Skewness (statistics)

The skewness ( English skewness or skew ) is a statistical indicator that describes the type and strength of the asymmetry of a probability distribution . It shows whether and to what extent the distribution is inclined to the right (right-hand part, left-skewed, negative skewness) or to the left (left-hand part, right-hand skewed, positive skewness). Any non- symmetrical distribution is called skewed .

definition

The skewness of a random variable is the central 3rd order moment (if the 3rd order moment exists), normalized to the standard deviation : ${\displaystyle \gamma _{m}}$ ${\displaystyle X}$ ${\displaystyle \mu _{3}}$ ${\displaystyle \sigma }$

${\displaystyle \gamma _{m}:={\frac {\mu _{3}}{\sigma ^{3}}}=\operatorname {E} \left[\left({\frac {X-\mu }{\sigma }}\right)^{3}\right]={\frac {\operatorname {E} \left[\left(X-\mu \right)^{3}\right]}{\sigma ^{3}}}={\frac {\operatorname {E} \left(X^{3}\right)-3\operatorname {E} \left(X^{2}\right)\mu +2\mu ^{3}}{\sigma ^{3}}}={\frac {\operatorname {E} \left(X^{3}\right)-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}}$.

with the expected value and the variance . This representation is also called the moment coefficient of skewness . The representation results from the cumulants ${\displaystyle \mu =\operatorname {E} (X)}$ ${\displaystyle \sigma ^{2}=\operatorname {Var} (X)}$ ${\displaystyle \kappa _{i}}$

${\displaystyle \gamma _{m}={\frac {\kappa _{3}}{\sqrt {\kappa _{2}^{3}}}}={\frac {\kappa _{3}}{\operatorname {Var} (X)^{\frac {3}{2}}}}}$.

The skewness can take on any real value.

• With negative skewness,, one speaks of a left- skewed or right- skewed distribution; in typical cases, it slopes more gently on the left than on the right. ${\displaystyle \gamma _{m}<0}$
• With positive skewness,, one speaks of a right- skewed or left- skewed distribution; it is typically the reverse of the slope on the right side than on the left. ${\displaystyle \gamma _{m}>0}$

Typical representatives of right-skewed distributions are the Bernoulli distribution for , the exponential distribution and the Pareto distribution for . ${\displaystyle p<1/2}$ ${\displaystyle k>3}$

The skewness is invariant under linear transformation with : ${\displaystyle a>0}$

${\displaystyle \gamma _{m}(aX+b)=\gamma _{m}(X)}$

The following applies to the sum of independent standardized random variables : ${\displaystyle X_{i}}$

${\displaystyle \gamma _{m}(X_{1}+X_{2}+\ldots +X_{n})=\left(\gamma _{m}(X_{1})+\gamma _{m}(X_{2})+\ldots +\gamma _{m}(X_{n})\right)/n^{\frac {3}{2}}}$,

d. H. the skewness of the sum of independent and identically distributed random variables is the original skewness divided by . ${\displaystyle {\sqrt {n}}}$

Empirical skew

The following formula is used to calculate the skewness of an empirical frequency distribution:

${\displaystyle g_{m}={\frac {m_{3}}{s^{3}}}={\frac {{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{3}}{{\sqrt {{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}^{3}}}={\frac {1}{n}}\sum _{i=1}^{n}\left({\frac {x_{i}-{\overline {x}}}{s}}\right)^{3}}$

So that the skewness is independent of the unit of measurement of the variable, the measured values ​​are calculated using the arithmetic mean and the empirical standard deviation of the observed values ${\displaystyle {\overline {x}}}$ ${\displaystyle s}$ ${\displaystyle x_{i}}$

${\displaystyle z_{i}={\frac {x_{i}-{\overline {x}}}{s}}}$

standardized . The standardization applies

${\displaystyle {\overline {z}}={\frac {1}{n}}\sum _{i=1}^{n}z_{i}=0\quad }$and . ${\displaystyle \quad s_{z}^{2}={\frac {1}{n}}\sum _{i=1}^{n}z_{i}^{2}=1}$

Estimating the skewness of a population

To estimate the unknown skewness of a population using sample data ( the sample size), the expected value and the variance from the sample must be estimated; H. the theoretical moments are replaced by the empirical ones: ${\displaystyle \gamma _{m}}$ ${\displaystyle x_{1},\ldots ,x_{n}}$ ${\displaystyle n}$

${\displaystyle {\tilde {\gamma }}_{m}={\frac {1}{n}}\sum _{i=1}^{n}\left({\frac {x_{i}-{\overline {x}}}{s}}\right)^{3}}$

with the sample mean and the sample standard deviation. However, this estimator is not unbiased for unlike ${\displaystyle {\overline {x}}}$ ${\displaystyle s}$ ${\displaystyle \gamma _{m}}$

${\displaystyle {\hat {\gamma }}_{m}={\frac {n}{(n-1)(n-2)}}\sum _{i=1}^{n}\left({\frac {x_{i}-{\overline {x}}}{s}}\right)^{3}}$.

More misalignment

Location of mean and median

The definition goes on Karl Pearson

${\displaystyle S={\frac {\mu -x_{\text{med}}}{\sigma }}}$

with the expected value , the median and the standard deviation . The range of values ​​of S is the interval . For symmetric distributions . Right-skewed distributions often have a positive , but there are exceptions to this rule of thumb. ${\displaystyle \mu }$ ${\displaystyle x_{\text{med}}}$ ${\displaystyle \sigma }$ ${\displaystyle [-1,1]}$ ${\displaystyle S=0}$ ${\displaystyle S}$

If the standard deviation diverges, Pearson's definition can be generalized by calling a distribution skewed to the right when the median is less than the expected value. In this sense, the Pareto distribution is skewed to the right for any parameter . ${\displaystyle k>1}$

Quantile coefficient of skewness

The quantile coefficient of skewness describes the normalized difference between the distance of the - and the - quantile to the median. So it is calculated as follows: ${\displaystyle \alpha }$ ${\displaystyle (1-\alpha )}$

${\displaystyle \gamma _{p}={\frac {(x_{1-\alpha }-x_{\text{med}})-(x_{\text{med}}-x_{\alpha })}{x_{1-\alpha }-x_{\alpha }}},\quad \alpha \in \left(0,{\tfrac {1}{2}}\right)}$

The quantile coefficient can assume values ​​between and . The quantile coefficient exists for any distribution, even if the expected value or the standard deviation should not be defined. ${\displaystyle -1}$ ${\displaystyle 1}$

A symmetrical distribution has the quantile coefficient ; a right-skewed (left-skewed) distribution usually has a positive (negative) quantile coefficient. For is the quartile coefficient. The Pareto distribution has positive quantile coefficients for any parameter . ${\displaystyle 0}$ ${\displaystyle \alpha ={\tfrac {1}{4}}}$ ${\displaystyle k>0}$

interpretation

Example of experimental data with a positive skew (right skew)

If , the distribution is skewed to the right , is , the distribution is skewed to the left . The following applies to benign distributions: In the case of right-skewed distributions, values ​​that are smaller than the mean value are observed more frequently, so that the peak ( mode ) is to the left of the mean value; the right part of the graph is flatter than the left. If so, the distribution is balanced on both sides. With symmetrical distributions, is always . Conversely, distributions with need not be symmetric. ${\displaystyle \gamma _{p}>0}$ ${\displaystyle \gamma _{p}<0}$ ${\displaystyle \gamma _{p}=0}$ ${\displaystyle \gamma _{p}=0}$ ${\displaystyle \gamma _{p}=0}$

The following rules of thumb can be stated for benign distributions:

• right skew: ${\displaystyle x_{\text{mod}}<x_{\text{med}}<{\overline {x}}}$
• symmetrical: ${\displaystyle x_{\text{mod}}=x_{\text{med}}={\overline {x}}}$
• left skewed: ${\displaystyle x_{\text{mod}}>x_{\text{med}}>{\overline {x}}}$

Skewness is a measure of the asymmetry of a probability distribution . Since the Gaussian distribution is symmetric, i. H. has zero skewness, skewness is a possible measure to compare a distribution with the normal distribution. (For a test of this property, see e.g. the Kolmogorow-Smirnow test .)

Interpretation of the skew

Right-skewed distributions can be found e.g. B. often in per capita income . There are a few people with extremely high incomes and very many people with rather low incomes. The third power gives the few very extreme values ​​a high weight and creates a skew with a positive sign . There are different formulas to calculate skewness. The common statistics packages such as SPSS , SYSTAT , Stata etc. use formulas that differ from the above torque-based calculation rules, especially in the case of a small number of cases.

See also

• Camber (statistics)

literature

• WH Press et al .: Numerical Recipes in C . 2nd Edition. Cambridge University Press, 1992, chapter 14.1.

Individual evidence

1. ↑ Bielefeld University: Andreas Handl - Symmetrie und Schiefe, p. 4 ( Memento from April 13, 2014 in the Internet Archive ) (PDF; 248 kB)
2. ^ "SPSS 16" by Felix Brosius, page 361
3. ^ Paul T. von Hippel: Mean, Median, and Skew: Correcting a Textbook Rule . In: Journal of Statistics Education . 13, No. 2, 2005.

Web links

• Schiefe explained using graphic examples