Positively skewed distribution

Environmental concentrations and other environmental variables tend to have positive skewness. Therefore, environmental statistics texts often focus on positive skew distributions such as the log-normal, gamma, and Weibull. Discussions of distributions with nonnormal kurtosis are somewhat more scarce. 

Fig. 2 Frequency distribution curves corresponding to (A) a normal distribution (B) a positively skewed distribution and (C) a bimodal distribution.

In the previous section it was shown that Vx 1.960s,t of the serum triglyceride data in Figure 16-3 resulted in biased reference limits (too low values), as was to be expected with this positively skewed distribution. However, it is often possible to transform data mathematically to obtain a distribution of transformed values that approximates a Gaussian distribution. With these new values, the 2.5 and 97.5 percentiles are localized at 2 standard deviations on both sides of the... 

It is frequently observed that logarithmically transformed values, y - ln(x), of a positively skewed distribution fit the Gaussian distribution rather closely. In other cases, square roots of the values, Vjc, result in a better approximation to the Gaussian distribution. This is the basis for the common use of the logarithmic and square root transformations when estimating reference limits. The method is applicable only to positively skewed distributions. The method is easily performed with a spreadsheet program. The procedure is as follows ... 

The well-known SD unit or normal equivalent deviate is such a measure. It is calculated as the difference between the observed value and the mean of the reference values divided by their standard deviation. Several similar ratios have been suggested/ but none has significant advantages over the other. All produce very confusing values if the reference distribution is very skewed. An observed value (e.g., with an SD unit of 2.2) would be above the 97,5 percentile if the reference distribution has a Gaussian shape, but might be well below the upper reference limit of a positively skewed distribution. Mathematical transformation of the reference distribution to the Gaussian shape may solve this problem. ... 

FIGURE 4.3 Graphical representation of kurtosis, K, and skewness, S, in comparison to the Gaussian (standard) distribution (upper left). The right-hand side shows leptokurtic (peaked) or platykurtic (flatted) distribution as well as positive skewed distribution (fronting) and negative skewed distribution (tailing). 

Skewness function is further applied to analyze the skewness of the speed distribution. As is known, skewness can be used as a statistic parameter to weigh the direction and degree of the distribution deflection. If the value of skewness SK is 0, it means that the distribution obeys a normal distribution. When SK is over 0, the distribution obeys positive skew distribution while it is under 0, it obeys negative skew distribution. The SK can be calculated as ... 

Similarily, the value of the H can be calculated as 1 under this circumstance, the value of the skewness is 0.7563, the ship speed obeys positive skew distribution. 

It can be seen that the ship speed of ships navigating in the bridge waterway obey positive skew distribution, which means that the low ship speed account for the majority, ships are more likely to choose a low speed when navigating in this special waterway. The shortest distance between ships and the bridge pier obeys negative skew distribution, which means that the bigger value of the shortest distance is dominated. The shortest distance between ships and the navigation buoy is almost... 

Ship speed Distribution Refer to figure 6 1 7.5789 2.3078 0.8374 Positive skew distribution... 

Feed Material Graphite powder -10% of particles under I Op and some 36% under 20p. Largest particles exceed lOOp, with average particle size 38.02p and median 25.2Ip, suggesting a positively skewed distribution (see Fig. 11 below). 

It is obvious that in Eqs. 1.10 - 1.13, when x is different, it will result in different values of the standard deviation, skewness and kurtosis. In general, these values are all different between arithmetic statistics and geometric statistics. From the definitions of arithmetic mean, median, and mode, one can predict that for positively skewed distributions that account for most of the industrial particulate systems, mode < median < mean. The degree of spread in these three values depends on the symmetry of the distribution. For completely symmetric distributions, these three values overlap. 

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