Skewness

The skewness ( English technical term: Skewness or skew) describes the " inclination strength" of a statistical distribution. It indicates whether and how strongly the distribution to the right ( positive skewness ) or left (negative skewness ) is inclined. Each non-symmetric distribution is called wrong.

• 2.1 Location of mean and median
• 2.2 Quantilskoeffizient the skewness

Definition

The skewness v ( X) of a random variable X is the central moment of order 3 μ3 (X ) (if the 3rd order moment exists), normalized to the standard deviation σX:

With the expected value mX: = E ( X). The skewness can take any real value.

• With negative skewness, v < 0, it is called left-skewed, the distribution falls flat on the left side than from on the right; There is relatively little low values. Mode> expected value.
• In case of positive skewness, v> 0, it is called skewed to the right, the distribution falls inversely flat on the right side from than on the left; there are relatively few high values. Mode < expected value.

The skewness is invariant under linear transformation with a> 0:

Empirical skewness

To calculate the skewness of an empirical frequency distribution, the following formula is used:

Thus, the skewness is independent of the unit of measurement of the variables, using the measured values ​​are the arithmetic mean and the standard deviation of the observed values

Standardized. Applies by standardizing

Estimate of the skewness of a population

In order to estimate the unknown skewness a population by sampling data ( the sample size ), the expected value and the variance of the sample to be estimated, ie the theoretical be replaced by the empirical moments:

With the sample mean and sample standard deviation. However, this estimator is not unbiased in the contrast to

More Inclined Dimensions

Position of mean and median

In Karl Pearson, the definition goes

With the expected value μ, the median and the standard deviation σ back. The range of S is the interval [-1,1]. For symmetric distributions is S = 0 Skewness distributions often have a positive S, but there are exceptions to this rule of thumb.

Quantilskoeffizient the skewness

The Quantilskoeffizient the skewness describes the normalized difference between the distance of the quantile and the median. It is therefore calculated as follows:

,

It can take values ​​between -1 and 1 of Quantilskoeffizient. A symmetric distribution has Quantilskoeffizienten 0; A steep (left parts ) distribution generally has a negative ( positive ) Quantilskoeffizienten. For results of the quartile coefficient.

Interpretation

( Quite steep also called ), so the distribution is skewed to the right, is the distribution left-skewed. At right leaning (or left divide ) distributions are values ​​that are smaller than the mean observed more often, so that the peak ( mode) is to the left of the mean; the right part of the graph is shallower than the left. Applies, so the distribution is balanced on both sides. For symmetric distributions is always. Conversely, do not have to be symmetric distributions.

As rules of thumb so you can hold:

• Skewed to the right:
• Symmetrical:
• Left-skewed:

The skewness is a measure of the asymmetry of a probability distribution. Since the Gaussian distribution is symmetrical, ie has a skewness of zero, the skewness is a possible measure to compare a distribution with the normal distribution. ( To test this property, see, eg, the Kolmogorov -Smirnov test. )

Interpretation of the skewness

Skewness distributions are found, for example, often in terms of per capita income. Here there are a few people with extremely high incomes and a lot of people with rather low incomes. By the 3rd power of the few very extreme values ​​get a high weight and it creates a Schiefemaß with a positive sign. There are different formulas to calculate the skewness. The standard statistical packages such as SPSS, SYSTAT, etc. Stata use, especially in the case of a small sample size of the above, moment -based computation rule different formulas.