Compute Skewness and (Excess) Kurtosis
Compute Skewness and (Excess) Kurtosis
skewness(x, na.rm = TRUE, type = "2", iterations = NULL, ...) kurtosis(x, na.rm = TRUE, type = "2", iterations = NULL, ...) ## 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, ...)
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 |
... |
Arguments passed to or from other methods. |
digits |
Number of decimal places. |
test |
Logical, if |
object |
An object returned by |
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, than 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, than 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 return
Type "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).
Values of skewness or kurtosis.
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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