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Compute Measures of Shape

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 x1, x2, ..., xn centered around X.

  • The stats::obliquity function that calculates the obliquity (skewness)

    ,

    where is the arithmetic average (mean) of the data sample x1, x2, ..., xn.

  • The stats::kurtosis function that calculates the kurtosis (excess)

    ,

    where is the arithmetic average (mean) of the data sample x1, x2, ..., xn.

The stats::moment function enables you to compute the kth moment of a data sample centered around an arbitrary value X. One of the popular measures in descriptive statistics is a central moment. The kth central moment of a data sample is the kth 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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