k = kurtosis(X) k = kurtosis(X,flag) k = kurtosis(X,flag,dim)

Description

k = kurtosis(X) returns the
sample kurtosis of X. For vectors, kurtosis(x) is
the kurtosis of the elements in the vector x.
For matrices kurtosis(X) returns the sample kurtosis
for each column of X. For N-dimensional
arrays, kurtosis operates along the first nonsingleton
dimension of X.

k = kurtosis(X,flag) specifies
whether to correct for bias (flag is 0)
or not (flag is 1, the default).
When X represents a sample from a population, the
kurtosis of X is biased, that is, it will tend
to differ from the population kurtosis by a systematic amount that
depends on the size of the sample. You can set flag to 0 to
correct for this systematic bias.

k = kurtosis(X,flag,dim) takes
the kurtosis along dimension dim of X.

kurtosis treats NaNs as
missing values and removes them.

Examples

X = randn([5 4])
X =
1.1650 1.6961 -1.4462 -0.3600
0.6268 0.0591 -0.7012 -0.1356
0.0751 1.7971 1.2460 -1.3493
0.3516 0.2641 -0.6390 -1.2704
-0.6965 0.8717 0.5774 0.9846
k = kurtosis(X)
k =
2.1658 1.2967 1.6378 1.9589

Kurtosis is a measure of how outlier-prone a distribution is.
The kurtosis of the normal distribution is 3. Distributions that are
more outlier-prone than the normal distribution have kurtosis greater
than 3; distributions that are less outlier-prone have kurtosis less
than 3.

The kurtosis of a distribution is defined as

where μ is the mean of x, σ is
the standard deviation of x, and E(t)
represents the expected value of the quantity t. kurtosis computes
a sample version of this population value.

Note
Some definitions of kurtosis subtract 3 from the computed value,
so that the normal distribution has kurtosis of 0. The kurtosis function
does not use this convention.

When you set flag to 1, the following equation
applies:

When you set flag to
0, the following equation applies:

This bias-corrected
formula requires that X contain at least four elements.