Skewness - Introduction

Introduction

Consider the distribution on the figure. The bars on the right side of the distribution taper differently than the bars on the left side. These tapering sides are called tails, and they provide a visual means for determining which of the two kinds of skewness a distribution has:

  1. negative skew: The left tail is longer; the mass of the distribution is concentrated on the right of the figure. It has relatively few low values. The distribution is said to be left-skewed, left-tailed, or skewed to the left. Example (observations): 1,1001,1002,1003.
  2. positive skew: The right tail is longer; the mass of the distribution is concentrated on the left of the figure. It has relatively few high values. The distribution is said to be right-skewed, right-tailed, or skewed to the right. Example (observations): 1,2,3,1000.

If the distribution is symmetric then the mean is equal to the median and the distribution will have close to zero skewness. (If, in addition, the distribution is unimodal, then the mean = median = mode.) This is the case of a coin toss or the series 1,2,3,4,... Note, however, that the converse is not true in general, i.e. zero skewness does not imply that the mean is equal to the median.

"Many textbooks," a 2005 article points out, "teach a rule of thumb stating that the mean is right of the median under right skew, and left of the median under left skew. this rule fails with surprising frequency. It can fail in multimodal distributions, or in distributions where one tail is long but the other is heavy. Most commonly, though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal. Such distributions not only contradict the textbook relationship between mean, median, and skew, they also contradict the textbook interpretation of the median."

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