Editing Skewness (section)

==Definition==

===Fisher's moment coefficient of skewness===
The skewness <math>\gamma_1</math> of a random variable {{mvar|X}} is the third [[standardized moment]] <math>\tilde{\mu}_3</math>, defined as:<ref name="StanBrown1"/><ref name="FXSolver1"/>

<math display="block">
    \gamma_1 := \tilde{\mu}_3 = \operatorname{E}\left[\left(\frac{X-\mu}{\sigma}\right)^3 \right]
             = \frac{\mu_3}{\sigma^3}
             = \frac{\operatorname{E}\left[(X-\mu)^3\right]}{\left( \operatorname{E}\left[ (X-\mu)^2 \right] \right)^{3/2}}
             = \frac{\kappa_3}{\kappa_2^{3/2}}
  </math>
where {{mvar|μ}} is the mean, {{mvar|σ}} is the [[standard deviation]], E is the [[expected value|expectation operator]], {{math|''μ''<sub>3</sub>}} is the third [[central moment]], and {{math|''κ''<sub>''t''</sub>}} are the {{mvar|t}}-th [[cumulant]]s. It is sometimes referred to as '''Pearson's moment coefficient of skewness''',<ref name="FXSolver1">[http://www.fxsolver.com/browse/formulas/Pearson's+moment+coefficient+of+skewness Pearson's moment coefficient of skewness], FXSolver.com</ref> or simply the '''moment coefficient of skewness''',<ref name="StanBrown1">[http://brownmath.com/stat/shape.htm "Measures of Shape: Skewness and Kurtosis"], 2008–2016 by Stan Brown, Oak Road Systems</ref> but should not be confused with Pearson's other skewness statistics (see below). The last equality expresses skewness in terms of the ratio of the third cumulant {{math|''κ''<sub>3</sub>}} to the 1.5th power of the second cumulant {{math|''κ''<sub>2</sub>}}. 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 {{math|Skew[''X'']}}.

If {{mvar|σ}} is finite and {{mvar|μ}} is finite too, then skewness can be expressed in terms of the non-central moment {{math|E[''X''<sup>3</sup>]}} by expanding the previous formula:
<math display="block">
\begin{align}
    \tilde{\mu}_3
     &= \operatorname{E}\left[\left(\frac{X-\mu}{\sigma}\right)^3 \right] \\
     &= \frac{\operatorname{E}[X^3] - 3\mu\operatorname E[X^2] + 3\mu^2\operatorname E[X] - \mu^3}{\sigma^3}\\
     &= \frac{\operatorname{E}[X^3] - 3\mu(\operatorname E[X^2] -\mu\operatorname E[X]) - \mu^3}{\sigma^3}\\
     &= \frac{\operatorname{E}[X^3] - 3\mu\sigma^2 - \mu^3}{\sigma^3}.
\end{align}
  </math>
<!-- EDITORS BEWARE: DO NOT CHANGE THIS INTO   E[X^3] - 3\mu E[X^2] + 2\mu^3  /// SEE TALK PAGE -->

===Examples===
Skewness can be infinite, as when
<math display="block">\Pr \left[ X > x \right]=x^{-2}\mbox{ for }x>1,\ \Pr[X<1]=0</math>
where the third cumulants are infinite, or as when
<math display="block">\Pr[X<x] = \begin{cases}
\frac{1}{2} (1-x)^{-3} & \text{ for } x < 0, \\[2pt]
\frac{1}{2} (1+x)^{-3} & \text{ for } x > 0.
\end{cases}</math>
where the third cumulant is undefined.

Examples of distributions with finite skewness include the following.

*  A [[normal distribution]] and any other symmetric distribution with finite third moment has a skewness of 0
*  A [[half-normal distribution]] has a skewness just below 1
*  An [[exponential distribution]] has a skewness of 2
*  A [[lognormal distribution]] can have a skewness of any positive value, depending on its parameters

===Sample skewness===
For a sample of ''n'' values, two natural estimators of the population skewness are<ref name=JG>{{cite journal |last=Joanes<!--sic--> |first=D. N. |last2=Gill |first2=C. A. |year=1998 |title=Comparing measures of sample skewness and kurtosis |journal=[[Journal of the Royal Statistical Society, Series D]] |volume=47 |issue=1 |pages=183–189 |doi=10.1111/1467-9884.00122}}</ref>

<math display="block">
b_1 = \frac{m_3}{s^3}
    = \frac{\tfrac{1}{n} \sum_{i=1}^n \left(x_i-\bar{x}\right)^3}{\left[\tfrac{1}{n-1} \sum_{i=1}^n \left(x_i-\bar{x}\right)^2 \right]^{3/2}}
</math>

and

<math display="block">
g_1 = \frac{m_3}{m_2^{3/2}}
    = \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i-\bar{x})^3}{\left[\tfrac{1}{n} \sum_{i=1}^n \left(x_i-\bar{x}\right)^2 \right]^{3/2}},
</math>

where <math>\bar{x}</math> is the [[sample mean]], {{mvar|s}} is the [[Standard deviation#Corrected sample standard deviation|sample standard deviation]], {{math|''m''<sub>2</sub>}} is the (biased) sample second central [[Moment (mathematics)|moment]], and {{math|''m''<sub>3</sub>}} is the (biased) sample third central moment.<ref name=JG/> <math>g_1</math> is a  [[Method of moments (statistics)|method of moments]] estimator.  

Another common definition of the ''sample skewness'' is<ref name=JG/><ref name=Doane2011>Doane, David P., and Lori E. Seward. [http://jse.amstat.org/v19n2/doane.pdf "Measuring skewness: a forgotten statistic."] Journal of Statistics Education 19.2 (2011): 1-18. (Page 7)</ref>

<math display="block">\begin{align}
G_1 & = \frac{k_3}{k_2^{3/2}} = \frac{n^2}{(n-1)(n-2)}\; b_1 = \frac{\sqrt{n(n-1)}}{n-2}\; g_1, \\
\end{align}</math>

where <math>k_3</math> is the unique symmetric unbiased estimator of the third [[cumulant]] and <math>k_2 = s^2</math> is the symmetric unbiased estimator of the second cumulant (i.e. the [[Variance#Population variance and sample variance|sample variance]]). This adjusted Fisher–Pearson standardized moment coefficient <math> G_1 </math> is the version found in [[Microsoft Excel|Excel]] and several statistical packages including [[Minitab]], [[SAS (software)|SAS]] and [[SPSS]].<ref name=Doane2011/>

Under the assumption that the underlying random variable <math>X</math> is normally distributed, it can be shown that all three ratios <math>b_1</math>, <math>g_1</math> and <math>G_1</math> are unbiased and [[Consistent estimator|consistent]] estimators of the population skewness <math>\gamma_1=0</math>, with <math>\sqrt{n} b_1 \mathrel{\xrightarrow{d}} N(0, 6)</math>, i.e., their distributions converge to a normal distribution with mean 0 and variance 6 ([[Ronald Fisher|Fisher]], 1930).<ref name=JG/> The variance of the sample skewness is thus approximately <math>6/n</math> for sufficiently large samples. More precisely, in a random sample of size ''n'' from a normal distribution,<ref name=Duncan1997>Duncan Cramer (1997) Fundamental Statistics for Social Research. Routledge. {{isbn|9780415172042}} (p 85)</ref><ref>Kendall, M.G.; Stuart, A. (1969) ''The Advanced Theory of Statistics, Volume 1: Distribution Theory, 3rd Edition'', Griffin. {{isbn|0-85264-141-9}} (Ex 12.9)</ref>

<math display="block"> \operatorname{var}(G_1)= \frac{6n ( n - 1 )}{ ( n - 2 )( n + 1 )( n + 3 ) } .</math>

In normal samples, <math>b_1</math> has the smaller variance of the three estimators, with<ref name=JG/>

<math display="block"> \operatorname{var}(b_1) < \operatorname{var} (g_1) < \operatorname{var}(G_1).</math>

For non-normal distributions, <math>b_1</math>, <math>g_1</math> and <math>G_1</math> are generally [[Bias of an estimator|biased estimators]] of the population skewness <math>\gamma_1</math>; their expected values can even have the opposite sign from the true skewness. For instance, a mixed distribution consisting of very thin Gaussians centred at −99, 0.5, and 2 with weights 0.01, 0.66, and 0.33 has a skewness <math>\gamma_1</math> of about −9.77, but in a sample of 3 <math>G_1</math> has an expected value of about 0.32, since usually all three samples are in the positive-valued part of the distribution, which is skewed the other way.