Skewness Statistics
Skewness, Kurtosis, and the Normal Curve
Skewness In 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. 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 infinity 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 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). Skewness 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 “Fisher’s skewness.” When the deviations from the mean are greater in one direction than in the other direction, this statistic will deviate from zero in the direction of the larger deviations. From sample data, Fisher’s 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 could use this sampling distribution to construct confidence intervals for or tests of hypotheses about 1, there is rarely any value in doing so. The most commonly
Skewness In 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. 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 infinity 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 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). Skewness 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 “Fisher’s skewness.” When the deviations from the mean are greater in one direction than in the other direction, this statistic will deviate from zero in the direction of the larger deviations. From sample data, Fisher’s 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 could use this sampling distribution to construct confidence intervals for or tests of hypotheses about 1, there is rarely any value in doing so. The most commonly
References: Balanda & MacGillivray. (1988). Kurtosis: A critical review. American Statistician, 42: 111-119. Blest, D.C. (2003). A new measure of kurtosis adjusted for skewness. Australian &New Zealand Journal of Statistics, 45, 175-179. Darlington, R.B. (1970). Is kurtosis really “peakedness?” The American Statistician, 24(2), 19-22. DeCarlo, L. T. (1997). On the meaning and use of kurtosis. Psychological Methods, 2, 292-307. Groeneveld, R.A. & Meeden, G. (1984). Measuring skewness and kurtosis. The Statistician, 33, 391-399. Hildebrand, D. K. (1986). Statistical thinking for behavioral scientists. Boston: Duxbury. Hopkins, K.D. & Weeks, D.L. (1990). Tests for normality and measures of skewness and kurtosis: Their place in research reporting. Educational and Psychological Measurement, 50, 717-729. Loether, H. L., & McTavish, D. G. (1988). Descriptive and inferential statistics: An introduction , 3rd ed. Boston: Allyn & Bacon. Moors, J.J.A. (1986). The meaning of kurtosis: Darlington reexamined. The American Statistician, 40, 283-284. Pearson, K. (1895) Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material. Philosophical Transactions of the Royal Society of London, 186, 343-414. Pearson, K. (1905). Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und Pearson. A Rejoinder. Biometrika, 4, 169-212. Snedecor, G.W. & Cochran, W.G. (1967). Statistical methods (6th ed.), Iowa State University Press, Ames, Iowa. Stuart, A. & Ord, J.K. (1994). Kendall’s advanced theory of statistics. Volume 1. Distribution Theory. Sixth Edition. Edward Arnold, London. Wuensch, K. L. (2005). Kurtosis. In B. S. Everitt & D. C. Howell (Eds.), Encyclopedia of statistics in behavioral science (pp. 1028 - 1029). Chichester, UK: Wiley. Wuensch, K * http://core.ecu.edu/psyc/WuenschK/docs30/Platykurtosis.jpg -- distribution of final grades in PSYC 2101 (undergrad stats), Spring, 2007.