skewed adj : having an oblique or slanting direction or position; "the picture was skew" [syn: skew]
- past of skew
Skewness, the third standardized moment, is written as \gamma_1 and defined as
- \gamma_1 = \frac, \!
where \mu_3 is the third moment about the mean and \sigma is the standard deviation. Equivalently, skewness can be defined as the ratio of the third cumulant \kappa_3 and the third power of the square root of the second cumulant \kappa_2:
- \gamma_1 = \frac. \!
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.
For a sample of n values the sample skewness is
- g_1 = \frac = \frac, \!
where x_i is the ith value, \bar 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
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.
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:
- An Asymmetry Coefficient for Multivariate Distributions by Michel Petitjean
skewed in Czech: Koeficient šikmosti
skewed in German: Schiefe (Statistik)
skewed in French: Asymétrie (statistique)
skewed in Spanish: Asimetría
skewed in Finnish: Vinous
skewed in Hebrew: צידוד (סטטיסטיקה)
skewed in Hungarian: Ferdeség
skewed in Italian: Simmetria (statistica)
skewed in Japanese: 歪度
skewed in Latvian: Asimetrijas koeficients
skewed in Lithuanian: Asimetrijos koeficientas
skewed in Dutch: Scheefheid
skewed in Polish: Współczynnik skośności
skewed in Portuguese: Obliquidade
skewed in Russian: Коэффициент асимметрии
skewed in Sundanese: Skewness
skewed in Turkish: Çarpıklık
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