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The measures of shape indicate the symmetry and flatness of
the distribution of a data sample. MuPAD^{®} provides the following
functions for calculating the measures of shape:

The

`stats::moment`

function that calculates the`k`

-th momentof the data sample

*x*_{1},*x*_{2},*...*,*x*_{n}centered around`X`

.The

`stats::obliquity`

function that calculates the obliquity (skewness),

where is the arithmetic average (mean) of the data sample

*x*_{1},*x*_{2},*...*,*x*_{n}.The

`stats::kurtosis`

function that calculates the kurtosis (excess),

where is the arithmetic average (mean) of the data sample

*x*_{1},*x*_{2},*...*,*x*_{n}.

The `stats::moment`

function
enables you to compute the `k`

th moment of a data
sample centered around an arbitrary value `X`

. One
of the popular measures in descriptive statistics is a central moment.
The `k`

th central moment of a data sample is the `k`

th
moment centered around the arithmetic average (mean) of that data
sample. The following statements are valid for any data sample:

The zero central moment is always 1.

The first central moment is always 0.

The second central moment is equal to the variance computed by using a divisor

`n`

, rather than`n - 1`

(available via the`Population`

option of`stats::variance`

).

For example, create the lists `L`

and `S`

as
follows:

L := [1, 1, 1, 1, 1, 1, 1, 1, 100.0]: S := [100, 100, 100, 100, 100, 100, 100, 100, 1.0]:

Calculate the arithmetic average of each list by using the `stats::mean`

function:

meanL := stats::mean(L); meanS := stats::mean(S)

Calculate the first four central moments of the list `L`

:

stats::moment(0, meanL, L), stats::moment(1, meanL, L), stats::moment(2, meanL, L), stats::moment(3, meanL, L)

The zero and first central moments are the same for any data
sample. The second central moment is the variance computed with the
divisor `n`

:

stats::variance(L, Population)

Now, calculate the first four central moments of the list `S`

:

stats::moment(0, meanS, S), stats::moment(1, meanS, S), stats::moment(2, meanS, S), stats::moment(3, meanS, S)

Again, the zero central moment is 1, the first central moment
is 0, and the second central moment is the variance computed with
the divisor `n`

:

stats::variance(S, Population)

The obliquity (skewness) is a measure of the symmetry of a distribution.
If the distribution is close to symmetrical around its mean, the value
of obliquity is close to zero. Positive values of obliquity indicate
that the distribution function has a longer tail to the right of the
mean. Negative values indicate that the distribution function has
a longer tail to the left of the mean. For example, calculate the
obliquity of the lists `L`

and `S`

:

stats::obliquity(L); stats::obliquity(S)

The kurtosis measures the flatness of a distribution. For normally distributed data, the kurtosis is zero. Negative kurtosis indicates that the distribution function has a flatter top than the normal distribution. Positive kurtosis indicates that the peak of the distribution function is sharper than it is for the normal distribution:

stats::kurtosis(-2, -1, -0.5, 0, 0.5, 1, 2), stats::kurtosis(-1, 0.5, 0, 0, 0, 0, 0.5, 1)

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