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Encyclopedia > Skewness
Example of experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790)
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. Image File history File links SkewedDistribution. ... Image File history File links SkewedDistribution. ... Species T. aestivum T. boeoticum T. dicoccoides T. dicoccon T. durum T. monococcum T. spelta T. sphaerococcum T. timopheevii References:   ITIS 42236 2002-09-22 Wheat Wheat For the indie rock group, see Wheat (band). ... Schematic image of wheat coleoptile (above) and flag leave (below) Coleoptile is the pointed protective sheath covering the emerging shoot in monocotyledons such as oats and grasses. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... This article is about the field of statistics. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...

Contents

Introduction

Consider the distribution in 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 (or snakes), and they provide a visual means for determining which of the two kinds of skewness a distribution has:

  1. positive skew: The right tail is longer; the mass of the distribution is concentrated on the left of the figure. The distribution is said to be right-skewed.
  2. negative skew: The left tail is longer; the mass of the distribution is concentrated on the right of the figure. The distribution is said to be left-skewed.

Image:Skew.png Image File history File links No higher resolution available. ...


Definition

Skewness, the third standardized moment, is written as γ1 and defined as In probability theory and statistics, the kth standardized moment of a probability distribution is μk/σk, where μk is the kth moment about the mean and σ is the standard deviation. ...

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

where μ3 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 κ3 and the third power of the square root of the second cumulant κ2: In probability theory and statistics, the kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity E[(X − E[X])k], where E is the expectation operator. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... // Cumulants of probability distributions In probability theory and statistics, the cumulants κn of the probability distribution of a random variable X are given by In other words, κn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ...

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 far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ...


For a sample of n values the sample skewness is

g_1 = frac{m_3}{m_2^{3/2}} = frac{sqrt{n,}sum_{i=1}^n (x_i-bar{x})^3}{left(sum_{i=1}^n (x_i-bar{x})^2right)^{3/2}}, !

where xi is the ith value, bar{x} is the sample mean, m3 is the sample third central moment, and m2 is the sample variance. In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. ... The kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity E[(X − E[X])k], where E is the expectation operator. ... This article is about mathematics. ...


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 skewness is In statistics, a biased estimator is one that for some reason on average over_ or underestimates what is being estimated. ...

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.


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


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.


Pearson skewness coefficients

Karl Pearson suggested two simpler calculations as a measure of skewness: Karl Pearson FRS (March 27, 1857 – April 27, 1936) established the discipline of mathematical statistics. ...

There is no guarantee that these will be the same sign as each other or as the ordinary definition of skewness. In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... In probability theory and statistics, a median is a type of average that is described as the number dividing the higher half of a sample, a population, or a probability distribution, from the lower half. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ...


See also


  Results from FactBites:
 
Skewness (286 words)
Skewness is a parameter that describes asymmetry in a random variable’s probability distribution.
The one on the left is positively skewed.
Evans, Hastings and Peacock (2000) is a handy reference with detailed information on numerous probability distributions, including formulas for the skewness of each.
  More results at FactBites »

 
 

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