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1、Skewness, Kurtosis, and the Normal CurveSkewnessIn everyday language, the terms “skewed” and “askew” are used to refer to something that is out of line or distorted on one side. When referring to the shape of frequency or probability distributions, “skewness” refers to asymmetry of the distribution.
2、 A distribution with an asymmetric tail extending out to the right is referred to as “positively skewed” or “skewed to the right,” while a distribution with an asymmetric tail extending out to the left is referred to as “negatively skewed” or “skewed to the left.” Skewness can range from minus infin
3、ity to positive infinity.Karl Pearson (1895) first suggested measuring skewness by standardizing the difference between the mean and the mode, that is, . Population modes are not well estimated from sample modes, but one can estimate the difference between the mean and the mode as being three times
4、the difference between the mean and the median (Stuart & Ord, 1994), leading to the following estimate of skewness: . Many statisticians use this measure but with the 3 eliminated, that is, . This statistic ranges from -1 to +1. Absolute values above 0.2 indicate great skewness (Hildebrand, 1986).Sk
5、ewness has also been defined with respect to the third moment about the mean: , which is simply the expected value of the distribution of cubed z scores. Skewness measured in this way is sometimes referred to as “Fishers skewness.” When the deviations from the mean are greater in one direction than
6、in the other direction, this statistic will deviate from zero in the direction of the larger deviations. From sample data, Fishers skewness is most often estimated by: . For large sample sizes (n 150), g1 may be distributed approximately normally, with a standard error of approximately . While one c
7、ould use this sampling distribution to construct confidence intervals for or tests of hypotheses about g1, there is rarely any value in doing so.The most commonly used measures of skewness (those discussed here) may produce surprising results, such as a negative value when the shape of the distribut
8、ion appears skewed to the right. There may be superior alternative measures not commonly used (Groeneveld & Meeden, 1984).It is important for behavioral researchers to notice skewness when it appears in their data. Great skewness may motivate the researcher to investigate outliers. When making decis
9、ions about which measure of location to report (means being drawn in the direction of the skew) and which inferential statistic to employ (one which assumes normality or one which does not), one should take into consideration the estimated skewness of the population. Normal distributions have zero s
10、kewness. Of course, a distribution can be perfectly symmetric but far from normal. Transformations commonly employed to reduce (positive) skewness include square root, log, and reciprocal transformations.Also see Skewness and the Relative Positions of Mean, Median, and ModeKurtosisKarl Pearson (1905
11、) defined a distributions degree of kurtosis as , where , the expected value of the distribution of Z scores which have been raised to the 4th power. b2 is often referred to as “Pearsons kurtosis,” and b23 (often symbolized with g2 ) as “kurtosis excess” or “Fishers kurtosis,” even though it was Pea
12、rson who defined kurtosis as b23. An unbiased estimator for g2 is . For large sample sizes (n 1000), g2 may be distributed approximately normally, with a standard error of approximately (Snedecor, & Cochran, 1967). While one could use this sampling distribution to construct confidence intervals for
13、or tests of hypotheses about g2, there is rarely any value in doing so.Pearson (1905) introduced kurtosis as a measure of how flat the top of a symmetric distribution is when compared to a normal distribution of the same variance. He referred to more flat-topped distributions (g2 0) as “leptokurtic,
14、” and equally flat-topped distributions as “mesokurtic” (g2 0). Kurtosis is actually more influenced by scores in the tails of the distribution than scores in the center of a distribution (DeCarlo, 1967). Accordingly, it is often appropriate to describe a leptokurtic distribution as “fat in the tail
15、s” and a platykurtic distribution as “thin in the tails.”Student (1927, Biometrika, 19, 160) published a cute description of kurtosis, which I quote here: “Platykurtic curves have shorter tails than the normal curve of error and leptokurtic longer tails. I myself bear in mind the meaning of the word
16、s by the above memoria technica, where the first figure represents platypus and the second kangaroos, noted for lepping.” Please point your browser to , scroll down to “kurtosis,” and look at Students drawings.Moors (1986) demonstrated that. Accordingly, it may be best to treat kurtosis as the extent to which scores are dispersed away from the s
