measures of skewness and kurtosis pdf

Measures of skewness and kurtosis pdf

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Are the Skewness and Kurtosis Useful Statistics?


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Note: This article was originally published in April and was updated in February The original article indicated that kurtosis was a measure of the flatness of the distribution — or peakedness. This is technically not correct see below.

Exploratory Data Analysis 1. EDA Techniques 1. Quantitative Techniques 1. A fundamental task in many statistical analyses is to characterize the location and variability of a data set. A further characterization of the data includes skewness and kurtosis.

Are the Skewness and Kurtosis Useful Statistics?

Like skewness , kurtosis describes the shape of a probability distribution and there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. Different measures of kurtosis may have different interpretations. The standard measure of a distribution's kurtosis, originating with Karl Pearson , [1] is a scaled version of the fourth moment of the distribution. This number is related to the tails of the distribution, not its peak; [2] hence, the sometimes-seen characterization of kurtosis as "peakedness" is incorrect. For this measure, higher kurtosis corresponds to greater extremity of deviations or outliers , and not the configuration of data near the mean.

The third moment measures skewness , the lack of symmetry, while the fourth moment measures kurtosis , roughly a measure of the fatness in the tails. The actual numerical measures of these characteristics are standardized to eliminate the physical units, by dividing by an appropriate power of the standard deviation. In the unimodal case, if the distribution is positively skewed then the probability density function has a long tail to the right, and if the distribution is negatively skewed then the probability density function has a long tail to the left. A symmetric distribution is unskewed. We proved part a in the section on properties of expected Value. The converse is not true—a non-symmetric distribution can have skewness 0. Examples are given in Exercises 30 and 31 below.


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This content cannot be displayed without JavaScript. Please enable JavaScript and reload the page. This article defines MAQL to calculate skewness and kurtosis that can be used to test the normality of a given data set. In statistics, normality tests are used to determine whether a data set is modeled for normal distribution. Many statistical functions require that a distribution be normal or nearly normal.

Measures of multivariate skewness and kurtosis are developed by extending certain studies on robustness of the t statistic. These measures are shown to possess desirable properties. The asymptotic distributions of the measures for samples from a multivariate normal population are derived and a test of multivariate normality is proposed. The effect of nonnormality on the size of the one-sample Hotelling's T 2 test is studied empirically with the help of these measures, and it is found that Hotelling's T 2 test is more sensitive to the measure of skewness than to the measure of kurtosis. Most users should sign in with their email address.

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About the Measures of Skewness and Kurtosis

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On measuring skewness and kurtosis


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