Skewness deviations

Paralytic strabismus (III, IV, VI nerve palsy, internuclear opthalmoplegia, skew deviation)... 

Skew deviation may be seen by various brainstem lesions and in increased intracranial pressure as well as in hepatic coma. Skew deviations are ipsiversive (ipsilateral eye undermost) with caudal pontomedullary lesions and contraversive (contralateral eye lowermost) with rostral pontomesencephalic lesions. They are associated with concomitant ocular torsion and tilts of the subjective visual vertical toward the undermost eye (Brandt and Dieterich 1993). 

Bornstein NM, Norris JW (1986) Subclavian steal a harmless haemodynamic phenomenon Lancet 2 303-305 Bousser MG, Chiras J, Bories J et al (1985) Cerebral venous thrombosis-a review of 38 cases. Stroke 16 199-213 Brandt T, Dieterich M (1993) Skew deviation with ocular torsion a vestibular brainstem sign of topographic diagnostic value. Ann Neurol 33 528-534... 

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). 

The variability or spread of the data does not always take the form of the true Normal distribution of course. There can be skewness in the shape of the distribution curve, this means the distribution is not symmetrical, leading to the distribution appearing lopsided . However, the approach is adequate for distributions which are fairly symmetrical about the tolerance limits. But what about when the distribution mean is not symmetrical about the tolerance limits A second index, Cp, is used to accommodate this shift or drift in the process. It has been estimated that over a very large number of lots produced, the mean could expect to drift about 1.5cr (standard deviations) from the target value or the centre of the tolerance limits and is caused by some problem in the process, for example tooling settings have been altered or a new supplier for the material being processed. 

If, as is usual, standard deviations are inserted for e cr has a similar interpretation. Examples are provided in Refs. 23, 75, 89, 93, 142, 169-171 and in Section 4.17. In complex data evaluation schemes, even if all inputs have Gaussian distribution functions, the output can be skewed,however. 

The principle of Maximum Likelihood is that the spectrum, y(jc), is calculated with the highest probability to yield the observed spectrum g(x) after convolution with h x). Therefore, assumptions about the noise n x) are made. For instance, the noise in each data point i is random and additive with a normal or any other distribution (e.g. Poisson, skewed, exponential,...) and a standard deviation s,. In case of a normal distribution the residual e, = g, - g, = g, - (/ /i), in each data point should be normally distributed with a standard deviation j,. The probability that (J h)i represents the measurement g- is then given by the conditional probability density function Pig, f) ... 

Figure 6.1 shows the gamma density with mean 100 and standard deviation 50. It shows that gamma densities are skewed, i.e., the typical outcome is different from the mean. 

The results obtained by a number of workers, using 5 different methods, are represented in Fig. 4. The method of Kirkman and Bynum (K10), and method D of Hsia et al. (H12), give such similar results that they are combined. In Fig. 4, normal distribution curves are drawn for normal homozygotes and heterozygotes, using published values for the means and standard deviations (no allowance is made for possible skewness or kurtosis) as far as possible, the same scale is used for all the methods. For galactosemics a smooth curve was drawn from the values published for individual patients. 

The first is to normalize the data, making them suitable for analysis by our most common parametric techniques such as analysis of variance ANOYA. A simple test of whether a selected transformation will yield a distribution of data which satisfies the underlying assumptions for ANOYA is to plot the cumulative distribution of samples on probability paper (that is a commercially available paper which has the probability function scale as one axis). One can then alter the scale of the second axis (that is, the axis other than the one which is on a probability scale) from linear to any other (logarithmic, reciprocal, square root, etc.) and see if a previously curved line indicating a skewed distribution becomes linear to indicate normality. The slope of the transformed line gives us an estimate of the standard deviation. If... 

Skewness Ks characterizes the symmetry of the distribution. A value of 0 characterizes the distribution as symmetric for asymmetric (skewed) distributions, it will be positive or negative, depending on whether the larger deviations from the mean are in the positive or negative direction (5). 

If the data distribution is extremely skewed it is advisable to transform the data to approach more symmetry. The visual impression of skewed data is dominated by extreme values which often make it impossible to inspect the main part of the data. Also the estimation of statistical parameters like mean or standard deviation can become unreliable for extremely skewed data. Depending on the form of skewness (left skewed or right skewed), a log-transformation or power transformation (square root, square, etc.) can be helpful in symmetrizing the distribution. 

The standard deviation is very sensitive to outliers if the data are skewed, not only the mean will be biased, but also s will be even more biased because squared deviations are used. In the case of normal or approximately normal distributions,, v is the best measure of spread because it is the most precise estimator for standard deviation is often uncritically used instead of robust measures for the spread. 

As already noted in Section 1.6.1, many statistical estimators rely on symmetry of the data distribution. For example, the standard deviation can be severely increased if the data distribution is much skewed. It is thus often highly recommended to first transform the data to approach a better symmetry. Unfortunately, this has to be done for each variable separately, because it is not sure if one and the same transformation will be useful for symmetrizing different variables. For right-skewed data, the log transformation is often useful (that means taking the logarithm of the data values). More flexible is the power transformation which uses a power p to transform values x into xp. The value of p has to be optimized for each variable any real number is reasonable for p, except p 0 where a log-transformation has to be taken. A slightly modified version of the power transformation is the Box Cox transformation, defined as... 

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