# Skewness: Wikis

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

# Encyclopedia

Example of experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790)

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.

## 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. Example (observations): 1,1000,1001,1002,1003
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. Example (observations): 1,2,3,4,100.

If there is zero skewness (i.e., the distribution is symmetric) then the mean = median. (If, in addition, the distribution is unimodal, then the mean = median = mode.)

"Many textbooks," a recent 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. [But] 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."[1]

## Definition

The skewness of a random variable X is the third standardized moment, denoted γ1 and defined as

$\gamma_1 = \operatorname{E}\Big[\big(\tfrac{X-\mu}{\sigma}\big)^{\!3}\, \Big] = \frac{\mu_3}{\sigma^3} = \frac{\operatorname{E}\big[(X-\mu)^3\big]}{\ \ \ \operatorname{E}\big[(X-\mu)^2\big]^{3/2}} = \frac{\kappa_3}{\kappa_2^{3/2}}\ ,$

where μ3 is the third moment about the mean μ, and σ is the standard deviation. The last equality expresses skewness in terms of the ratio of the third cumulant κ3 and the 1.5th power of the second cumulant κ2. This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant.

The skewness is also sometimes denoted Skew[X]. Older textbooks used to denote the skewness as $\scriptstyle\sqrt{\beta_1}$, which was rather inconvenient since skewness can be negative.

The formula expressing skewness in terms of the non-central moment E[X3] is

$\gamma_1 = \frac{\operatorname{E}[X^3] - 3\mu\sigma^2 - \mu^3}{\sigma^3}\ .$

### Sample skewness

For a sample of n values the sample skewness is

$g_1 = \frac{m_3}{m_2^{3/2}} = \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^3}{\left(\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^2\right)^{3/2}}\ ,$

where $\scriptstyle\overline{x}$ is the sample mean, m3 is the sample third central moment, and m2 is the sample variance.

Given samples from a population, the equation for the sample skewness g1 above is a biased estimator of the population skewness. The usual estimator of population skewness is[citation needed]

$G_1 = \frac{k_3}{k_2^{3/2}} = \frac{\sqrt{n\,(n-1)}}{n-2}\; g_1,$

where k3 is the unique symmetric unbiased estimator of the third cumulant and k2 is the symmetric unbiased estimator of the second cumulant. Unfortunately G1 is, nevertheless, generally biased. Its expected value can even have the opposite sign from the true skewness; compare unbiased estimation of standard deviation.

### Properties

If Y is the sum of n independent random variables, all with the same distribution as X, then it can be shown that Skew[Y] = Skew[X] / √n.

## Applications

Skewness has benefits in many areas. Many simplistic models assume normal distribution; i.e., data are symmetric about the mean. The normal distribution has a skewness of zero. But in reality, data points may not be perfectly symmetric. So, an understanding of the skewness of the dataset indicates whether deviations from the mean are going to be positive or negative.

D'Agostino's K-squared test is a goodness-of-fit normality test based on sample skewness and sample kurtosis.

## Pearson's skewness coefficients

Karl Pearson suggested simpler calculations as a measure of skewness: The Pearson mode skewness[2], defined by

Pearson's first skewness coefficient [3], defined by

as well as Pearson's second skewness coefficient, defined by

Starting from a standard cumulant expansion around a Normal distribution, one can actually show that skewness = 6 (meanmedian) / standard deviation ( 1 + kurtosis / 8) + O(skewness^2)

There is no guarantee that these will be the same sign as each other or as the ordinary definition of skewness.

# Study guide

Up to date as of January 14, 2010

### From Wikiversity

Skewness refers to asymmetry (or "tapering") in the distribution of sample data:

1. negative skew: The left tail is longer; the mass of the distribution is concentrated on the right of the figure. It has a few relatively low values. The distribution is said to be left-skewed. In such a distribution, the mean is lower than median which in turn is lower than the mode (i.e.; mean < median < mode); in which case the skewness is lower than zero.
2. positive skew: The right tail is longer; the mass of the distribution is concentrated on the left of the figure. It has a few relatively high values. The distribution is said to be right-skewed. In such a distribution, the mean is greater than median which in turn is greater than the mode (i.e.; mean > median > mode); in which case the skewness is greater than zero.

In a skewed (unbalanced, lopsided) distribution, the mean is farther out in the long tail than is the median. If there is no skewness or the distribution is symmetric like the bell-shaped normal curve then the mean = median = mode.