Skewness Measure

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A Skewness Measure is a measure of the degree of asymmetry of a probability distribution.



References

2016

The qualitative interpretation of the skew is complicated. For a unimodal distribution, negative skew indicates that the tail on the left side of the probability density function is longer or fatter than the right side – it does not distinguish these shapes. Conversely, positive skew indicates that the tail on the right side is longer or fatter than the left side. In cases where one tail is long but the other tail is fat, skewness does not obey a simple rule. For example, a zero value indicates that the tails on both sides of the mean balance out, which is the case for a symmetric distribution, but is also true for an asymmetric distribution where the asymmetries even out, such as one tail being long but thin, and the other being short but fat. Further, in multimodal distributions and discrete distributions, skewness is also difficult to interpret. Importantly, the skewness does not determine the relationship of mean and median (...)
The skewness of a random variable X is the third standardized moment γ1, defined as
[math] \gamma_1 = \operatorname{E}\left[\left(\frac{X-\mu}{\sigma}\right)^3 \right] = \frac{\mu_3}{\sigma^3} = \frac{\operatorname{E}\left[(X-\mu)^3\right]}{\ \ \ ( \operatorname{E}\left[ (X-\mu)^2 \right] )^{3/2}} = \frac{\kappa_3}{\kappa_2^{3/2}}, [/math]
where μ is the mean, σ is the standard deviation, E is the expectation operator, and μ3 is the third central moment. It is sometimes referred to as Pearson's moment coefficient of skewness, or simply the moment coefficient of skewness, 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 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].
Skewness can be expressed in terms of the non-central moment E[X3] by expanding the previous formula,
[math] \begin{align} \gamma_1 &= \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]

2016

  • (Weisstein, 2016) ⇒ Weisstein, Eric W. "Central Moment." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Skewness.html Retrieved 2016-08-21
    • Skewness is a measure of the degree of asymmetry of a distribution. If the left tail (tail at small end of the distribution) is more pronounced than the right tail (tail at the large end of the distribution), the function is said to have negative skewness. If the reverse is true, it has positive skewness. If the two are equal, it has zero skewness.

      Several types of skewness are defined, the terminology and notation of which are unfortunately rather confusing. "The" skewness of a distribution is defined to be

[math] \gamma_1=\frac{\mu_3}{\mu_2^{3/2}}[/math]
where [math]\mu_i[/math] is the ith central moment.