Computes the kurtosis.
kurtosis(x, na.rm = FALSE, type = 3)The estimated kurtosis of x.
a numeric vector containing the values whose kurtosis is to be computed.
a logical value indicating whether NA values
should be stripped before the computation proceeds.
an integer between 1 and 3 selecting one of the algorithms for computing kurtosis detailed below.
If x contains missings and these are not removed, the kurtosis
is NA.
Otherwise, write \(x_i\) for the non-missing elements of x,
\(n\) for their number, \(\mu\) for their mean, \(s\) for
their standard deviation, and
\(m_r = \sum_i (x_i - \mu)^r / n\)
for the sample moments of order \(r\).
Joanes and Gill (1998) discuss three methods for estimating kurtosis:
\(g_2 = m_4 / m_2^2 - 3\). This is the typical definition used in many older textbooks.
\(G_2 = ((n+1) g_2 + 6) * (n-1) / ((n-2)(n-3))\). Used in SAS and SPSS.
\(b_2 = m_4 / s^4 - 3 = (g_2 + 3) (1 - 1/n)^2 - 3\). Used in MINITAB and BMDP.
Only \(G_2\) (corresponding to type = 2) is unbiased under
normality.
D. N. Joanes and C. A. Gill (1998), Comparing measures of sample skewness and kurtosis. The Statistician, 47, 183--189.