Skewness

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Skewness

In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of Statistical randomness phenomena. The central objects of probability theory are random variables, stochastic processes, and event s: mathematical abstractions of determinism events or measured quantities that may either be single occurrences or evolve over time in an a...
and statistics

Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
, skewness is a measure of the asymmetry of the probability distribution

Probability distribution

In probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval ....
of a real

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
-valued random variable

Random variable

In mathematics, random variables are used in the study of Randomness and probability. They were developed to assist in the analysis of Game of chance, stochastic events, and the results of experiment by capturing only the mathematical properties necessary to answer probability questions....
.

Introduction
Consider the distribution in the figure. The bars on the right side of the distribution taper differently than the bars on the left side.

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Introduction

Consider the distribution in the figure. The bars on the right side of the distribution taper differently than the bars on the left side. These tapering sides are called tails, and they provide a visual means for determining which of the two kinds of skewness a distribution has:

negative skew: The left tail is longer; the mass of the distribution is concentrated on the right of the figure. It has relatively few low values. The distribution is said to be left-skewed. In such a distribution, the mean
Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
is lower than median
Median

In probability theory and statistics, a median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half....
which in turn is lower than the mode
Mode (statistics)

In statistics, the mode is the value that occurs the most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....
(i.e.; mean < median < mode); in which case the skewness coefficient is lower than zero. Example (observations): 1,1000,1001,1002,1003
positive skew: The right tail is longer; the mass of the distribution is concentrated on the left of the figure. It has relatively few high values. The distribution is said to be right-skewed. In such a distribution, the mean
Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
is greater than median
Median

In probability theory and statistics, a median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half....
which in turn is greater than the mode
Mode (statistics)

In statistics, the mode is the value that occurs the most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....
(i.e.; mean > median > mode); in which case the skewness coefficient is greater than zero. Example (observations): 1,2,3,4,100

In a skewed (unbalanced, lopsided) distribution, the mean

Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
is farther out in the long tail than is the median

Median

In probability theory and statistics, a median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half....
. If there is no skewness or the distribution is symmetric like the bell-shaped normal curve then the mean = median = mode.

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.

Definition

Skewness, the third standardized moment

Standardized moment

In probability theory and statistics, the kthstandardized moment of a probability distribution is where is the kth moment about the mean and σ is the standard deviation....
, is written as and defined as

where is the third moment about the mean and is the standard deviation

Standard deviation

In statistics, standard deviation is a simple measure of the variability or statistical dispersion of a data set. A low standard deviation indicates that all of the data points are very close to the same value , while high standard deviation indicates that the data are ?spread out? over a large range of values....
. Equivalently, skewness can be defined as the ratio of the third cumulant

Cumulant

In probability theory and statistics, if a random variable X admits an expected value ? = E and a variance s2 = E, then these are the first two cumulants: ? = ?1 and s2 = ?2....
and the third power of the square root of the second cumulant :

This is analogous to the definition of kurtosis

Kurtosis

In probability theory and statistics, kurtosis is a measure of the "peakedness" of the probability distribution of a real number-valued random variable....
, 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

where is the i^th value, is the sample mean, is the sample third central moment

Central moment

In probability theory and statistics, the kth moment about the mean of a real-valued random variable X is the quantity μk := E[k], where E is the expected value....
, and is the sample variance.

Given samples from a population, the equation for the sample skewness above is a biased estimator of the population skewness. The usual estimator of skewness is

where is the unique symmetric unbiased estimator of the third cumulant and is the symmetric unbiased estimator of the second cumulant. Unfortunately 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

Statistical independence

In probability theory, to say that two event s are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs....
random variables, all with the same distribution as X, then it can be shown that Skew[Y] = Skew[X] / vn.

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

Karl Pearson Fellow of the Royal Society established the disciplineof mathematical statistics.In 1911 he founded the world's first university statistics department at University College London....
suggested two simpler calculations as a measure of skewness:

(mean
Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
− mode
Mode (statistics)

In statistics, the mode is the value that occurs the most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....
) / standard deviation
Standard deviation

In statistics, standard deviation is a simple measure of the variability or statistical dispersion of a data set. A low standard deviation indicates that all of the data points are very close to the same value , while high standard deviation indicates that the data are ?spread out? over a large range of values....
3 (mean
Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
− median
Median

In probability theory and statistics, a median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half....
) / standard deviation
Standard deviation

In statistics, standard deviation is a simple measure of the variability or statistical dispersion of a data set. A low standard deviation indicates that all of the data points are very close to the same value , while high standard deviation indicates that the data are ?spread out? over a large range of values....

There is no guarantee that these will be the same sign as each other or as the ordinary definition of skewness.

External links

by Michel Petitjean

Skewness

Introduction

Definition

Pearson skewness coefficients

See also

External links