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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><title>R: Kurtosis</title> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <link rel="stylesheet" type="text/css" href="R.css" /> </head><body> <table width="100%" summary="page for kurtosis {e1071}"><tr><td>kurtosis {e1071}</td><td style="text-align: right;">R Documentation</td></tr></table> <h2>Kurtosis</h2> <h3>Description</h3> <p>Computes the kurtosis. </p> <h3>Usage</h3> <pre> kurtosis(x, na.rm = FALSE, type = 3) </pre> <h3>Arguments</h3> <table summary="R argblock"> <tr valign="top"><td><code>x</code></td> <td> <p>a numeric vector containing the values whose kurtosis is to be computed.</p> </td></tr> <tr valign="top"><td><code>na.rm</code></td> <td> <p>a logical value indicating whether <code>NA</code> values should be stripped before the computation proceeds.</p> </td></tr> <tr valign="top"><td><code>type</code></td> <td> <p>an integer between 1 and 3 selecting one of the algorithms for computing skewness detailed below.</p> </td></tr> </table> <h3>Details</h3> <p>If <code>x</code> contains missings and these are not removed, the skewness is <code>NA</code>. </p> <p>Otherwise, write <i>x_i</i> for the non-missing elements of <code>x</code>, <i>n</i> for their number, <i>mu</i> for their mean, <i>s</i> for their standard deviation, and <i>m_r = ∑_i (x_i - mu)^r / n</i> for the sample moments of order <i>r</i>. </p> <p>Joanes and Gill (1998) discuss three methods for estimating kurtosis: </p> <dl> <dt>Type 1:</dt><dd> <p><i>g_2 = m_4 / m_2^2 - 3</i>. This is the typical definition used in many older textbooks.</p> </dd> <dt>Type 2:</dt><dd> <p><i>G_2 = ((n+1) g_2 + 6) * (n-1) / ((n-2)(n-3))</i>. Used in SAS and SPSS. </p> </dd> <dt>Type 3:</dt><dd> <p><i>b_2 = m_4 / s^4 - 3 = (g_2 + 3) (1 - 1/n)^2 - 3</i>. Used in MINITAB and BMDP.</p> </dd> </dl> <p>Only <i>G_2</i> (corresponding to <code>type = 2</code>) is unbiased under normality. </p> <h3>Value</h3> <p>The estimated kurtosis of <code>x</code>. </p> <h3>References</h3> <p>D. N. Joanes and C. A. Gill (1998), Comparing measures of sample skewness and kurtosis. <em>The Statistician</em>, <b>47</b>, 183–189. </p> <h3>Examples</h3> <pre> x <- rnorm(100) kurtosis(x) </pre> <hr /><div style="text-align: center;">[Package <em>e1071</em> version 1.7-3 <a href="00Index.html">Index</a>]</div> </body></html>