What is Skewness and Kurtosis?
Introduction
Skewness is a measure of the asymmetry of data distribution. It can be quantified as a statistical moment, or as the degree of asymmetry of a probability density function. Data distribution is symmetric if it looks the same to the left and right of the center. If the distribution is not symmetric, it is said to be skewed.
The direction of the skew is determined by which side of the distribution is longer. A distribution can be both skewed and kurtotic. Kurtosis is a measure of the peakedness of data distribution. A distribution that is more peaked than the normal distribution is said to be leptokurtic. A distribution that is less peaked than the normal distribution is said to be platykurtic.
The skewness of a data distribution can be affected by outliers. Outliers are data points that are far from the rest of the data. They can cause a distribution to be skewed to the left or the right, depending on their location. There are many different ways to measure skewness. The most common is the Pearson mode skewness coefficient. This measures the skewness of data distribution by dividing the difference between the mode and the mean by the standard deviation.
What is skewness?
In statistics, skewness is a measure of the asymmetry of a distribution. A distribution is skewed if it is not symmetric about its mean. In other words, if the mean is not equal to the median, then the distribution is skewed. There are two types of skewness: positive and negative. A distribution is positively skewed if the mean is greater than the median. This is often the case with data that has a long tail on the right side. For example, data on incomes are often positively skewed because there are a few people with very high incomes. On the other hand, distribution is negatively skewed if the mean is less than the median. This is often the case with data that has a long tail on the left side. The skewness of a distribution can be measured using the following formula:
Skewness = (3 * (Mean – Median)) / Standard Deviation
A distribution is considered to be symmetric if the skewness is between -1 and 1. If the skewness is greater than 1, then the distribution is positively skewed. If the skewness is less than -1, then the distribution is negatively skewed. The skewness of distribution can also be visualized using a histogram. A positively skewed distribution will have a long tail on the right side, while a negatively skewed distribution will have a long tail on the left side.
What is the Pearson mode skewness coefficient?
The Pearson mode skewness coefficient is a measure of skewness or the degree of asymmetry of a distribution. It is based on the mode, or most common value, of the distribution. The mode skewness coefficient can be positive or negative, depending on whether the mode is to the left or right of the mean. A positive coefficient indicates that the mode is to the right of the mean, while a negative coefficient indicates that the mode is to the left of the mean. The mode skewness coefficient is used to assess the symmetry of a distribution. Distribution is symmetric if the mean, mode, and median are all equal. If the mean and mode are equal but the median is not, then the distribution is said to be mode-skewed. The mode skewness coefficient is a measure of the degree of mode-skewness and is calculated as:
Skewness Coefficient = (Mode – Mean) / Standard Deviation
The mode skewness coefficient can be used to compare the symmetry of two or more distributions. A distribution with a higher mode skewness coefficient is more mode-skewed than a distribution with a lower mode skewness coefficient
What is Kurtosis?
Kurtosis is a statistical measure that quantifies the degree of peakedness of a distribution. It is often used to identify outliers in data sets, as well as to characterize the overall shape of a distribution. Kurtosis is calculated by taking the standardized moment of the fourth order about the mean, divided by the square of the standardized moment of the second order about the mean. This can be expressed as:
Kurtosis = μ4 / σ4
Where μ4 is the fourth central moment of the distribution and σ4 is the fourth central standardized moment. The kurtosis of a normal distribution is 3. A distribution with kurtosis less than 3 is said to be platykurtic, while a distribution with kurtosis greater than 3 is said to be leptokurtic.
Platykurtic distributions are relatively flat, while leptokurtic distributions are relatively peaked. Outliers are more likely to occur in leptokurtic distributions than in platykurtic distributions. Kurtosis can be used to identify whether a data set is normal or not. A data set is considered to be non-normal if its kurtosis is significantly different from 3.
Kurtosis is also a useful tool for identifying outliers in data sets. Outliers are values that are significantly different from the rest of the data. In a leptokurtic distribution, outliers are more likely to occur. Kurtosis can also be used to characterize the overall shape of a distribution.
Kurtosis is important in financial markets because it can help to identify distributions that are fat-tailed and therefore may be more likely to experience extreme events. Fat-tailed distributions are more likely to have outliers, and extreme events are more likely to occur in the tails of a distribution. Kurtosis can be measured in a number of ways, but the most common measure is excess kurtosis.
Excess kurtosis is simply the kurtosis of distribution minus the kurtosis of the normal distribution. A distribution with excess kurtosis of 0 is exactly normal, while a distribution with a positive excess kurtosis is leptokurtic and distribution with a negative excess kurtosis is platykurtic.
Excess kurtosis is a measure of the combined weight of the tails of distribution relative to the normal distribution. A distribution with a positive excess kurtosis will have heavier tails than the normal distribution, while a distribution with a negative excess kurtosis will have lighter tails.
Also, read about skewness and how data get affected.