Skewness formulas

In the formulas below N represents the population size, n the sample size, μ the population mean, $\bar{x}$ the sample mean and $\tilde{x}$ the sample median.

Methods based on third moment
An explanation why the third moment could be used as a measure of skewness can be found at stats exchange (Glen_b, 2014).

Fisher-Pearson coefficient of Skewness for population data (1895)
$\gamma_1=\frac{\mu_3}{(\mu_2)^{3/2}}=\frac{\mu_3}{\mu_2\sqrt{\mu_2}}$
With $\mu_3=\frac{\sum_{i=1}^{N}(x_i-\mu)^{3}}{N}$ , and $\mu_2=\frac{\sum_{i=1}^{N}(x_i-\mu)^{2}}{N}$

Biased Fisher-Pearson coefficient of Skewness for sample data (1895)
$g_1=\frac{m_3}{(m_2)^{3/2}}$
With $m_3=\frac{\sum_{i=1}^{n}(x_i-\bar{x})^{3}}{n}$ , and $m_2=\frac{\sum_{i=1}^{n}(x_i-\bar{x})^{2}}{n-1}$

Adjusted Fisher-Pearson standardized moment coefficient (1895)
$G_1=\frac{\sqrt{n(n-1)}}{n-2}\gamma_1=\frac{n^2}{(n-1)(n-2)}g_1=\frac{n}{(n-1)(n-2)}\sum_{i=1}^{n}\left(\frac{x_i-\bar{x}}{s} \right)^{3}$

Methods based on tiles
There are several methods based on tiles (quartiles, decentiles, etc.). Hinkley (1975) created a general formula for these.

Hinkley (1975)
$G_1=\frac{F^{-1}(1-p)+F^{-1}(p)-2\tilde{x}}{F^{-1}(1-p)-F^{-1}(p)}$
With $F^{-1}(i)$ the value of the i-th percentile, and 1 - p > p.

Bowley (1920)
Bowley skewness is equal to using the quartiles in Hinkley's formula
$G_1=\frac{Q_3+Q_1-2\tilde{x}}{Q_3-Q_1}$
With Q_i being the i'th quartile.

Kelly (Unknown)
Kelley Skewness is equal to using the first decile in Hinkley's formula:
$G_1=\frac{F^{-1}(0.9)+F^{-1}(0.1)-2\tilde{x}}{F^{-1}(0.9)-F^{-1}(0.1)}$

Yule (1911)
Used a very similar formula as Bowley, but his definition of the quartile range was to divide it by two, yielding:
$G_1=\frac{Q_3+Q_1-2\tilde{x}}{\frac{Q_3-Q_1}{2}}=\frac{2(Q_3+Q_1-2\tilde{x})}{Q_3-Q_1}$

Methods based on mode or median
Groeneveld & Meedan (1984)
Groeneveld & Meedan compared various values of Hinkley's formula and averaged it out on:
$\gamma_1=\frac{\mu-median}{\frac{\sum_{i=1}^{N}|x_i-median|}{N}}=\frac{N(\mu-median)}{\sum_{i=1}^{N}|x_i-median|}$

Pearson Mode Skewness (1895)
$S_k=\frac{\bar{x}-Mode}{s}$

Pearson Median Skewness (1895)
$S_k=\frac{3(\bar{x}-\tilde{x})}{s}$

Hotelling & Solomons (1932)
Use the Pearson median skewness, but do not multiply the difference by three.
$S_k=\frac{\bar{x}-\tilde{x}}{s}$

References

Bowley, A. L. (1920). Elements of Statistics (4th ed.). London: P. S. King & Son.

Glen_b. (2014, August 13). Proof / derivation of skewness and kurtosis formulas - Cross Validated. Retrieved April 6, 2015, from http://stats.stackexchange.com/a/111660

Groeneveld, R. A., & Meeden, G. (1984). Measuring Skewness and Kurtosis. Journal of the Royal Statistical Society. Series D (The Statistician), 33(4), 391–399. http://doi.org/10.2307/2987742

Hinkley, D. V. (1975). On Power Transformations to Symmetry. Biometrika, 62(1), 101–111. http://doi.org/10.2307/2334491

Hotelling, H., & Solomons, L. M. (1932). The Limits of a Measure of Skewness. The Annals of Mathematical Statistics, 3(2), 141–142. http://doi.org/10.1214/aoms/1177732911

Pearson, K. (1895). Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material. Philosophical Transactions of the Royal Society of London. (A.), 186, 343–414. http://doi.org/10.1098/rsta.1895.0010

Yule, G. U. (1911). An Introduction to the Theory of Statistics. London: Charles Griffin. Retrieved from http://archive.org/details/anintroductiont01yulegoog

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vrijdag 2 mei 2014

Skewness formulas

Skewness formulas

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