Compute Skewness and (Excess) Kurtosis
skewness(x, ...)# S3 method for numeric
skewness(
x,
remove_na = TRUE,
type = "2",
iterations = NULL,
verbose = TRUE,
...
)
kurtosis(x, ...)
# S3 method for numeric
kurtosis(
x,
remove_na = 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, ...)
Values of skewness or kurtosis.
A numeric vector or data.frame.
Arguments passed to or from other methods.
Logical. Should NA values be removed before computing (TRUE)
or not (FALSE, default)?
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'.
The number of bootstrap replicates for computing standard
errors. If NULL (default), parametric standard errors are computed.
Toggle warnings and messages.
Number of decimal places.
Logical, if TRUE, tests if skewness or kurtosis is
significantly different from zero.
An object returned by 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 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.
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, then adjusts the result:
G2 = ((n + 1) * g2 + 6) * (n - 1)/((n - 2) * (n - 3)). This is what
SAS and SPSS usually return
Type "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.
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.
skewness(rnorm(1000))
kurtosis(rnorm(1000))
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