Compute Skewness and (Excess) Kurtosis

## Usage

```
skewness(x, na.rm = TRUE, type = "2", iterations = NULL, verbose = TRUE, ...)
kurtosis(x, na.rm = TRUE, type = "2", iterations = NULL, verbose = TRUE, ...)
# S3 method for parameters_kurtosis
print(x, digits = 3, test = FALSE, ...)
# S3 method for parameters_skewness
print(x, digits = 3, test = FALSE, ...)
# S3 method for parameters_skewness
summary(object, test = FALSE, ...)
# S3 method for parameters_kurtosis
summary(object, test = FALSE, ...)
```

## Arguments

- x
A numeric vector or data.frame.

- na.rm
Remove missing values.

- type
Type of algorithm for computing skewness. May be one of

`1`

(or`"1"`

,`"I"`

or`"classic"`

),`2`

(or`"2"`

,`"II"`

or`"SPSS"`

or`"SAS"`

) or`3`

(or`"3"`

,`"III"`

or`"Minitab"`

). See 'Details'.- iterations
The number of bootstrap replicates for computing standard errors. If

`NULL`

(default), parametric standard errors are computed.- verbose
Toggle warnings and messages.

- ...
Arguments passed to or from other methods.

- digits
Number of decimal places.

- test
Logical, if

`TRUE`

, tests if skewness or kurtosis is significantly different from zero.- object
An object returned by

`skewness()`

or`kurtosis()`

.

## Details

### Skewness

Symmetric distributions have a `skewness`

around zero, while
a negative skewness values indicates a "left-skewed" distribution, and a
positive skewness values indicates a "right-skewed" distribution. Examples
for the relationship of skewness and distributions are:

Normal distribution (and other symmetric distribution) has a skewness of 0

Half-normal distribution has a skewness just below 1

Exponential distribution has a skewness of 2

Lognormal distribution can have a skewness of any positive value, depending on its parameters

(https://en.wikipedia.org/wiki/Skewness)

### Types of Skewness

`skewness()`

supports three different methods for estimating skewness,
as discussed in Joanes and Gill (1988):

Type "1" is the "classical" method, which is

`g1 = (sum((x - mean(x))^3) / n) / (sum((x - mean(x))^2) / n)^1.5`

Type "2" first calculates the type-1 skewness, then adjusts the result:

`G1 = g1 * sqrt(n * (n - 1)) / (n - 2)`

. This is what SAS and SPSS usually return.Type "3" first calculates the type-1 skewness, then adjusts the result:

`b1 = g1 * ((1 - 1 / n))^1.5`

. This is what Minitab usually returns.

### Kurtosis

The `kurtosis`

is a measure of "tailedness" of a distribution. A
distribution with a kurtosis values of about zero is called "mesokurtic". A
kurtosis value larger than zero indicates a "leptokurtic" distribution with
*fatter* tails. A kurtosis value below zero indicates a "platykurtic"
distribution with *thinner* tails
(https://en.wikipedia.org/wiki/Kurtosis).

### Types of Kurtosis

`kurtosis()`

supports three different methods for estimating kurtosis,
as discussed in Joanes and Gill (1988):

Type "1" is the "classical" method, which is

`g2 = n * sum((x - mean(x))^4) / (sum((x - mean(x))^2)^2) - 3`

.Type "2" first calculates the type-1 kurtosis, then adjusts the result:

`G2 = ((n + 1) * g2 + 6) * (n - 1)/((n - 2) * (n - 3))`

. This is what SAS and SPSS usually returnType "3" first calculates the type-1 kurtosis, then adjusts the result:

`b2 = (g2 + 3) * (1 - 1 / n)^2 - 3`

. This is what Minitab usually returns.