Note: Use only in the MuPAD Notebook Interface. This functionality does not run in MATLAB. |
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 moment
of
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)