Statistics negatively skewed distribution

Empirical evidence indicates that the majority of the respondents locate their service quality score at the right-hand side of the scale (Brown et al. 1993 Parasuraman et al. 1988 1991 Peterson and Wilson 1992). This distribution is referred to as negatively skewed. A skewed distribution contains several serious implications for statistical analysis. To begin with, the mean might not be a suitable measure of central tendency. In a negatively skewed distribution, the mean is typictilly to the left of the median and the mode and thus excludes considerable information about the variable under study (Peterson and Wilson 1992). Skewness also attenuates the correlation between variables. Consequently, the true relationship between variables in terms of a correlation coefficient may be understated (Peterson and Wilson 1992). Einally, parametric tests (e.g., t-test, F-test) assume that the population is normally or at least symmetrically distributed. 

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 ... 

Su and are upper and lower limits., 2.B-i Xi-ai2 n-i are Chi-squared statistics for o /2 and (1-0/2) confidence levels and (n-1) degrees of freedom Skewness is positive if in comparison to the normal distribution the peak of the distribution is shifted to the left with a low tail extending to the right, and negative if the peak is shifted to the right and a long low tail extends to the left towards smaller values Kurtosis is positive if in comparison to the normal distribution there is a more acute peak around the mean and thicker tails, and negative if distribution has a lower, wider peak and thinner tails... 

Small departures from zero skew are quite likely to occur in samples from symmetrically distributed populations, but large positive or negative values are less likely. How large a departure from zero indicates that the skew is real Fortunately, most statistical packages give information relevant to this question, either by giving the standard error (SE) of gl or the probability (P) that the observed value of gl might have occurred by chance if the parent population is in fact symmetrically distributed. 

At times, particularly if Y is negative, such as may be the case when modeling percent change from baseline as the dependent variable, it may necessary to add a constant c to the data. Berry (1987) presents a method that may be used to define the optimal value of c while still ensuring a symmetrical distribution. The method makes use of skewness and kurtosis, which characterize the degree of asymmetry of a distribution. These statistics will be used to identify a transformation constant c such that the new data set will have better normality properties than the original scores. Let 7, and y2 be the skewness and kurtosis of the distribution as defined by Eqs. (4.36) and (4.37), respectively, substituting Ln(Y + c) for the variable e in Eq. (4.38). If g is defined as... 

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