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 variability of a data set. A 

further characterization of the data includes skewness and 

kurtosis. 

Skewness is a measure of symmetry, or more precisely, the 

lack of symmetry. A distribution, 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 histogram is an effective graphical technique for showing 

both the skewness and kurtosis of data set. 

Definition 

of Skewness 

For univariate data Y1, Y2, ..., YN, 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 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. Some 

measurements have a lower bound and are skewed right. For 

example, in reliability studies, failure times cannot be 

negative. 

For univariate data Y1, Y2, ..., YN, the formula for kurtosis is: 

skewness = ∑ ( − N

i=1 Yi Y

 ̄)

3

(N − 1)s3

Y

 ̄

http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm
Definition 

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 addition, 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 shows histograms for 10,000 random 

numbers generated from a normal, a double exponential, a 

Cauchy, and a Weibull distribution. 

kurtosis = ∑ ( − N

i=1 Yi Y

 ̄)

4

(N − 1)s4 Y

 ̄

kurtosis = −3 ∑ ( − N

i=1 Yi Y ̄)

4

(N − 1)s4

http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm
Normal

Distribution 

The first histogram is a sample from a normal distribution. 

The normal distribution is a symmetric distribution with wellbehaved tails. This is indicated 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 normal, it has a stronger peak, 

more rapid decay, and heavier tails. 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 the Cauchy distribution to values 

between -10 and 10. The full data set 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 skewness near zero. 

Due to the heavier tails, we might expect the kurtosis to be 

larger than for a normal distribution. In fact the skewness is 

69.99 and the kurtosis is 6,693. These 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.htm
transformation is 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 techniques 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 coefficients are available in most 

general purpose statistical software programs. 

http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm
Privacy 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.gov is also much 

appreciated. 

http://www.itl.nist.gov/div898/handbook/index.htm