In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimodal distribution, negative skew commonly indicates that the tail 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 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.
Consider the two distributions in the figure just below. 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 called tails, and they provide a visual means to determine which of the two kinds of skewness a distribution has:
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 negative outlier, 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 of nonparametric 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.
In the older notion of nonparametric skew, defined as where is the mean, is the median, and is the standard 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 is symmetric, then the mean is equal to the median, and the distribution has zero skewness.If the distribution is both symmetric and unimodal, then the mean = 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 does not imply that the mean is equal to the median.
A 2005 journal article points out:
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 in multimodal distributions, or in distributions where one tail is long but the other is heavy. 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.
The skewness of a random variable X is the third standardized moment , defined as:
where μ is the mean, σ is the standard deviation, E is the expectation operator, μ3 is the third central moment, and κt are the t-th cumulants. It is sometimes referred to as Pearson's moment coefficient of skewness,or simply the moment coefficient of skewness, 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 cumulant κ3 to the 1.5th power of the second cumulant κ2. This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant. The skewness is also sometimes denoted Skew[X].
If σ is finite, μ is finite too and skewness can be expressed in terms of the non-central moment E[X3] by expanding the previous formula,
Skewness can be infinite, as when
where the third cumulants are infinite, or as when
where the third cumulant is undefined.
Examples of distributions with finite skewness include the following.
For a sample of n values, a natural method of moments estimator of the population skewness is
where is the sample mean, s is the sample standard deviation, and the numerator m3 is the sample third central moment.
Another common definition of the sample skewness is
where is the unique symmetric unbiased estimator of the third cumulant and is the symmetric unbiased estimator of the second cumulant (i.e. the sample variance). This adjusted Fisher–Pearson standardized moment coefficient is the version found in Excel and several statistical packages including Minitab, SAS and SPSS.
In general, the ratios and are both biased 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.) Nevertheless, in random samples from a normal distribution, both and have the correct expected value of zero (Fisher, 1930).
Under the assumption that the underlying random variable is normally distributed, it can be shown that , i.e., its distribution converges to a normal distribution with mean 0 and variance 6. The variance of the sample skewness is thus approximately for sufficiently large samples. More precisely, in a random sample of size n from a normal distribution,
In normal samples, has the smaller variance of the two estimators, with
where in the denominator
is the (biased) sample second central moment.
Skewness is a descriptive statistic that can be used in conjunction with the histogram and the normal quantile 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 a confidence 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 as value at risk in finance) via the Cornish-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 a goodness-of-fit normality test based on sample skewness and sample kurtosis.
Other measures of skewness have been used, including simpler calculations suggested by Karl Pearson(not to be confused with Pearson's moment coefficient of skewness, see above). These other measures are:
The Pearson mode skewness,or first skewness coefficient, is defined as
The Pearson median skewness, or second skewness coefficient,is defined as
Which is a simple multiple of the nonparametric skew.
Bowley's measure of skewness (from 1901),also called Yule's coefficient (from 1912) is defined as:
When writing it as , it is easier to see that 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 the semi-interquartile range (Q3-Q1)/2, which for symmetric distributions is the MAD measure of dispersion.
Other names for this measure are Galton's measure of skewness,the Yule–Kendall index and the quartile skewness,
A more general formulation of a skewness function was described by Groeneveld, R. A. and Meeden, G. (1984):
where F is the cumulative distribution function. This leads to a corresponding overall measure of skewness ≤ u < 1. Another measure can be obtained by integrating the numerator and denominator of this expression. The function γ(u) satisfies −1 ≤ γ(u) ≤ 1 and is well defined without requiring the existence of any moments of the distribution. 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.defined as the supremum of this over the range 1/2
Bowley's measure of skewness is γ(u) evaluated at u = 3/4. Kelley's measure of skewness uses u = 0.1.
Groeneveld and Meeden have suggested, as an alternative measure of skewness,
where μ is the mean, ν is the median, |...| is the absolute value, and E() is the expectation operator. This is closely related in form to Pearson's second skewness coefficient.
Use of L-moments in place of moments provides a measure of skewness known as the L-skewness.
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. ' is an independent identically distributed copy of X, and denotes the norm in the Euclidean space, then a simple measure of asymmetry with respect to location parameter θ isIt is called distance skewness and denoted by dSkew. If X is a random variable taking values in the d-dimensional Euclidean space, X has finite expectation, X
and dSkew(X) := 0 for X = θ (with probability 1). Distance skewness is always between 0 and 1, equals 0 if and only if X is diagonally symmetric with respect to θ (X and 2θ−X have the same probability distribution) and equals 1 if and only if X is a constant c () with probability one. Thus there is a simple consistent statistical test of diagonal symmetry based on the sample distance skewness:
The medcouple is a scale-invariant robust measure of skewness, with a breakdown point of 25%.It is the median of the values of the kernel function
taken over all couples such that , where is the median of the sample . It can be seen as the median of all possible quantile skewness measures.
In probability theory and statistics, kurtosis is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosis describes the shape of a probability distribution and there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. Different measures of kurtosis may have different interpretations.
There are several kinds of mean in mathematics, especially in statistics.
In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random variable from the mean. The various moments form one set of values by which the properties of a probability distribution can be usefully characterized. Central moments are used in preference to ordinary moments, computed in terms of deviations from the mean instead of from zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its location.
In probability theory, a normaldistribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean. Informally, it measures how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , or .
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In probability and statistics, Student's t-distribution is any member of a family of continuous probability distributions that arise when estimating the mean of a normally-distributed population in situations where the sample size is small and the population's standard deviation is unknown. It was developed by English statistician William Sealy Gosset under the pseudonym "Student".
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In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive is because of randomness or because the estimator does not account for information that could produce a more accurate estimate.
In probability theory and statistics, a standardized moment of a probability distribution is a moment that is normalized. The normalization is typically a division by an expression of the standard deviation which renders the moment scale invariant. This has the advantage that such normalized moments differ only in other properties than variability, facilitating e.g. comparison of shape of different probability distributions.
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. The generalization to multiple variables is called a Dirichlet distribution.
In mathematics, the moments of a function are quantitative measures related to the shape of the function's graph. The concept is used in both mechanics and statistics. If the function represents mass, then the zeroth moment is the total mass, the first moment divided by the total mass is the center of mass, and the second moment is the rotational inertia. If the function is a probability distribution, then the zeroth moment is the total probability, the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics.
In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Note that a limit distribution need to exist, which requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables.
In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object.
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can be either be closed or open. Therefore, the distribution is often abbreviated U, where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution's support.
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In statistics, the Jarque–Bera test is a goodness-of-fit test of whether sample data have the skewness and kurtosis matching a normal distribution. The test is named after Carlos Jarque and Anil K. Bera. The test statistic is always nonnegative. If it is far from zero, it signals the data do not have a normal distribution.
In statistics and probability theory, the nonparametric skew is a statistic occasionally used with random variables that take real values. It is a measure of the skewness of a random variable's distribution—that is, the distribution's tendency to "lean" to one side or the other of the mean. Its calculation does not require any knowledge of the form of the underlying distribution—hence the name nonparametric. It has some desirable properties: it is zero for any symmetric distribution; it is unaffected by a scale shift; and it reveals either left- or right-skewness equally well. In some statistical samples it has been shown to be less powerful than the usual measures of skewness in detecting departures of the population from normality.
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