## `stats::kurtosis` -- kurtosis (excess)

#### Introduction

`stats::kurtosis``(data)` returns the kurtosis (the coefficient of excess) of the data.

#### Call(s)

`stats::kurtosis(x1, x2, ..)`
`stats::kurtosis([x1, x2, ..])`
`stats::kurtosis(s <, c>)`

#### Parameters

`x1, x2, ..` | - | the statistical data: arithmetical expressions.
`s` | - | a sample of domain type `stats::sample`.
`c` | - | an integer representing a column index of the sample `s`. This column provides the data `x1`, `x2` etc.

#### Returns

an arithmetical expression. `FAIL` is returned, if the kurtosis does not exist.

#### Related Functions

`stats::obliquity`

#### Details

- The kurtosis of n values x[1],..,x[n] is

      (1/n*((x[1] - X)^4 + .. +(x[n] - X)^4)) /
                    (1/n*((x[1] - X)^2 + .. +(x[n] - X)^2))^2 - 3 ,

  where X is the mean value of the x[i]. The kurtosis measures whether a distribution is ``flat'' or ``peaked''. For normally distributed data the kurtosis is zero. If the distribution function of the data has a flatter top than the normal distribution, then the kurtosis is negative. The kurtosis is positive, if the distribution function has a high peak, compared to the normal distribution.
- The column index `c` is optional, if the data are given by a `stats::sample` object containing only one non-string column. Cf. example 3.

#### Example 1

We calculate the kurtosis of some values:

    >> stats::kurtosis(0, 7, 7, 6, 6, 6, 5, 5, 4, 1)

                                   -74146/271441

Alternatively, the data may be passed as a list:

    >> stats::kurtosis([2, 2, 4, 6, 8, 10, 10])

                                      -85/54

#### Example 2

We create a sample:

    >> stats::sample([[a, 5, 8], [b, 3, 7], [c, d, 0]])

                                     a  5  8
                                     b  3  7
                                     c  d  0

The kurtosis of the second column is:

    >> stats::kurtosis(%, 2)

                /       d \4     /       d \4     / 2 d       \4
              3 | 1/3 - - |  + 3 | 7/3 - - |  + 3 | --- - 8/3 |
                \       3 /      \       3 /      \  3        /
              -------------------------------------------------- - 3
              / /       d \2   /       d \2   / 2 d       \2 \2
              | | 1/3 - - |  + | 7/3 - - |  + | --- - 8/3 |  |
              \ \       3 /    \       3 /    \  3        /  /

#### Example 3

We create a sample consisting of one string column and one non-string column:

    >> stats::sample([["1996", 1242], ["1997", 1353], ["1998", 1142]])

                                  "1996"  1242
                                  "1997"  1353
                                  "1998"  1142

We compute the kurtosis of the second column. In this case this column does not have to be specified, since it is the only non-string column:

    >> stats::kurtosis(%)

                                       -3/2

#### Changes

- new function

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