Standard deviation robust

Chapter 7 Response 7.1 (a) The mean, standard deviation, robust average (median) and robust standard deviation (MADe) are shown in Table SAQ 7.1. Table SAQ 7.1 Data from a proficiency testing round for the determination of moisture in barley ... 

Random errors Relative standard deviation Robust variance Samples and populations Standard deviation of the mean (standard error of the mean)... 

The standard deviation, Sj, is the most commonly used measure of dispersion. Theoretically, the parent population from which the n observations are drawn must meet the criteria set down for the normal distribution (see Section 1.2.1) in practice, the requirements are not as stringent, because the standard deviation is a relatively robust statistic. The almost universal implementation of the standard deviation algorithm in calculators and program packages certainly increases the danger of its misapplication, but this is counterbalanced by the observation that the consistent use of a somewhat inappropriate statistic can also lead to the right conclusions. 

The population from which the observations were obtained is normally distributed with an unknown mean p and standard deviation G. In actual practice, this is a robust test, in the sense that in most types of problems it is not sensitive to the normality assumption when the sample size is 10 or greater. 

To calculate the robust standard deviation for this data set, you first have to calculate the absolute difference between each result and the median, x, — median, and then find the median of these values. The median absolute deviation (MAD) is 0.02 %abv. This is converted to a standard deviation equivalent (MADe) by multiplying by 1.483 ... 

For comparison, the mean of the data set is 40.02 %abv and the standard deviation is 0.05 %abv. You can see that the robust standard deviation is substantially smaller than the standard deviation. The use of robust statistics has reduced the influence of the extreme values in the data set. 

Section 1.6.2 discussed some theoretical distributions which are defined by more or less complicated mathematical formulae they aim at modeling real empirical data distributions or are used in statistical tests. There are some reasons to believe that phenomena observed in nature indeed follow such distributions. The normal distribution is the most widely used distribution in statistics, and it is fully determined by the mean value p. and the standard deviation a. For practical data these two parameters have to be estimated using the data at hand. This section discusses some possibilities to estimate the mean or central value, and the next section mentions different estimators for the standard deviation or spread the described criteria are fisted in Table 1.2. The choice of the estimator depends mainly on the data quality. Do the data really follow the underlying hypothetical distribution Or are there outliers or extreme values that could influence classical estimators and call for robust counterparts ... 

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. 

A robust counterpart to the standard deviation s is the median absolute deviation (MAD, x mad). 

MAD is based on the median xM as central value the absolute differences x — xM are calculated and MAD is defined as the median of these differences. In the case of a normal distribution MAD can be used for a robust estimation of the theoretical standard deviation cr by... 

Robust estimations of the variance are squared standard deviations as obtained from IQR or MAD, (siqr)2, and (smad)2 respectively. 

If the data include outliers, it is advisable to use robust versions of centering and scaling. The simplest possibility is to replace the arithmetic means of the columns by the column medians, and the standard deviations of the columns by the median absolute deviations (MAD), see Sections 1.6.3 and 1.6.4, as shown in the following M-code for a matrix X. 

The corresponding robust correlation matrix R MCD, containing the elements y,, is obtained from the robust covariance matrix by dividing by robust standard deviations in R by... 

Note that there are again different options for scaling the data. The variables could be scaled using the empirical standard deviations of the variables, or by using robust versions (see Section 2.2.2). The latter option should be preferred if outliers are present or if the data are inhomogeneous (for instance, divided into groups). 

Instead of the classical estimation of the standard deviation, SEP, robust methods can be applied as described in Section 1.6.4, for instance the spread measures, VjqR... 

Regression M-estimates minimize 11p(ei/cr) to obtain the estimated regression coefficients, where the choice of the function p determines the robustness of the estimation, and cr is a robust estimation of the standard deviation of the residuals (Maronna et al. 2006). Note that both the residuals e, and the residual scale cr depend on the regression coefficients, and several procedures have been proposed to estimate both quantities. 

The CE method was validated in terms of accuracy, precision, linearity, range, limit of detection, limit of quantitation, specificity, system suitability, and robustness. Improved reproducibility of the CZE method was obtained using area normalization to determine the purity and levels of potential impurities and degradation products of IB-367 drug substance. The internal standard compensated mainly for injection variability. Through the use of the internal standard, selected for its close mobility to IB-367, the method achieved reproducibility in relative migration time of 0.13% relative standard deviation (RSD), and relative peak area of 2.75% RSD. 

For robustness testing, it is important to observe all relevant parameters, which include resolution of critical peak pairs and efficiencies. The means and additionally the standard deviations for all relevant peaks should be given for migration times (especially Jeof)) peak areas (PAs), and relative PAs (Table... 

For proficiency tests z-seores have been widely nsed for mat r years. Z-scores represent the deviation from the assigned valne in standard deviation units. The standard deviation may be calculated after exelusion of outiiers or with robust statistios. In some eases it is set to a eertain value aeeording to the quality targets of the PT provider. 

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