The histogram is an effective graphical technique for showing both the skewness and kurtosis of data set. For univariate data Y 1 , Y 2 , Note that in computing the skewness, the s is computed with N in the denominator rather than N - 1. The above formula for skewness is referred to as the Fisher-Pearson coefficient of skewness.
The adjustment approaches 1 as N gets large. For reference, the adjustment factor is 1. The skewness for a normal distribution is zero, and any symmetric data should have a skewness near zero.
Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right. By skewed left, we mean that the left tail is long relative to the right tail. Similarly, skewed right means that the right tail is long relative to the left tail. If the data are multi-modal, then this may affect the sign of the skewness. Some measurements have a lower bound and are skewed right. For example, in reliability studies, failure times cannot be negative.
It should be noted that there are alternative definitions of skewness in the literature. There are many other definitions for skewness that will not be discussed here. Note that in computing the kurtosis, the standard deviation is computed using N in the denominator rather than N - 1.
The kurtosis for a standard normal distribution is three. In addition, with the second definition positive kurtosis indicates a "heavy-tailed" distribution and negative kurtosis indicates a "light tailed" distribution.
Which definition of kurtosis is used is a matter of convention this handbook uses the original definition. When using software to compute the sample kurtosis, you need to be aware of which convention is being followed.
Many sources use the term kurtosis when they are actually computing "excess kurtosis", so it may not always be clear. The following example shows histograms for 10, random numbers generated from a normal, a double exponential, a Cauchy, and a Weibull distribution.
The kurtosis of a mesokurtic distribution is neither high nor low, rather it is considered to be a baseline for the two other classifications. A platykurtic distribution is flatter less peaked when compared with the normal distribution, with fewer values in its shorter i. Platykurtic distributions have negative kurtosis values.
Leptokurtic distributions have positive kurtosis values. A leptokurtic distribution has a higher peak thin bell and taller i. An extreme positive kurtosis indicates a distribution where more of the values are located in the tails of the distribution rather than around the mean.
The normal distribution has kurtosis of zero. Kurtosis characterizes the shape of a distribution - that is, its value does not depend on an arbitrary change of the scale and location of the distribution. For example, kurtosis of a sample or population of temperature values in Fahrenheit will not change if you transform the values to Celsius the mean and the variance will, however, change. The kurtosis of a distribution or sample is equal to the 4th central moment divided by the 4th power of the standard deviation , minus 3.
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