Kurtosis
In
probability theory and statistics, Kurtosis is a measure of the tailedness of
the probability distribution of a real valued random variable. In a similar way to the concept of skewness,
kurtosis is a descriptor of the shape of a probability distribution and, just
as for skewness, there are different ways of quantifying it for a theoretical
distribution and curresponding ways of estimating it from a sample from a
population. Depending on the particular
measure of kurtosis that is used, there are various interpretations of
kurtosis, and of how particular measure should be interpreted.
The
standard measure of kurtosis, originating with Karl Pearson, is based on a
scaled version of the fourth moment of the data or population. This number is related to the tails of the
distribution, not its peak, hence, the sometimes seen characterization as
peakedness is mistaken. For this measure
higher kurtosis is the result of infrequent extreme deviations, as opposed to
frequent modestly sized deviations.
The kurtosis of any univariate normal
distribution is 3. It is common to compare the kurtosis of a
distribution to this value. Distributions
with kurtosis less than 3 are said to be platykurtic, although this does not imply the distribution is
"flat-topped" as sometimes reported. Rather, it means the
distribution produces fewer and less extreme outliers than does the normal
distribution. An example of a
platykurtic distribution is the uniform distribution, which does not produce
outliers. Distributions with kurtosis
greater than 3 are said to be leptokurtic.
An example of a leptokurtic distribution is the Laplace distribution,
which has tails that asymptotically approach zero more slowly than a Gaussian,
and therefore produces more outliers than the normal distribution. It is also common practice to use an adjusted
version of Pearson's kurtosis, the excess kurtosis, which is the kurtosis minus
3, to provide the comparison to the normal distribution.
Types
of Kurtosis
There
are three types kurtosis, first one is Mesokurtic, second one is Leptokurtic
and third one is Platykurtic.
Figure.1 Types of Kurtosis
Figure.1 Types of Kurtosis
Distribution with zero excess kurtosis are called mesokurtic, or mesokurtotic.
The most prominent example of a mesokurtic distribution is the normal
distribution family, regardless of the values of its parameters. A few other well-known distributions can be
mesokurtic, depending on parameter values.
A distribution with positive excess kurtosis is
called leptokurtic, or
leptokurtotic. "Lepto-" means "slender." In terms of shape, a leptokurtic
distribution has fatter tails.
Examples of leptokurtic distributions include the Student’s t-distribution, Rayleigh
distribution, Laplace distribution, exponential distribution, Poisson distribution and
the logistic distribution. Such
distributions are sometimes termed super-Gaussian.
A distribution with negative excess kurtosis is called platykurtic, or platykurtotic. "Platy-" means
"broad" In terms of shape, a platykurtic distribution has thinner tails. Examples of
platykurtic distributions include the continuous and discrete uniform distribution, and the raised cosine
distribution. The most platykurtic distribution of all is the Bernoulli distribution with p = 1/2 (for example the number
of times one obtains "heads" when flipping a coin once, a coin toss),
for which the excess kurtosis is −2. Suchdistributions are sometimes
termed sub-Gaussian.
Table.1
Types of kurtosis and values
Sl.No.
|
Types of kurtosis
|
Values
|
1
|
Mesokurtic
|
0.263
|
2
|
Leptokurtic
|
<0.263
|
3
|
Platykurtic
|
Applications of kurtosis.
The sample kurtosis is a useful measure of
whether there is a problem with outliers in a data set. Larger kurtosis
indicates a more serious outlier problem, and may lead the researcher to choose
alternative statistical methods.
POWER POINT ON KURTOSIS
