Compute Skewness and (Excess) Kurtosis

```
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, ...)
```

x | A numeric vector or data.frame. |
---|---|

na.rm | Remove missing values. |

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

iterations | The number of bootstrap replicates for computing standard
errors. If |

verbose | Toggle warnings and messages. |

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

digits | Number of decimal places. |

test | Logical, if |

object | An object returned by |

Values of skewness or kurtosis.

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)

`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 returnType "3" first calculates the type-1 skewness, then adjusts the result:

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

. This is what Minitab usually returns.

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).

`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, than 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, than adjusts the result:

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

. This is what Minitab usually returns.

It is recommended to compute empirical (bootstrapped) standard errors (via
the `iterations`

argument) than relying on analytic standard errors
(Wright & Herrington, 2011).

D. N. Joanes and C. A. Gill (1998). Comparing measures of sample skewness and kurtosis. The Statistician, 47, 183–189.

Wright, D. B., & Herrington, J. A. (2011). Problematic standard errors and confidence intervals for skewness and kurtosis. Behavior research methods, 43(1), 8-17.