# Documentation

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English verison of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

## Compute Measures of Shape

MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.

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 `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)```

Was this topic helpful?

Watch now