Skewness inprobability theory andstatistics is a measure of the asymmetry of theprobability distribution of areal-valuedrandom variable about its mean. Similarly tokurtosis, it provides insights into characteristics of a distribution. The skewness value can be positive, zero, negative, or undefined.
For aunimodal distribution (a distribution with a single peak), negative skew commonly indicates that thetail is on the left side of the distribution, and positive skew indicates that the tail is on the right. 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 in skewness means that the tails on both sides of the mean balance out overall; this is the case for a symmetric distribution but can also be true for an asymmetric distribution where one tail is long and thin, and the other is short but fat. Thus, the judgement on the symmetry of a given distribution by using only its skewness is risky; the distribution shape must be taken into account.
Consider the two distributions in the figure. Within each graph, the values on the right side of the distribution taper differently from the values on the left side. These tapering sides are calledtails, and they provide a visual means to determine which of the two kinds of skewness a distribution has:
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Skewness in a data series may sometimes be observed not only graphically but by simple inspection of the values. For instance, consider the numeric sequence (49, 50, 51), whose values are evenly distributed around a central value of 50. We can transform this sequence into a negatively skewed distribution by adding a value far below the mean, which is probably a negativeoutlier, e.g. (40, 49, 50, 51). Therefore, the mean of the sequence becomes 47.5, and the median is 49.5. Based on the formula ofnonparametric skew, defined as the skew is negative. Similarly, we can make the sequence positively skewed by adding a value far above the mean, which is probably a positive outlier, e.g. (49, 50, 51, 60), where the mean is 52.5, and the median is 50.5.
As mentioned earlier, a unimodal distribution with zero value of skewness does not imply that this distribution is symmetric necessarily. However, a symmetric unimodal or multimodal distribution always has zero skewness.
The skewness is not directly related to the relationship between the mean and median: a distribution with negative skew can have its mean greater than or less than the median, and likewise for positive skew.[2]
In the older notion ofnonparametric skew, defined as where is themean, is themedian, and is thestandard deviation, the skewness is defined in terms of this relationship: positive/right nonparametric skew means the mean is greater than (to the right of) the median, while negative/left nonparametric skew means the mean is less than (to the left of) the median. However, the modern definition of skewness and the traditional nonparametric definition do not always have the same sign: while they agree for some families of distributions, they differ in some of the cases, and conflating them is misleading.
If the distribution issymmetric, then the mean is equal to the median, and the distribution has zero skewness.[3] If the distribution is both symmetric andunimodal, then themean =median =mode. This is the case of a coin toss or the series 1,2,3,4,... Note, however, that the converse is not true in general, i.e. zero skewness (defined below) does not imply that the mean is equal to the median.
A 2005 journal article points out:[2]
Many textbooks teach a rule of thumb stating that the mean is right of the median under right skew, and left of the median under left skew. This rule fails with surprising frequency. It can fail inmultimodal distributions, or in distributions where one tail islong but the other isheavy. Most commonly, though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal. Such distributions not only contradict the textbook relationship between mean, median, and skew, they also contradict the textbook interpretation of the median.
For example, in the distribution of adult residents across US households, the skew is to the right. However, since the majority of cases is less than or equal to the mode, which is also the median, the mean sits in the heavier left tail. As a result, the rule of thumb that the mean is right of the median under right skew failed.[2]
The skewness of a random variableX is the thirdstandardized moment, defined as:[4][5]
whereμ is the mean,σ is thestandard deviation, E is theexpectation operator,μ3 is the thirdcentral moment, andκt are thet-thcumulants. It is sometimes referred to asPearson's moment coefficient of skewness,[5] or simply themoment coefficient of skewness,[4] 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 cumulantκ3 to the 1.5th power of the second cumulantκ2. This is analogous to the definition ofkurtosis as the fourth cumulant normalized by the square of the second cumulant. The skewness is also sometimes denotedSkew[X].
Ifσ is finite andμ is finite too, then skewness can be expressed in terms of the non-central momentE[X3] by expanding the previous formula:
Skewness can be infinite, as whenwhere the third cumulants are infinite, or as whenwhere the third cumulant is undefined.
Examples of distributions with finite skewness include the following.
For a sample ofn values, two natural estimators of the population skewness are[6]
and
where is thesample mean,s is thesample standard deviation,m2 is the (biased) sample second centralmoment, andm3 is the (biased) sample third central moment.[6] is amethod of moments estimator.
Another common definition of thesample skewness is[6][7]
where is the unique symmetric unbiased estimator of the thirdcumulant and is the symmetric unbiased estimator of the second cumulant (i.e. thesample variance). This adjusted Fisher–Pearson standardized moment coefficient is the version found inExcel and several statistical packages includingMinitab,SAS andSPSS.[7]
Under the assumption that the underlying random variable is normally distributed, it can be shown that all three ratios, and are unbiased andconsistent estimators of the population skewness, with, i.e., their distributions converge to a normal distribution with mean 0 and variance 6 (Fisher, 1930).[6] The variance of the sample skewness is thus approximately for sufficiently large samples. More precisely, in a random sample of sizen from a normal distribution,[8][9]
In normal samples, has the smaller variance of the three estimators, with[6]
For non-normal distributions,, and are generallybiased estimators of the population skewness; their expected values can even have the opposite sign from the true skewness. For instance, a mixed distribution consisting of very thin Gaussians centred at −99, 0.5, and 2 with weights 0.01, 0.66, and 0.33 has a skewness of about −9.77, but in a sample of 3 has an expected value of about 0.32, since usually all three samples are in the positive-valued part of the distribution, which is skewed the other way.
Skewness is a descriptive statistic that can be used in conjunction with thehistogram and the normalquantile plot to characterize the data or distribution.
Skewness indicates the direction and relative magnitude of a distribution's deviation from the normal distribution.
With pronounced skewness, standard statistical inference procedures such as aconfidence interval for a mean will be not only incorrect, in the sense that the true coverage level will differ from the nominal (e.g., 95%) level, but they will also result in unequal error probabilities on each side.
Skewness can be used to obtain approximate probabilities and quantiles of distributions (such asvalue at risk in finance) via theCornish–Fisher expansion.
Many models assume normal distribution; i.e., data are symmetric about the mean. The normal distribution has a skewness of zero. But in reality, data points may not be perfectly symmetric. So, an understanding of the skewness of the dataset indicates whether deviations from the mean are going to be positive or negative.
D'Agostino's K-squared test is agoodness-of-fitnormality test based on sample skewness and sample kurtosis.
Other measures of skewness have been used, including simpler calculations suggested byKarl Pearson[10] (not to be confused with Pearson's moment coefficient of skewness, see above). These other measures are:
The Pearson mode skewness,[11] or first skewness coefficient, is defined as
The Pearson median skewness, or second skewness coefficient,[12][13] is defined as
Which is a simple multiple of thenonparametric skew.
Bowley's measure of skewness (from 1901),[14][15] also calledYule's coefficient (from 1912)[16][17] is defined as:whereQ is thequantile function (i.e., the inverse of thecumulative distribution function). The numerator is difference between the average of the upper and lower quartiles (a measure of location) and the median (another measure of location), while the denominator is thesemi-interquartile range, which for symmetric distributions is equal to theMAD measure ofdispersion.[citation needed]
Other names for this measure are Galton's measure of skewness,[18] the Yule–Kendall index[19] and the quartile skewness,[20]
Similarly, Kelly's measure of skewness is defined as[21]
A more general formulation of a skewness function was described by Groeneveld, R. A. and Meeden, G. (1984):[22][23][24]The functionγ(u) satisfies−1 ≤γ(u) ≤ 1 and is well defined without requiring the existence of any moments of the distribution.[22] Bowley's measure of skewness isγ(u) evaluated atu = 3/4 while Kelly's measure of skewness isγ(u) evaluated atu = 9/10. This definition leads to a corresponding overall measure of skewness[23] defined as thesupremum of this over the range1/2 ≤u < 1. Another measure can be obtained by integrating the numerator and denominator of this expression.[22]
Quantile-based skewness measures are at first glance easy to interpret, but they often show significantly larger sample variations than moment-based methods. This means that often samples from a symmetric distribution (like the uniform distribution) have a large quantile-based skewness, just by chance.
Groeneveld and Meeden have suggested, as an alternative measure of skewness,[22]
whereμ is the mean,ν is the median,|...| is theabsolute value, andE() is the expectation operator. This is closely related in form toPearson's second skewness coefficient.
Use ofL-moments in place of moments provides a measure of skewness known as the L-skewness.[25]
A value of skewness equal to zero does not imply that the probability distribution is symmetric. Thus there is a need for another measure of asymmetry that has this property: such a measure was introduced in 2000.[26] It is calleddistance skewness and denoted bydSkew. IfX is a random variable taking values in thed-dimensional Euclidean space,X has finite expectation,X' is an independent identically distributed copy ofX, and denotes the norm in the Euclidean space, then a simplemeasure of asymmetry with respect to location parameterθ isanddSkew(X) := 0 forX =θ (with probability 1). Distance skewness is always between 0 and 1, equals 0 if and only ifX is diagonally symmetric with respect toθ (X and2θ −X have the same probability distribution) and equals 1 if and only ifX is a constantc () with probability one.[27] Thus there is a simple consistentstatistical test of diagonal symmetry based on thesample distance skewness:
Themedcouple is a scale-invariant robust measure of skewness, with abreakdown point of 25%.[28] It is themedian of the values of the kernel functiontaken over all couples such that, where is the median of thesample. It can be seen as the median of all possible quantile skewness measures.
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