Mathematics
Skewness
Skewness is a measure of the asymmetry of a probability distribution. In a skewed distribution, the tail on one side of the peak is longer or fatter than on the other, indicating that the distribution is not symmetrical. A positive skewness indicates a longer or fatter tail on the right side, while a negative skewness indicates a longer or fatter tail on the left side.
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eBook - PDFQuantitative Techniques in Business, Management and Finance
A Case-Study Approach
- Umeshkumar Dubey, D P Kothari, G K Awari(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
4.7.2 Application This formula applies to differences greater than one standard deviation about the mean, and k must be greater than 1. In the case of a symmetrical bell-shaped curve (Figure 4.1), we can say that 1. Approximately 68% of the observations in the population fall within ± 1 standard deviations from the mean. 2. Approximately 95% of the observations in the population fall within ± 2 standard deviations from the mean. 3. Approximately 99% of the observations in the population fall within ± 3 standard deviations from the mean. 97 Measures of Variation and Skewness 4.8 Skewness The measures of central tendency and variation do not reveal all the characteristics of a given set of data. For example, two distributions may have the same mean and standard deviation but may differ widely in the shape of their distribution. Either the distribution of data is symmetrical or it is not. If the distribution of data is not symmetrical, it is called asymmetrical or skewed. Thus, Skewness refers to the lack of symmetry in distribution. A simple method of detecting the direction of Skewness is to consider the tails of the distribution. Rules 1. Data is symmetrical when there are no extreme values in a particular direc-tion so that low and high values balance each other (Figure 4.2). In this case, mean = median = mode. 2. If the longer tail is towards the lower value or left-hand side, the Skewness is nega-tive (Figure 4.3). Negative Skewness arises when the mean is decreased by some extremely low values, thus making mean < median < mode. If the longer tail of the distribution is towards the higher values or right-hand side, the Skewness is positive (Figure 4.4). Positive Skewness occurs when the mean is increased by some unusually high values, thereby making mean > median > mode.
eBook - PDFStatistics Using Stata
An Integrative Approach
- Sharon Lawner Weinberg, Sarah Knapp Abramowitz(Authors)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
Often one may be interested in a numerical summary of Skewness for the purpose of comparing the Skewness of two or more distributions or for evaluating the degree to which a single distribution is skewed. The Skewness statistic is such a numerical summary. Equation 3.6 gives the expression for the Skewness statistic as calculated by Stata and the standard error of the Skewness. Skewness = ( ) / ( ) X X N X X N i i - - ∑ ∑ 3 2 3 (3.6) Standard Error Skewness = 6 1 2 1 3 N N N N N ( ) ( )( )( ) - - + + While the variance is based on the sum of squared deviations about the mean, the skew- ness statistic is based on the sum of cubed deviations about the mean. And while the vari- ance can only be positive or zero (since it is based on the sum of squared deviations), the Skewness statistic may be either positive, zero, or negative (since it is based on the sum of cubed deviations). The Skewness statistic is positive when the distribution is skewed posi- tively and it is negative when the distribution is skewed negatively. If the distribution is perfectly symmetric, the Skewness statistic equals zero. The more severe the skew, the more the Skewness statistic departs from zero. To compare the Skewness of two distributions of relatively equal size, one may use the Skewness values themselves. To compare the Skewness of two distributions that are quite unequal in size, or to evaluate the severity of Skewness for a particular distribution MEASURES OF LOCATION, SPREAD, AND Skewness 93 when that distribution is small or moderate in size, one should compute a Skewness ratio. The Skewness ratio is obtained as the Skewness statistic divided by the standard error of the Skewness statistic. Although the meaning of the standard error will be discussed in Chapter 9, we note simply for now that the standard error is a function of the sample size.
- Dror G. Feitelson(Author)
- 0(Publication Date)
- Cambridge University Press(Publisher)
Order statistics such as the median and other percentiles are much more stable and therefore can be estimated with a higher degree of confidence. 3.1.5 Focus on Skew The classic approach to describing distributions is to first use the central tendency and possibly augment it with some measure of dispersion. This implicitly assumes a roughly symmetric distribution. As we saw earlier, many of the distributions encountered in workload examples are not symmetric: they are skewed. This means that the right tail is much longer than the left tail. In fact, in many cases there is no left tail, and the relevant values are bounded by 0. (In principle distributions can also be skewed to the left, but in workloads this does not happen.) Skewed distributions have, of course, been recognized for many years. An indi- cation that a distribution is skewed is that the median is significantly different from the mean. The conventional metric for Skewness is γ 1 = μ 3 σ 3 = 1 n n i=1 (X i − ¯ X) 3 1 n n i=1 (X i − ¯ X) 2 3/2 (3.19) This may be interpreted as the weighted average of the signed distances from the mean, where large distances get much higher weights: the weights are equal to the distance squared. This is then normalized by the standard deviation raised to the appropriate power. Positive values of γ 1 indicate that the distribution is skewed to the right, and negative values indicate that it is skewed to the left. The definition of Skewness still retains the notion of a central mode and character- izes the asymmetry according to distances from the mean. An alternative approach suggested by the Faloutsos brothers is to forgo the conventional metrics of central tendency and dispersion and to focus exclusively on the shape of the tail [226]. For example, as we see later in Chapter 5, an important class of skewed distributions have a heav y tail, which is defined as a tail that decays polynomially.
eBook - PDF- T.-W. Lee(Author)
- 2008(Publication Date)
- CRC Press(Publisher)
Skewness represents the degree of distortion from a symmetrical distribution. For a perfectly symmetrical distribution, the Skewness is zero. Kurtosis is a measure of the smoothness of the data distribution. A data distribution with a sharp peak at some point has a positive kurtosis, whereas a flat distribution has a negative kurtosis. Skewness = = --- ≈ ∑ = = ∑ g N N N y y i i N i ( )( ) 1 2 1 3 s 1 3 3 N i y y N ( ) -s (1.24a) where standard deviation s = ∑ --= = i N i y y N 1 2 1 ( ) (1.24b) kurtosis = K N N N N N y y i = + ---- ( ) ( )( )( ) 1 1 2 3 s ----= ∑ i N N N N 1 4 2 3 1 2 3 ( ) ( )( ) (1.25) Thermal and Flow Measurements To determine the relationship between two variables, x and y , in the data, the correlation coefficient is used. R x x y y x x y y xy i N i i i N i i N i = ∑ --∑ -∑ -= = = 1 1 2 1 ( )( ) ( ) ( ) / 2 1 2 (1.26) Looking at Equation 1.26, it is evident that if the variable x is exactly fol-lowed by y, then the correlation coefficient would be 1. Conversely, if the variable x was completely independent of y , then R xy would be zero. Other variations of the above correlation coefficient are also used to characterize turbulence flows. In addition to the foregoing statistical parameters, an important approach for time-series data is frequency analysis, which gives the frequency content of the signal. An example of signal containing many frequency components is turbu-lent flow, where different size eddies in the turbulent flow produce various time scales of flow fluctuations and therefore frequency content in velocity measure-ments. Another example is sound spectrum, where many frequency components may contribute to the overall sound level. The most common method to analyze the frequency content is Fourier analysis, or discrete Fourier analysis for digi-tized data acquisition.
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