# Moments Skewness And Kurtosis In Statistics Pdf

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Why do we care? One application is testing for normality : many statistics inferences require that a distribution be normal or nearly normal.

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## On measuring skewness and kurtosis

The third moment measures skewness , the lack of symmetry, while the fourth moment measures kurtosis , roughly a measure of the fatness in the tails. The actual numerical measures of these characteristics are standardized to eliminate the physical units, by dividing by an appropriate power of the standard deviation.

In the unimodal case, if the distribution is positively skewed then the probability density function has a long tail to the right, and if the distribution is negatively skewed then the probability density function has a long tail to the left.

A symmetric distribution is unskewed. We proved part a in the section on properties of expected Value. The converse is not true—a non-symmetric distribution can have skewness 0.

Examples are given in Exercises 30 and 31 below. Since skewness is defined in terms of an odd power of the standard score, it's invariant under a linear transformation with positve slope a location-scale transformation of the distribution. On the other hand, if the slope is negative, skewness changes sign. Recall that location-scale transformations often arise when physical units are changed, such as inches to centimeters, or degrees Fahrenheit to degrees Celsius.

Kurtosis comes from the Greek word for bulging. In the unimodal case, the probability density function of a distribution with large kurtosis has fatter tails, compared with the probability density function of a distribution with smaller kurtosis. Since kurtosis is defined in terms of an even power of the standard score, it's invariant under linear transformations. We will show in below that the kurtosis of the standard normal distribution is 3.

Some authors use the term kurtosis to mean what we have defined as excess kurtosis. As always, be sure to try the exercises yourself before expanding the solutions and answers in the text. Recall that an indicator random variable is one that just takes the values 0 and 1. Indicator variables are the building blocks of many counting random variables. The corresponding distribution is known as the Bernoulli distribution , named for Jacob Bernoulli. Parts a and b have been derived before.

Recall that a fair die is one in which the faces are equally likely. In addition to fair dice, there are various types of crooked dice. Here are three:. A flat die, as the name suggests, is a die that is not a cube, but rather is shorter in one of the three directions. Flat dice are sometimes used by gamblers to cheat. Select each of the following, and note the shape of the probability density function in comparison with the computational results above. In each case, run the experiment times and compare the empirical density function to the probability density function.

Recall that the continuous uniform distribution on a bounded interval corresponds to selecting a point at random from the interval. Continuous uniform distributions arise in geometric probability and a variety of other applied problems. Parts a and b we have seen before.

Open the special distribution simulator, and select the continuous uniform distribution. Vary the parameters and note the shape of the probability density function in comparison with the moment results in the last exercise.

For selected values of the parameter, run the simulation times and compare the empirical density function to the probability density function. This distribution is widely used to model failure times and other arrival times. The exponential distribution is studied in detail in the chapter on the Poisson Process. Vary the rate parameter and note the shape of the probability density function in comparison to the moment results in the last exercise.

For selected values of the parameter, run the experiment times and compare the empirical density function to the true probability density function. The Pareto distribution is named for Vilfredo Pareto. It is a heavy-tailed distribution that is widely used to model financial variables such as income. The Pareto distribution is studied in detail in the chapter on Special Distributions. Open the special distribution simulator and select the Pareto distribution.

Vary the shape parameter and note the shape of the probability density function in comparison to the moment results in the last exercise. Normal distributions are widely used to model physical measurements subject to small, random errors and are studied in detail in the chapter on Special Distributions.

Parts a and b were derived in the previous sections on expected value and variance. Part c follows from symmetry. The results follow immediately from the formulas for skewness and kurtosis under linear transformations and the previous result. Open the special distribution simulator and select the normal distribution.

Vary the parameters and note the shape of the probability density function in comparison to the moment results in the last exercise. For selected values of the parameters, run the experiment times and compare the empirical density function to the true probability density function.

The beta distribution is studied in detail in the chapter on Special Distributions. Find each of the following:. Open the special distribution simulator and select the beta distribution. Select the parameter values below to get the distributions in the last three exercises. In each case, note the shape of the probability density function in relation to the calculated moment results.

Run the simulation times and compare the empirical density function to the probability density function. The particular beta distribution in the last exercise is also known as the standard arcsine distribution. The arcsine distribution is studied in more generality in the chapter on Special Distributions.

Open the Brownian motion experiment and select the last zero. Note the shape of the probability density function in relation to the moment results in the last exercise. The following exercise gives a simple example of a discrete distribution that is not symmetric but has skewness 0. The following exercise gives a more complicated continuous distribution that is not symmetric but has skewness 0.

It is one of a collection of distributions constructed by Erik Meijer. However, it's best to work with the random variables. Again, the mean is the only possible point of symmetry. Computational Exercises As always, be sure to try the exercises yourself before expanding the solutions and answers in the text.

Indicator Variables Recall that an indicator random variable is one that just takes the values 0 and 1. Dice Recall that a fair die is one in which the faces are equally likely. Uniform Distributions Recall that the continuous uniform distribution on a bounded interval corresponds to selecting a point at random from the interval.

Counterexamples The following exercise gives a simple example of a discrete distribution that is not symmetric but has skewness 0.

## INDUSTRIAL STATISTICS

Central Moments- The average of all the deviations of all observations in a dataset from the mean of the observations raised to the power r In the previous equation, n is the number of observations, X is the value of each individual observation, m is the arithmetic mean of the observations, and r is a positive integer. Measures of Skewness And Kurtosis Chapter 9. Moments Moments are a set of statistical parameters to measure a distribution. In statistics the values measure something relative to the center of the values. High Kurtosis Exhibit 1 These graphs illustrate the notion of kurtosis.

## Measures of Shape: Skewness and Kurtosis

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:.

The third moment measures skewness , the lack of symmetry, while the fourth moment measures kurtosis , roughly a measure of the fatness in the tails. The actual numerical measures of these characteristics are standardized to eliminate the physical units, by dividing by an appropriate power of the standard deviation.

The degree of tailedness of a distribution is measured by kurtosis. It tells us the extent to which the distribution is more or less outlier-prone heavier or light-tailed than the normal distribution. It is difficult to discern different types of kurtosis from the density plots left panel because the tails are close to zero for all distributions.

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