Quantile

<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">Quantile</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="#percentiles">percentile</a><p class="word_list"> <a href="#quantile">quantile</a><p class="word_list"> <a href="#quartiles">quartile</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">Consider a random variable <i>X</i>. A 0.75-<b><a name="quantile">quantile</a></b> of <i>X </i>is any value <i>x</i> such that <i>Pr</i>(<i>X <font face="Times New Roman">≤</font></i> <i>x</i>) = .75. This is illustrated with a hypothetical probability density function for <i>X</i> in Exhibit 1.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="300" cellpadding="0" id="table5"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Example: 0.75-Quantile</span><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_quantile.gif" width="300" height="149"></td> </tr> <tr> <td> <p class="exhibit_legend" style="margin-bottom: 24">A 0.75-quantile of <i>X </i>is any value <i>x</i> such that <i><br> Pr</i>(<i>X <font face="Times New Roman">≤</font></i> <i>x</i>) = .75.</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table6"> <tr> <td align="left" valign="top"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" id="table7"> <tr> <td width="336"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="9"> </td> </tr> </table> </td> <td align="left" valign="top" width="325"> <p class="body">More generally, a <i>q</i>-quantile of a random variable <i>X </i>is any value <i>x</i> such that <i>Pr</i>(<i>X <font face="Times New Roman">≤</font></i> <i>x</i>) = <i>q</i>. If a random variable is not continuous, or if it has zero probability density on some interval, quantiles may not be unique or may not exist.</p> <p class="body">Certain quantiles have special names. The 0.25-, 0.50-, and 0.75-quantiles are called the first, second and third <b> <a name="quartiles">quartiles</a></b>. The 0.01-, 0.02-, 0.03-, ... , 0.98-, 0.99-quantiles are called the first, second, third, ... , ninety-eighth, and ninety-ninth <b><a name="percentiles">percentiles</a></b>.</p> </td> </tr> </table> <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"><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 distribution.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/kurtosis.htm">kurtosis</a> A parameter describing the peakedness and tails of a distribution.<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 continuous probability distribution whose probability density function has a "bell" shape.<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 distribution.<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 distribution.<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 continuous probability distribution that 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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