Compute Skewness and (Excess) Kurtosis
Usage
skewness(x, ...)
# S3 method for class 'numeric'
skewness(
x,
remove_na = TRUE,
type = "2",
iterations = NULL,
verbose = TRUE,
...
)
kurtosis(x, ...)
# S3 method for class 'numeric'
kurtosis(
x,
remove_na = TRUE,
type = "2",
iterations = NULL,
verbose = TRUE,
...
)
# S3 method for class 'parameters_kurtosis'
print(x, digits = 3, test = FALSE, ...)
# S3 method for class 'parameters_skewness'
print(x, digits = 3, test = FALSE, ...)
# S3 method for class 'parameters_skewness'
summary(object, test = FALSE, ...)
# S3 method for class 'parameters_kurtosis'
summary(object, test = FALSE, ...)Arguments
- x
A numeric vector or data.frame.
- ...
Arguments passed to or from other methods.
- remove_na
Logical. Should
NAvalues be removed before computing (TRUE) or not (FALSE, default)?- 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") or3(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.
- 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()orkurtosis().
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)^1.5Type "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.