Kurtosis

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue" stylesrc="../_private/template.htm" bgcolor="#FFFFFF" link="#669933" vlink="#000000" alink="#000000"> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table1"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table2"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table3"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Kurtosis</td> <td width="45" align="right" valign="bottom"> <map name="FPMap6"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap6" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table4"> <tr> <td width="0%" align="left" valign="top"> <p class="header2_black">Explained:<br> <img border="0" src="../images/blank_1x100.gif" width="100" height="1"></td> <td width="0%" align="left" valign="top"><p class="word_list"> <a href="#Kurtosis">kurtosis</a><p class="word_list"> <a href="#leptokurtic">leptokurtosis</a><p class="word_list"> <a href="#platykurtic">platykurtosis</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> <td width="100%" align="left" valign="top"><p class="word_list"> <br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <p class="body"><b><a name="Kurtosis">Kurtosis</a></b> is a parameter that describes the shape of a random variable’s probability density function (PDF). Consider the two PDFs in Exhibit 1:</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="400" cellpadding="0" id="table5"> <tr> <td> <p class="exhibit_header">Low vs. High Kurtosis<br> <span class="exhibit_subheader">Exhibit 1</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td> <img border="0" src="../images/ex1_kurtosis.gif" width="350" height="110"></td> </tr> <tr> <td> <p class="exhibit_legend">These graphs illustrate the notion of kurtosis. The PDF on the right has higher kurtosis than the PDF on the left. It is more peaked at the center, and it has fatter tails.</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table6"> <tr> <td align="right" width="9" height="14"></td> <td width="300" height="14"> <font size="2"> </font><font size="4"> </font></td> </tr> <tr> <td align="right" width="9"> </td> <td width="300"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> </table> <p class="body">Which would you say has the greater <a href="../articles/standard_deviation.htm">standard deviation</a>? It is impossible to say. The PDF on the right is more peaked at the center, which might lead us to believe that it has a lower standard deviation. It has fatter tails, which might lead us to believe that it has a higher standard deviation. If the effect of the peakedness exactly offsets that of the fat tails, the two PDFs will have the same standard deviation. The different shapes of the two PDFs illustrate kurtosis. The PDF on the right has a greater kurtosis than the one on the left. </p> <p class="body">The kurtosis of a random variable <i>X</i> is denoted <img border="0" src="../formulas/kurtosis_notation.gif" align="absbottom" width="15" height="12"> or <i>kurt</i>(<i>X</i>). It is defined as</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table7"> <tr> <td width="100%" align="center"> <p style="margin-top: 6; margin-bottom: 6"> <img border="0" src="../formulas/kurtosis_1.gif" width="154" height="51"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">where <img border="0" src="../formulas/aaa_mu.gif" align="absbottom" width="8" height="12"> and <img border="0" src="../formulas/aaa_sigma.gif" align="absbottom" width="8" height="11"> are the <a href="../articles/mean.htm">mean</a> and standard deviation of <i>X</i>. </p> <p class="body">A <a href="../articles/normal_distribution.htm">normal</a> random variable has a kurtosis of 3 irrespective of its mean or standard deviation. <span style="font-size: 12.0pt; font-family: Times New Roman">If a random variable’s kurtosis is greater than 3, it is said to be <b> <a name="leptokurtic">leptokurtic</a></b>. If its kurtosis is less than 3, it is said to be<b> <a name="platykurtic">platykurtic</a></b>. Leptokurtosis is associated with PDFs that are simultaneously “peaked” and have “fat tails.” Platykurtosis is associated with PDFs that are simultaneously less peaked and have thinner tails. They are said to have "shoulders." In Exhibit 1, the PDF on the left is platykurtic. The one on the right is leptokurtic.</span></p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table8"> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Internal Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table9"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><span class="verdana10"><strong style="font-weight: 400"><a href="../articles/chi-squared_distribution.htm">chi-squared distribution</a></strong></span><strong style="font-weight: 400"> If you square a normal random variable, the result is a chi-squared random variable.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/cornish_fisher.htm">Cornish-Fisher expansion</a> A formula for approximating quintiles of a random variable based only on its first few cumulants.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/mean.htm">expected value</a> A parameter describing the "center of gravity" of a PDF.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/joint_normal_distribution.htm">joint normal distribution</a> A multivariate distribution with normal marginal distributions.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/lognormal_distribution.htm"><strong style="font-weight: 400">lognormal distribution</strong></a><strong style="font-weight: 400"> A random variable is lognormal if its logarithm is normal.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/normal_distribution.htm">normal distribution</a> A normal random variable has a "bell shaped" PDF.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/quantile.htm">quantile</a> A notion from probability that can be used as a parameter.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/skewness.htm">skewness</a> A parameter that describes the lack of symmetry of a PDF.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/standard_deviation.htm">standard deviation</a> A parameter describing the dispersion of a PDF.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/uniform_distribution.htm">uniform distribution</a> <strong style="FONT-WEIGHT: 400">A random variable is uniform if it has constant probability on a finite interval.</strong></td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Sponsored Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" width="75"> <img border="0" src="../images/ad_side_banner_3.jpg" width="103" height="285"></td> <td bgcolor="#FFFFCC" width="99%">  <p class="ad_legend" style="margin-left: 0; 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