Gaussian distribution median value

In particle size analysis it is important to define three terms. The three important measures of central tendency or averages, the mean, the median, and the mode are depicted in Figure 2.4. The mode, it may be pointed out, is the most common value of the frequency distribution, i.e., it corresponds to the highest point of the frequency curve. The distribution shown in Figure 2.4 (A) is a normal or Gaussian distribution. In this case, the mean, the median and the mode are found to fie in exactly the same position. The distribution shown in Figure 2.4 (B) is bimodal. In this case, the mean diameter is almost exactly halfway between the two distributions as shown. It may be noted that there are no particles which are of this mean size The median diameter lies 1% into the higher of the two distri-... 

A very important probability distribution is the normal or Gaussian distribution (after the German mathematician, Karl Friedrich Gauss, 1777-1855). The normal distribution has the same value for the mean, median and mode. The equation describing this distribution (the probability density function)... 

Random error — The difference between an observed value and the mean that would result from an infinite number of measurements of the same sample carried out under repeatability conditions. It is also named indeterminate error and reflects the - precision of the measurement [i]. It causes data to be scattered according to a certain probability distribution that can be symmetric or skewed around the mean value or the median of a measurement. Some of the several probability distributions are the normal (or Gaussian) distribution, logarithmic normal distribution, Cauchy (or Lorentz) distribution, and Voigt distribution. Voigt distribution is... 

The median, though less efficient than the mean for observations with a gaussian distribution, often is recommended because of its insensitivity to a divergent value especially when dealing with small numbers of observations. 

The distribution of random errors should follow the Gaussian or normal curve if the number of measurements is large enough. The shape of Gaussian distribution was given in Chapter 3 (Fig. 3.4). It can be characterized by two variables—the central tendency and the symmetrical variation about tjie central tendency. Two measures of the central tendency are the mean, X, and the median. One of these values is usually taken as the correct value for an analysis, although statistically there is no correct value but rather the most probable value. The ability of an analyst to determine this most probable value is referred to as his accuracy. 

By laser diffraction a size distribution by volume is primary obtained, that can be transferred to the intensity and number distribution. Although laser diffraction is sensitive over a broad particle size range, a small number of microparticles will escape determination. It should therefore kept in mind, that the upper limit of the size distribution does not naturally present the real situation, e.g. that particles in the lower m-range may be present although the D99-value was, e.g. determined at 400 nm. Different diameters are normally obtained from size distributions (e.g. mean, mode, median) and only if the particle sizes are Gaussian distributed all these diameters will have the same value. 

The likelihood of a particular event is generally described by the mean, median, and standard deviation (tr). The word generally is used as Lorentzian distributions do not have a mean, and hence a values, whereas Gaussian and Poisson distributions do. The median represents the middle of the range of values modeled. Note A mean is needed to derive average value exhibited by the distribution and a describes the likelihood from the mean value over which a particular event will occur, i.e. 68.3% occur within lmean value, 95.5% occur within 2a, and 99.7% occur within 3[Pg.294]

FIGURE 2.6 Illustration of the median value for a normal (Gaussian) distribution. 

FIGURE 2.10 Representation of the normalized normal (Gaussian) distribution and its probability intervals in terms of its standard deviation respecting the median value. 

Population distributions. (A) Gaussian or "normal" distribution. This is symmetrical about the mean and the values of the mean, median, and mode are the same. (B) A long "tail" of higher values, this represents a nonsymmetrical distribution in which there are different values for the mean, median, and mode (C). 

Furthermore, the mean square approach is sensitive to residual outliers (outliers are data points that lie far from the mean or median). Another possibility is to use the variance of the residuals. Since the variance is approximately equal to the mean squared residuals, the problem with the non-Gaussian residual distribution persists and the use of the Li norm is preferable. The Li norm minimizes the absolute values of the residuals and is less sensitive to outliers. For further approaches concerning the measurement of the best agreement between observed and calculated traveltime, refer to Ruzek and Kvasnicka [2001]. 

A. Each epistemic variable is represented by its median and a range We have supposed that the true and unknown distributions of the epistemic variables are Gaussian. The results will depend in the capacity of the expert to give quantiles and interval close to the actual values. We have supposed first that the expert gives the true median and the true 95% interval for the epistemic variables (Figure 3). Belief and plausibility functions of the response Pf are described here by 256 focal intervals indeed, the 8 epistemic variables are each described by two intervals, that means 2 (= 256) hypercubes to be evaluated in the space of epistemic variables. For each hypercube, the evaluation of the minimum and maximum values of the response provides the corresponding... 

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