1. Exploratory Data Analysis 1.3. EDA Techniques 1.3.5. Quantitative Techniques 

## 1.3.5.11. Measures of Skewness and Kurtosis

Skewness and Kurtosis A fundamental task in many statistical analyses is to characterize the location and variabilityof a data set. A further characterization of th e data includes skewness and kurtosis. Skewness is a measure of symm etry, or more precisely, the lack of symmetry. A di stribution, or data set, is symmetric if it looks the same to the left and right of the center point.
Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. That is, data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case.
The histogramis an effective graphical technique for showing both the skewness and kurtosis of data set. Definition of Skewness For univariate data Y 1 , Y 2 , ..., Y N

, the formula for skewness is:

where is the mean, s is the standard deviation, and N is the number of data points. The skewness for a normal distribution is zero, and any symmetric datashould 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 ri ght tail. Similarly, skewed right means that the right tail is long relative to the left tail. Some measurements have a lower bound and are skewed right. For example, in reliability studi es, failure times cannot be negative.
For univariate data Y 1 , Y 2 , ..., Y N

, the formula for kurtosis is: 

### skewness = ( − ∑ N i =1 

### Y i 

### Y  ̄ ) 3 

### ( N − 1) s 3


### Y

 ̄ http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htmDefinition of Kurtosis
where is the mean, s is the standard deviation, and N is the number of data points.

Alternative Definition of Kurtosis The kurtosis for a standard normal distribution is three. For this reason, some sources use the following definition of kurtosis (often referred to as "excess kurtosis"):

This definition is used so that the standard normal distribution has a kurtosis of zero. In addi tion, with the second definition positive kurtosis indicates a "peaked" distribution and negative kurtosis indicates a "flat" 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. Examples The following example show s histograms for 10,000 random numbers generated from a normal, a double exponential, a Cauchy, and a Weibull distribution. 

### kurtosis = ( − ∑ N i =1 

### Y i 

### Y  ̄ ) 4 

### ( N − 1) s 4 

### Y

 ̄

kurtosis = −3 ( − ∑ N i =1 

### Y i 

### Y  ̄ ) 4 

### ( N − 1) s 4 http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htmNorma l Distribution The first histogram is a sample from a normal distributio n
. The normal distribution is a symm etric distribution with wellbehaved tails. This is indicat ed by the skewness of 0.03. The kurtosis of 2.96 is near the expected value of 3. The histogram verifies the symmetry.
Double Exponential Distribution The second histogram is a sample from a double exponential distribution . The double exponential is a symmetric distribution. Compared to the no rmal, it has a stronger peak, more rapid decay, and heavier tail s. That is, we would expect a skewness near zero and a kurtosis higher than 3. The skewness is 0.06 and the kurtosis is 5.9. Cauchy Distribution The third histogram is a sample from a Cauchy distribution

. For better visual comparison with the other data sets, we restricted the histogram of th e Cauchy distribution to values between -10 and 10. The full dataset for the Cauchy data in fact has a minimum of approximately -29,000 and a maximum of approximately 89,000. The Cauchy distribution is a symmetric distribution with heavy tails and a single peak at the center of the distribution. Since it is symmetric, we would expect a skewne ss near zero. Due to the heavier tails, we mi ght expect the kurtosis to be larger than for a normal distribution. In fact the skewness is 69.99 and the kurtosis is 6,693. Th ese extremely high values can be explained by the heavy tails. Just as the mean and standard deviation can be distorted by extreme values in the tails, so too can the skewness and kurtosis measures.
Weibull Distribution The fourth histogram is a sample from a Weibull distribution with shape parameter 1.5. The Weibull distribution is a skewed distribution with the amount of skewness depending on the value of the shape parameter. The degree of decay as we move away from the center also depends on the value of the shape parameter. For this data set, the skewness is 1.08 and the kurtosis is 4.46, which indicates moderate skewness and kurtosis. Dealing with Skewness and Kurtosis Many classical statistical tests and intervals depend on normality assumptions. Significant skewness and kurtosis clearly indicate that data are not normal. If a data set exhibits significant skewness or kurtosis (as indicated by a histogram or the numerical measures), what can we do about it? One approach is to apply some type of transformation to try to make the data normal, or more nearly normal. The Box-Cox http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htmtransformatio nis a useful technique for trying to normalize a data set. In particular, taking the log or square root of a data set is often useful for data that exhibit moderate right skewness. Another approach is to use te chniques based on distributions other than the normal. For example, in reliability studies, the exponential, Weibull, and lognormal distributions are typically used as a basis for modeling rather than using the normal distribution. The probability plot correlation coefficient plot and the probability plot are useful tools for determining a good distributional model for the data. Software The skewness and kurtosis coeffi cients are available in most general purpose statistical software programs. http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htmPrivacy Policy/Security Notice Disclaimer | FOIA NIST is an agency of the U.S. Commerce Department.
Date created: 6/01/2003 Last updated: 10/30/2013 To reference the Handbook please use a citation of the form: NIST/SEMATECH e-Handbook of Statistical Methods , http://www.itl.nist.gov/div898/handbook/, date. (Links to specific pages can also be referenced this way, if suitable.)
A significant update was made to the Handbook April, 2012 Printer friendly versions of each chapter in the Handbook can be found here

. Feedback on the Handbook sent to handbook@nist.govis also much appreciated. http://www.itl.nist .gov/div898/handbook/index.htm