<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article  PUBLIC '-//OASIS//DTD DocBook XML V4.4//EN'  'http://www.docbook.org/xml/4.4/docbookx.dtd'><article><articleinfo><title>FAQ/eorder</title><revhistory><revision><revnumber>23</revnumber><date>2013-03-08 10:17:56</date><authorinitials>localhost</authorinitials><revremark>converted to 1.6 markup</revremark></revision><revision><revnumber>22</revnumber><date>2012-07-09 12:39:40</date><authorinitials>PeterWatson</authorinitials><revremark>Revert to revision 19.</revremark></revision><revision><revnumber>21</revnumber><date>2012-07-07 19:41:55</date><authorinitials>ec2-75-101-167-21.compute-1.amazonaws.com</authorinitials><revremark>S1fkGa , [url=http://ivejrpypbtcr.com/]ivejrpypbtcr[/url], [link=http://krsvfvzikkcq.com/]krsvfvzikkcq[/link], http://bokpitlsmfty.com/</revremark></revision><revision><revnumber>20</revnumber><date>2012-07-06 22:25:54</date><authorinitials>ks203878.kimsufi.com</authorinitials><revremark>he disagreed with the cnucoil plan to introduce the 40km/h speed zone across the city, saying it would further congest traffic.&quot; If this is true then Walt Secord should resign his position on the Stay</revremark></revision><revision><revnumber>19</revnumber><date>2009-01-13 12:12:57</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>18</revnumber><date>2009-01-13 10:33:40</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>17</revnumber><date>2009-01-12 17:38:12</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>16</revnumber><date>2009-01-12 17:29:52</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>15</revnumber><date>2009-01-12 17:29:40</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>14</revnumber><date>2009-01-12 17:26:24</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>13</revnumber><date>2009-01-12 17:25:13</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>12</revnumber><date>2009-01-12 17:25:01</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>11</revnumber><date>2009-01-12 17:02:04</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>10</revnumber><date>2009-01-12 17:01:43</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>9</revnumber><date>2009-01-12 16:58:45</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>8</revnumber><date>2009-01-12 16:58:03</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>7</revnumber><date>2009-01-12 16:57:34</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>6</revnumber><date>2009-01-12 16:54:42</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>5</revnumber><date>2009-01-12 16:54:17</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>4</revnumber><date>2009-01-12 16:38:36</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>3</revnumber><date>2009-01-12 16:37:02</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>2</revnumber><date>2009-01-12 16:36:43</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>1</revnumber><date>2009-01-12 16:36:30</date><authorinitials>PeterWatson</authorinitials></revision></revhistory></articleinfo><section><title>Quick formula for the expected value of a Normal order statistic</title><para>The expected value of  the i-th Normal Order Statistic from a sample of size, n, may be computed as the absolute value of </para><para>$$\Phi^{-1}(U(i,n))$$ </para><para>where $$\Phi^{-1}$$ is the inverse Normal cumulative density function which converts a probability, such as U(i,n), to a z-score. </para><para>where the U(i,n) are the U(i) taken from <ulink url="http://en.wikipedia.org/wiki/Normal_probability_plot">here</ulink> and reproduced in the table below. </para><itemizedlist><listitem override="none"><informaltable><tgroup cols="2"><colspec colname="col_0" colwidth="50*"/><colspec colname="col_1" colwidth="50*"/><tbody><row rowsep="1"><entry colsep="1" rowsep="1"><para> <emphasis role="strong">U(i,n)</emphasis> </para></entry><entry colsep="1" rowsep="1"><para> <emphasis role="strong">i</emphasis> </para></entry></row><row rowsep="1"><entry colsep="1" rowsep="1"><para> $$ 1- 0.5^{1/n} </para></entry><entry colsep="1" rowsep="1"><para>i=1</para></entry></row><row rowsep="1"><entry colsep="1" rowsep="1"><para> $$\frac{i-0.3175}{n+0.365}$$ </para></entry><entry colsep="1" rowsep="1"><para> $$i=2, \ldots, n-1$$ </para></entry></row><row rowsep="1"><entry colsep="1" rowsep="1"><para> $$0.5^{1/n}$$ </para></entry><entry colsep="1" rowsep="1"><para> i=n</para></entry></row></tbody></tgroup></informaltable></listitem></itemizedlist><para>For example U(1,3)=0.82 (using above approximation) compared to an exact value of 0.85. </para><para>An exact version written in the C language programming language (which may be run on Unix) is obtained using a version of Algorithm AS 177 of the Royal Statistical Applied Statistics algorithm page. The C program code is listed <ulink url="https://lsr-wiki-01.mrc-cbu.cam.ac.uk/statswiki/FAQ/eorder/statswiki/FAQ/Corder#">here.</ulink> </para><para>The algorithm is labelled Expected Normal Order Statistics (Exact and Approximate) and is based on the original FORTRAN code by Royston, 1982. </para><para>A table of expected values of normal order statistics (2.4(a)) are in Neave HR (1978). </para><para><emphasis role="underline">References</emphasis> </para><para>Neave HR (1978) Statistics tables for mathematicians, engineers, economists and the behavioural and management sciences. Unwin Hyman:London. </para><para>Royston, JP (1982), Algorithm AS 177, <emphasis>Applied Statistics</emphasis>, <emphasis role="strong">31(2)</emphasis>:161-165. </para></section></article>