. In addition, cumulants analysis is always repeatable, though for At2/2>0.3, the polydispersity index is increasingly less sensitive to variations in the width of the true distribution. 

The variance of a distribution cr, the usual measure of distribution breadth for small populations, depends strongly on the average molecular weight of the distribution ... 

It has been established that the variance of a rectangular distribution of sample... 

Here, f(x) is tlie probability of x occurrences of an event tliat occurs on the average p times per unit of space or time. Both tlie mean and tlie variance of a random variable X liaving a Poisson distribution are (i. 

The mean and variance of a random variable X having a log-normal distribution are given by... 

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 preceding discussion of the variance as a measure of the spread of a distribution about its mean has been largely qualitative. One way in which the variance can be used to give quantitative information about the distribution is through use of the Chebychev inequality, which gives an upper bound on the amount of area contained outside an interval centered at the mean. The formal statement of the Chebychev inequality is... 

Variance, 269 of a distribution, 120 significance of, 123 of a Poisson distribution, 122 Variational equations of dynamical systems, 344 of singular points, 344 of systems with n variables, 345 Vector norm, 53 Vector operators, 394 Vector relations in particle collisions, 8 Vectors, characteristic, 67 Vertex, degree of, 258 Vertex, isolated, 256 Vidale, M. L., 265 Villars, P.,488 Von Neumann, J., 424 Von Neumann projection operators, 461... 

Let a volume (Vi) be injected onto a column resulting in a rectangular distribution of sample at the front of the column. According to the principle of the Summation of Variances, the variance of the final peak will be the sum of the variances of the sample volume plus the normal variance of a peak for a small sample. 

The subscript f on aj denotes that this variance has units of time squared. The dimensionless variance measures the breadth of a distribution in a way... 

Secondly, knowledge of the estimation variance E [P(2c)-P (2c)] falls short of providing the confidence Interval attached to the estimate p (3c). Assuming a normal distribution of error In the presence of an Initially heavily skewed distribution of data with strong spatial correlation Is not a viable answer. In the absence of a distribution of error, the estimation or "krlglng variance o (3c) provides but a relative assessment of error the error at location x Is likely to be greater than that at location 2 " if o (2c)>o (2c ). Iso-varlance maps such as that of Figure 1 tend to only mimic data-posltlon maps with bull s-eyes around data locations. 

These weights depend on several characteristics of the data. To understand which ones, let us first consider the univariate case (Fig. 33.7). Two classes, K and L, have to be distinguished using a single variable, Jt,. It is clear that the discrimination will be better when the distance between and (i.e. the mean values, or centroids, of 3 , for classes K and L) is large and the width of the distributions is small or, in other words, when the ratio of the squared difference between means to the variance of the distributions is large. Analytical chemists would be tempted to say that the resolution should be as large as possible. 

This choice of Qj yields ML estimates of the parameters if the error terms in each response variable and for each experiment (Ey, i=I,...N j=l,...,m) are independently distributed normally with zero mean and constant variance. Namely, the variance of a particular response variable is constant from experiment to experiment however, different response variables have different variances, i.e.,... 

The effects of simulated annealing on the distribution of curvature in the simulation box are shown in Fig. 3.2.B. To capture the changes in the distribution of curvature, the variance of this distribution is monitored, which turns out to be - for a large number of points Np - very well approximated by... 

Lambda (A), however, is not restricted to integer values. Since A represents the mean value of the data, and in fact is equal to both the mean and the variance of the distribution, there is no reason this mean value has to be restricted to integer values, even though the data itself is. We have already used this property of the Poisson distribution in plotting the curves in Figure 49-20b. 

Previously, in the case of constant detector noise, we then set Var(A s) and Var(A r) equal to the same value. This is the point at which must we now depart from the previous derivation, since in the case of Poisson-distributed noise the sample and reference noise levels will rarely, if ever, be the same. However, we are fortunate in this case that Poisson-distributed noise has a unique and very useful property that we have indeed previously made use of the variance of Poisson-distributed noise is equal to the mean signal value. Hence we can substitute Es for Var(A s) and Er for Var(A r) ... 

The most important measure for the dispersion of a data distribution is the variance. The variance of a population (with all possible data being known) is the mean of the squares of the deviations of the individual values from the population mean. 

A statistical term referring to a monoparametric distribution used to obtain confidence intervals for the variance of a normally distributed random variable. The so-called chi-square (x ) test is a protocol for comparing the goodness of fit of observed and theoretical frequency distributions. 

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)... 

This result is obtained from the binomial distribution if we let p approach 0 and n approach infinity. In this case, the mean fx = p approaches a finite value. The variance of a Poisson distribution is given as cr = fx. 

The variance of a quantity /, which is variously denoted by aj, var(/), or a1 if), measures the intrinsic range of fluctuations in a system. Given N properly distributed samples of /, the variance is defined as the average squared deviation from the mean ... 

Since the C curve for this vessel is broad and unsymmetrical, see Fig. 11.El, let us guess that dispersion is too large to allow use of the simplification leading to Fig. 13.4. We thus start with the variance matching procedure of Eq. 18. The mean and variance of a continuous distribution measured at a finite number of equidistant locations is given by Eqs. 3 and 4 as... 

Bias corrections are sometimes applied to MLEs (which often have some bias) or other estimates (as explained in the following section, [mean] bias occurs when the mean of the sampling distribution does not equal the parameter to be estimated). A simple bootstrap approach can be used to correct the bias of any estimate (Efron and Tibshirani 1993). A particularly important situation where it is not conventional to use the true MLE is in estimating the variance of a normal distribution. The conventional formula for the sample variance can be written as = SSR/(n - 1) where SSR denotes the sum of squared residuals (observed values, minus mean value) is an unbiased estimator of the variance, whether the data are from a normal distribution... 

Supposing that one has decided on bounds for a variable, one can fit a distribution that has a bounded support, such as the beta distribution or Johnson SB distribution. Alternatively, in a Monte Carlo implementation, one may sample the unbounded distribution and discard values that fall beyond the bounds. However, then a source of some discomfort is that the parameters of the distribution truncated in this way may deviate from the specification of the distribution (e.g., the mean and variance will be modified by truncation). It seems reasonable for Monte Carlo software to report the percentage discarded, and report means and variances of the distributions as truncated, for comparison to means and variances specified. 

The mean summarises only one aspect of a distribution. We also need some measure of spread or dispersion, the tendency for observations to depart from the central tendency. The standard measure of dispersion is the variance ... 

We consider forming a confidence interval for the variance of a normal distribution. As shown in Example 4.29, the interval is formed by finding ci wt,r and cuppel. such that Prob[c/ovltT < %2 -11 < cuppgr = 1 - a. The endpoints of the confidence interval are then (n-1 )s2/cupper and (n-l)s1/ciowet.. How do we find the narrowest interval Consider simply minimizing the width of the interval, cupper - ciower subject to the constraint that the probability contained in the interval is (1-a). Prove that for symmetric and asymmetric distributions alike, the narrowest interval will be such that the density is the same at the two endpoints. 

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