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  1. <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
  2. "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
  3. <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"><head>
  4. <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /><!-- <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" /> -->
  5. <meta name="Author" lang="en" content="Miguel Casquilho" />
  6. <title>M&middot;C, IST: error propagation</title>
  7. <link rel="stylesheet" type="text/css"
  8. href="http://web.tecnico.ulisboa.pt/~mcasquilho/commonstyle.css" />
  9. </head> <!-- 18-Aug-2008 -->
  10. <body class="sm">
  11. <form method="post" action=
  12. "http://web.tecnico.ulisboa.pt/~mcasquilho/cgi-bin/interfacepng.php?propag">
  13. <input name="executable" type="hidden" value="propag.exe" />
  15. <table width="90%" bgcolor="#eeeeee" align="center" border="1" class="sm">
  16. <tr><td colspan="3">
  17. <table width="100%" class="sm">
  18. <tr><td width="50px"><a href='https://fenix.ist.utl.pt/departamentos/deq'><img alt='&copy;'
  19. src="http://web.tecnico.ulisboa.pt/~mcasquilho/depot/MC.gif" border='0'
  20. width='40' height='41' /></a>
  21. </td>
  22. <td><div style="font-size: 13.5pt; text-align: center"><!-- title --><b
  23. onmouseout="this.style.color='blue', this.style.fontStyle='normal'"
  24. onmouseover="this.style.color='red',
  25. this.style.fontStyle='bold'">
  26. Error
  27. propagation</b></div>&emsp;
  28. Calculates errors for various operations on data.
  29. <!-- b>Under construction</b --></td>
  30. <td width="112px"><table width='100%'>
  31. <tr><td align='center' bgcolor='white'>
  32. <input type="submit"
  33. value="&nbsp; Execute &nbsp;" style="background-color: #00bfff"
  34. onmouseout="this.style.fontStyle='normal', this.style.fontWeight='normal'"
  35. onmouseover="this.style.fontStyle='normal', this.style.fontWeight='bold'" /></td>
  36. </tr><tr><td align='center'><span style='font-size:
  37. xx-small'>2020.Mar.27 08:38:43</span></td></tr>
  38. </table>
  39. </td></tr></table>
  40. </td></tr> <!-- ................................................... -->
  42. <tr><td width="15%" align="center">Operation</td>
  43. <td width="45%"><input type="radio" name="iop"
  44. value="1" checked="checked" />Addition <input type="radio" name="iop"
  45. value="2" />Product</td>
  46. <td width="40%"><i>Operation</i> chosen (other ignored).</td></tr>
  48. <tr><td align="center">ADDITION:<br /><i>n</i>, data</td>
  49. <td><table width="100%">
  50. <tr><td><input name="nadd" type="text" value="2" size="2" maxlength="4" /></td>
  51. <td><textarea name="xadd" style="font-size: small; color: red" cols="28"
  52. rows="2" >-14.0031 14.2047
  53. 0.0005 0.0005 </textarea></td></tr></table></td>
  54. <td><i>Number</i> of data;
  55. <br />and <i>addends</i>, and
  56. their&nbsp;respective <i>errors</i>. &bull;</td></tr>
  58. <tr><td align="center">PRODUCT:<br /><i>n</i>, data</td>
  59. <td><table width="100%">
  60. <tr><td><input name="nprod" type="text" value="3" size="2" maxlength="4" /></td>
  61. <td><textarea name="xprod" style="font-size: small; color: red" cols="28"
  62. rows="3" >100 1 100
  63. 1 2 1
  64. 1 0.01 1 </textarea></td></tr></table></td>
  65. <td><i>Number</i> of data;
  66. <br />and <i>factors</i>, and
  67. their&nbsp;respective <i>exponents</i> and <i>errors</i>. &bull;</td></tr>
  69. <tr><td align="center">Show values</td>
  70. <td><select name="ishow"><option value="0" selected="selected">No</option>
  71. <option value="1">Yes</option></select></td>
  72. <td>Shows the coordinates of the graph. &bull;</td></tr>
  74. <!-- ...fonts/freefont/florence luxembrg muriel PARKAVE.pfb -->
  75. <!-- png linetype: my designations
  76. 1 red ff0000, 2 green 00c000, 3 blue 0080ff, 4 c000ff, 5 00eeee, 6 brown c04000,
  77. 7 beige eeee00, 8 dark blue 2020c0, 9 peach ffc020, 10 dark green 008040,
  78. 11 violeta a080ff, 12 dark brown 804000, 13 pink ff80ff, 14 turquesa 00c060,
  79. 15 turquesa 00c0c0, 16 turq. escuro 006080, 17 escarlate c06080,
  80. 18 dark green 008000, 19 green 40ff80, 20 dark blue 306080, 21 dark red 806000,
  81. 22 dark gray 404040, 23 olive green 408000, 24 navy blue 000080, 25 806010,
  82. 26 806060, 27 ~806080, 28 ~5050c0, 29 0000ff, 30 006000, 31 ~eb0c0,
  83. 32 40c080, 33 60a0c0 -->
  84. <!-- plot gnudata.dat \ (or anything: this line is a MANDATORY dummy) -->
  86. <tr><td colspan="3"><input name="gnuplot" type="hidden"
  87. value="set terminal png enhanced font 'FreeSerif, 13' size 500, 380
  88. set timestamp font 'FreeSerif, 10' offset 5, -0.7
  89. set nokey; set title 'Result, uncertainty' offset 0, +.5
  90. set xlabel 'i' offset 12, .1; set ylabel 'y'
  91. plot 'anyname.dat' \
  92. using 1:2 with lines linetype 1 linewidth 4, \
  93. '' using 1:3 with lines linetype 2 linewidth 2, \
  94. '' using 1:4 with lines linetype 2 linewidth 2" />
  95. <table width="100%" class="sm">
  96. <tr><td valign="top" colspan="2">&emsp; From the <i>data</i>
  97. (set of observations), calculates some results for the propagation
  98. of errors (<i>propagation of uncertainty</i>). <i>Upper limit</i>
  99. of error ("large" but always valid) and <i>probable</i> error
  100. (smaller but valid only for independent errors) are addressed.
  101. <br />&emsp;It is assumed that, for a probability of 95 %, it is
  102. <i>&Delta; x</i> &cong; 2 <i>&sigma;<sub>x</sub></i>, from
  103. &Phi;(<i>k</i>) &minus; &Phi;(&minus;<i>k</i>) &cong;
  104. 95.4 % for <i>k</i>&nbsp;=&nbsp;2 (<a href=
  105. "http://web.tecnico.ulisboa.pt/~mcasquilho/compute/qc/F-Gauss.php">Gauss</a>). Other authors
  106. (such as Taylor [1997]) give
  107. <i>&Delta; x</i> &cong; <i>&sigma;<sub>x</sub></i>.
  108. <br />&emsp;A simple graph is shown with the <i>probable error</i> and its
  109. limits.
  110. <table class="sm">
  111. <tr valign="top"><td>&emsp;Other suggested data:</td>
  112. <td><i>&nbsp; Addition</i></td><td>(<i>n</i> = 4)<br />(<i>n</i> = 3)</td>
  113. <td> 540 -72 940 -97<br />200 50 -20</td><td>&nbsp;</td>
  114. <td align="right">with errors &nbsp; 10 1 20 1<br /> 2 2 1</td>
  115. <td>[Taylor, 1997, 50]<br />[<i>ibid.</i>, 62]</td></tr>
  116. <tr valign="top"><td>&nbsp;</td>
  117. <td><i>&nbsp;Product</i></td><td>(<i>n</i> = 3)<br />(<i>n</i> = 3)</td>
  118. <td> 200 5.5 10<br /> 200 50 20</td>
  119. <td>exp.s 1 1 -1 (a division)<br />exp.s 1 1 -1</td>
  120. <td align="right">with errors &nbsp; 2 0.1 0.4<br /> 4 4 2</td>
  121. <td>[Taylor, 1997, 53]<br />[<i>ibid.</i>, 62]</td></tr>
  122. </table></td></tr>
  123. </table></td></tr>
  125. <tr><td align="center"><input type="reset"
  126. style="font-size: smaller" value=" Reset " /></td><td><i>References:</i></td>
  127. <td align="center">Propag08823</td></tr>
  129. <tr><td colspan="3"><p class="ref">&bull;&nbsp;<a href=
  130. "http://web.mit.edu/trex/www/trex2/"><span class="sc">Nepf</span></a>, Heidi:
  131. <a href="http://web.mit.edu/fluids-modules/www/exper_techniques/
  132. 2.Propagation_of_Uncertaint.pdf">Propagation of Uncertainty</a>.<i>pdf</i>
  133. (accessed 2008-08-13) (<a class="decnone" href=
  134. "./,propagation/MIT.2.Propagation_of_Uncertainty.pdf">=</a>),
  135. <a href="http://web.mit.edu/">MIT</a>.</p>
  137. <p class="ref">&bull;&nbsp;<a href=
  138. "http://people.physics.tamu.edu/sante/index.htm"><span
  139. class="sc">Scuro</span></a>, Sante R.:
  140. <a href="http://people.physics.tamu.edu/sante/index_files/TAMU/error.pdf">Intro.
  141. to error theory</a>.<i>pdf</i> (accessed 2008-08-13) (<a class="decnone" href=
  142. "http://web.tecnico.ulisboa.pt/~mcasquilho/compute/errtheory/,basics/
  143. ScuroErrTheo.pdf">=</a>),
  144. <a href="http://www.physics.tamu.edu/">Texas A&amp;M Univ.</a>).
  145. &bull;&nbsp;<a href=
  146. "http://class.phys.psu.edu/p559/experiments/html/error.html">Err.
  147. analysis</a> (<a href="http://class.phys.psu.edu/">PennState U.</a>).
  148. &bull;&nbsp;<a href=
  149. "http://teacher.nsrl.rochester.edu/PHY_LABS/Labs_121.html">Wolfs</a>,&nbsp;F.,
  150. see "Error analysis"
  151. (Univ. of <a href="http://www.rochester.edu/">Rochester</a>).</p>
  153. <p class="ref">&bull;&nbsp;<span class="sc">Weisstein</span>, Eric W.,
  154. "<a href="http://mathworld.wolfram.com/ErrorPropagation.html">Error
  155. propagation</a>". From <i>MathWorld</i> &mdash; a Wolfram Web Resource
  156. (2008-08-04).</p>
  157. <p class="ref">&bull;&nbsp;<span class="sc">Taylor</span>, John R.,
  158. <acronym title="BibIST"><a href=
  159. "http://thesaurus.reitoria.utl.pt/search~S3*por?/cQC39.TAY.67208/
  160. cqc+++39+tay67208/-3,-1,0,B/
  161. frameset&amp;FF=cqc+++39+tay67208&amp;1,1,">1997</a></acronym>,
  162. "An introduction to error analysis", 2.nd ed., <acronym title="Amazon"><a href=
  163. "http://www.amazon.co.uk/In
  164. troduction-Error-Analysis-Uncertainties-Measurements/
  165. dp/093570275X/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1217804330&amp;sr=1-1"
  166. >Univ. Sci. Books</a></acronym>
  167. (<a href="http://www.uscibooks.com">USB</a>), Sausalito, CA (USA)
  168. (<a href="http://physics.gac.edu/~mellema/Taylor/
  169. Taylor_files/frame.htm">intro</a> IE).</p>
  171. <p class="ref">&bull;&nbsp;<span class="sc">Bevington</span>, Philip R. and
  172. D. Keith <span class="sc">Robinson</span>, <acronym title="BibIST"><a href=
  173. "http://thesaurus.reitoria.utl.pt/search~S8*por/?searchtype=
  174. c&amp;searcharg=00.BR.61056&amp;searchscope=3&amp;SORT=D&amp;extended=
  175. 0&amp;SUBMIT=Pesquisar&amp;searchlimits=&amp;searchorigarg=
  176. c00.BR.61056">2003</a></acronym>, "<acronym title="Amazon"><a href=
  177. "http://www.amazon.com/Reduction-Error-Analysis-Physical-Sciences/dp/
  178. 0072472278/ref=pd_bbs_sr_1?ie=UTF8&amp;s=books&amp;qid=1219893338&amp;sr=
  179. 1-1">Data reduction and error analysis for the physical
  180. sciences</a></acronym>", 3.rd ed., McGraw-Hill, New York, NY (USA).</p>
  182. <p class="ref">&bull;&nbsp;<span class="sc">Taylor</span>, John Keenan,
  183. 1987, "Quality assurance of chemical measurements", <acronym
  184. title="Amazon"><a href=
  185. "http://www.amazon.co.uk/Quality-Assurance-Chemical-Measurements-Keenan/dp/
  186. 0873710975/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1219100382&amp;sr=1-1"
  187. >Lewis Publishers, Inc.</a></acronym>, Chelsea, Michigan&nbsp;(USA).</p>
  189. <p class="ref">&bull;&nbsp;1778-08-23: <a href=
  190. "http://www-groups.dcs.st-and.ac.uk/~history/Biographies/
  191. Wronski.html">Wronski</a>, Josef-Maria Hoëné de, birthday.</p>
  192. </td></tr>
  193. </table>
  194. </form>
  196. <div style="line-height: 6pt">&nbsp;</div>
  197. <div style="line-height: 1.5pt; color: black;
  198. background-color: red">&nbsp;</div>
  200. <table width="100%" class="sm"><tr>
  201. <td width="58px"><a href="http://validator.w3.org/check?uri=
  202. http://web.tecnico.ulisboa.pt/~mcasquilho/compute/errtheory/Fx-propagation.php;verbose=1;ss=1"><img
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  204. alt="Valid HTML 4.01!" border="0" width="55px" height="19px" /></a></td>
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  208. <td width="70%" align=
  209. "right"><i>http://web.tecnico.ulisboa.pt/~mcasquilho/compute/errtheory/Fx-propagation.php</i><br />
  210. <i>Created:</i> 2008-08-23 &mdash; <i>Last modified:</i>
  211. 2008-09-26</td></tr></table>
  212. </body></html>

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