Skewness: Definition from Answers.com

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.

1 Introduction
2 Definition
- 2.1 Sample skewness
- 2.2 Properties
3 Applications
4 Pearson's skewness coefficients
5 See also
6 References
7 External links

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:

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
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.

In a skewed (unbalanced, lopsided) distribution, the mean is farther out in the long tail than is the median. If there is no skewness i.e. the distribution is symmetric then the mean = the median = the mode.

Many textbooks 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.^[1]

Definition

Skewness, the third standardized moment, is written as $γ 1$ and defined as

$\gamma_1 = \frac{\mu_3}{\sigma^3}, \!$

where $μ 3$ is the third moment about the mean and $σ$ is the standard deviation. Equivalently, skewness can be defined as the ratio of the third cumulant $κ 3$ and the third power of the square root of the second cumulant $κ 2$ :

$\gamma_1 = \frac{\kappa_3}{\kappa_2^{3/2}}. \!$

This is analogous to the definition of kurtosis, which is expressed as the fourth cumulant divided by the fourth power of the square root of the second cumulant.

The skewness of a random variable X is sometimes denoted Skew[X].

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 $x i$ is the i^th value, $\bar{x}$ is the sample mean, $m 3$ is the sample third central moment, and $m 2$ is the sample variance.

Given samples from a population, the equation for the sample skewness $g 1$ above is a biased estimator of the population skewness. The usual estimator of skewness is

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

where $k 3$ is the unique symmetric unbiased estimator of the third cumulant and $k 2$ is the symmetric unbiased estimator of the second cumulant. Unfortunately $G 1$ 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 is symmetric about the mean. The normal distribution has a skewness of zero. But in reality, data points are not 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

(mean − mode) / standard deviation,

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

3 (mean − mode) / standard deviation,

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

3 (mean − median) / standard deviation.

Starting from a standard cumulant expansion around a Normal distribution, one can actually show that skewness = 6 (mean − median) / 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.

References

^ Textbook rule often fails
^ Weisstein, Eric W., "Pearson Mode Skewness" from MathWorld.
^ Weisstein, Eric W., "Pearson's skewness coefficients" from MathWorld.

External links

An Asymmetry Coefficient for Multivariate Distributions by Michel Petitjean
On More Robust Estimation of Skewness and Kurtosis Comparison of skew estimators by Kim and White.

Statistics

Descriptive statistics

Continuous data

Location	Mean (Arithmetic, Geometric, Harmonic) · Median · Mode

Dispersion	Range · Standard deviation · Coefficient of variation · Percentile

Moments	Variance · Semivariance · Skewness · Kurtosis

Categorical data

Frequency · Contingency table

Inferential statistics
and
hypothesis testing

Inference	Confidence interval (Frequentist inference) · Credible interval (Bayesian inference) · Significance · Meta-analysis

Design of experiments	Population · Sampling · Stratified sampling · Replication · Blocking · Sensitivity and specificity

Sample size estimation	Statistical power · Effect size · Standard error

General estimation	Bayesian estimator · Maximum likelihood · Method of moments · Minimum distance · Maximum spacing

Specific tests	Z-test (normal) · Student's t-test · F-test · Chi-square test · Pearson's chi-square test · Wald test · Mann–Whitney U · Wilcoxon signed-rank test

Survival analysis	Survival function · Kaplan–Meier · Logrank test · Failure rate · Proportional hazards models

Correlation
and
regression analysis

Correlation	Pearson product-moment correlation · Rank correlation (Spearman's rho, Kendall's tau) · Confounding variable

Linear regression	Linear model · General linear model · Generalized linear model · Analysis of variance · Analysis of covariance

Nonlinear regression	Nonparametric · Semiparametric · Logistic

Statistical graphics

Bar chart · Biplot · Box plot · Control chart · Correlogram · Forest plot · Histogram · Q-Q plot · Run chart · Scatter plot · Stemplot

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