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- Symmetry, Skewness and Kurtosis
- How to Interpret Excess Kurtosis and Skewness
- Testing for Normality using Skewness and Kurtosis

By Dr. Saul McLeod , published Kurtosis is a statistical measure used to describe the degree to which scores cluster in the tails or the peak of a frequency distribution.

The mean and variance are called the first raw moment about zero and the second moment about the mean respectively. The third and fourth moments about the mean, called skewness and kurtosis , are also occasionally used in risk analysis as numerical descriptions of shape. They can also be applied when fitting a distribution to data through Method of Moments , if there are three or more parameters to estimate.

Discrete variable:. Continuous variable:. This is often called the standardised skewness , since it is divided by s 3 to give a unitless statistic. The skewness statistic refers to the lopsidedness of the distribution see left panel below. If a distribution has a negative skewness sometimes described as left skewed it has a longer tail to the left than to the right.

A positively skewed distribution right skewed has a longer tail to the right, and zero skewed distributions are usually symmetric.

This is often called the standardised kurtosis , since it is divided by s 4 , again to give a unitless statistic. The kurtosis statistic refers to the peakedness of the distribution see right panel above - the higher the kurtosis, the more peaked the distribution. A Normal distribution has a kurtosis of 3, so kurtosis values for a distribution are often compared to 3. For example, if a distribution has a kurtosis below 3 it is flatter than a Normal distribution.

The following table gives some examples of skewness and kurtosis for common distributions. Reference Number: M-MA. Monte Carlo simulation in Excel. Learn more. Adding risk and uncertainty to your project schedule. Pelican - in-depth video What is Enterprise Risk Management? Is Pelican right for you? What makes Pelican special?

Other moments measures of shape. ModelRisk Monte Carlo simulation in Excel. Learn more Tamara Adding risk and uncertainty to your project schedule. Learn more Navigation Risk management Risk management introduction What are risks and opportunities? Censored data Fitting a continuous non-parametric second-order distribution to data Goodness of Fit Plots Fitting a second order Normal distribution to data Using Goodness-of Fit Statistics to optimize Distribution Fitting Information criteria - SIC HQIC and AIC Fitting a second order parametric distribution to observed data Fitting a distribution for a continuous variable Does the random variable follow a stochastic process with a well-known model?

Fitting a distribution for a discrete variable Fitting a discrete non-parametric second-order distribution to data Fitting a continuous non-parametric first-order distribution to data Fitting a first order parametric distribution to observed data Fitting a discrete non-parametric first-order distribution to data Fitting distributions to data Technical subjects Comparison of Classical and Bayesian methods Comparison of classic and Bayesian estimate of Normal distribution parameters Comparison of classic and Bayesian estimate of intensity lambda in a Poisson process Comparison of classic and Bayesian estimate of probability p in a binomial process Which technique should you use?

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Note: This article was originally published in April and was updated in February The original article indicated that kurtosis was a measure of the flatness of the distribution — or peakedness. This is technically not correct see below. Kurtosis is a measure of the combined weight of the tails relative to the rest of the distribution. This article has been revised to correct that misconception. New information on both skewness and kurtosis has also been added. You have a set of samples.

Sign in. To go straight to the Python code that shows how to test for normality, scroll down to the section named Example. The data set used in the article can be downloaded from this link. Normality means that your data follows the normal distribution. While building a linear regression model, one assumes that Y depends on a matrix of regression variables X. This makes Y conditionally normal on X. Several statistical techniques and models assume that the underlying data is normally distributed.

*The data set can represent either the population being studied or a sample drawn from the population. Symmetry and Skewness. Definition 1 : We use skewness as a measure of symmetry.*

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