% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/skewness_kurtosis.R
\name{skewness}
\alias{skewness}
\alias{skewness.numeric}
\alias{kurtosis}
\alias{kurtosis.numeric}
\alias{print.parameters_kurtosis}
\alias{print.parameters_skewness}
\alias{summary.parameters_skewness}
\alias{summary.parameters_kurtosis}
\title{Compute Skewness and (Excess) Kurtosis}
\usage{
skewness(x, ...)

\method{skewness}{numeric}(
  x,
  remove_na = TRUE,
  type = "2",
  iterations = NULL,
  verbose = TRUE,
  ...
)

kurtosis(x, ...)

\method{kurtosis}{numeric}(
  x,
  remove_na = TRUE,
  type = "2",
  iterations = NULL,
  verbose = TRUE,
  ...
)

\method{print}{parameters_kurtosis}(x, digits = 3, test = FALSE, ...)

\method{print}{parameters_skewness}(x, digits = 3, test = FALSE, ...)

\method{summary}{parameters_skewness}(object, test = FALSE, ...)

\method{summary}{parameters_kurtosis}(object, test = FALSE, ...)
}
\arguments{
\item{x}{A numeric vector or data.frame.}

\item{...}{Arguments passed to or from other methods.}

\item{remove_na}{Logical. Should \code{NA} values be removed before computing (\code{TRUE})
or not (\code{FALSE}, default)?}

\item{type}{Type of algorithm for computing skewness. May be one of \code{1}
(or \code{"1"}, \code{"I"} or \code{"classic"}), \code{2} (or \code{"2"},
\code{"II"} or \code{"SPSS"} or \code{"SAS"}) or \code{3} (or  \code{"3"},
\code{"III"} or \code{"Minitab"}). See 'Details'.}

\item{iterations}{The number of bootstrap replicates for computing standard
errors. If \code{NULL} (default), parametric standard errors are computed.}

\item{verbose}{Toggle warnings and messages.}

\item{digits}{Number of decimal places.}

\item{test}{Logical, if \code{TRUE}, tests if skewness or kurtosis is
significantly different from zero.}

\item{object}{An object returned by \code{skewness()} or \code{kurtosis()}.}
}
\value{
Values of skewness or kurtosis.
}
\description{
Compute Skewness and (Excess) Kurtosis
}
\details{
\subsection{Skewness}{
Symmetric distributions have a \code{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:
\itemize{
\item Normal distribution (and other symmetric distribution) has a skewness
of 0
\item Half-normal distribution has a skewness just below 1
\item Exponential distribution has a skewness of 2
\item Lognormal distribution can have a skewness of any positive value,
depending on its parameters
}

(\cite{https://en.wikipedia.org/wiki/Skewness})
}

\subsection{Types of Skewness}{
\code{skewness()} supports three different methods for estimating skewness,
as discussed in \cite{Joanes and Gill (1988)}:
\itemize{
\item Type "1" is the "classical" method, which is \code{g1 = (sum((x - mean(x))^3) / n) / (sum((x - mean(x))^2) / n)^1.5}
\item Type "2" first calculates the type-1 skewness, then adjusts the result:
\code{G1 = g1 * sqrt(n * (n - 1)) / (n - 2)}. This is what SAS and SPSS
usually return.
\item Type "3" first calculates the type-1 skewness, then adjusts the result:
\code{b1 = g1 * ((1 - 1 / n))^1.5}. This is what Minitab usually returns.
}
}

\subsection{Kurtosis}{
The \code{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
\emph{fatter} tails. A kurtosis value below zero indicates a "platykurtic"
distribution with \emph{thinner} tails
(\cite{https://en.wikipedia.org/wiki/Kurtosis}).
}

\subsection{Types of Kurtosis}{
\code{kurtosis()} supports three different methods for estimating kurtosis,
as discussed in \cite{Joanes and Gill (1988)}:
\itemize{
\item Type "1" is the "classical" method, which is \code{g2 = n * sum((x - mean(x))^4) / (sum((x - mean(x))^2)^2) - 3}.
\item Type "2" first calculates the type-1 kurtosis, then adjusts the result:
\code{G2 = ((n + 1) * g2 + 6) * (n - 1)/((n - 2) * (n - 3))}. This is what
SAS and SPSS usually return
\item Type "3" first calculates the type-1 kurtosis, then adjusts the result:
\code{b2 = (g2 + 3) * (1 - 1 / n)^2 - 3}. This is what Minitab usually
returns.
}

}

\subsection{Standard Errors}{
It is recommended to compute empirical (bootstrapped) standard errors (via
the \code{iterations} argument) than relying on analytic standard errors
(\cite{Wright & Herrington, 2011}).
}
}
\examples{
skewness(rnorm(1000))
kurtosis(rnorm(1000))
}
\references{
\itemize{
\item D. N. Joanes and C. A. Gill (1998). Comparing measures of sample
skewness and kurtosis. The Statistician, 47, 183–189.
\item Wright, D. B., & Herrington, J. A. (2011). Problematic standard
errors and confidence intervals for skewness and kurtosis. Behavior
research methods, 43(1), 8-17.
}
}
