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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This tutorial explores some of the built-in curve fitting and data analysis functions you can use in a worksheet.</p>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here are the topics covered in this tutorial:</p>
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<p style="list" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<link href="#graphing data" title="Links to section" popup="false">To Plot Your Data</link>
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<link href="#single" title="Links to section" popup="false">To Plot a Single Data Variable from One Input Table or Vector</link>
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<link href="#two input tables" title="Links to section" popup="false">To Plot Two Data Variables from Two Input Tables or Vectors</link>
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<link href="#one input table" title="Links to section" popup="false">To Plot Two Data Variables from One Input Table, a File Read Component, or a Matrix</link>
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<link href="#lsfit" title="Links to section" popup="false">To Perform a Least Squares Fit</link>
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<link href="#line function" title="Links to section" popup="false">The <b>line</b> Function</link>
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<text use-page-width="false" push-down="false" lock-width="true">
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<link href="#special fit" title="Links to section" popup="false">To Perform a Specialized Curve Fit</link>
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<link href="#expfit function" title="Links to section" popup="false">The <b>expfit</b> Function</link>
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<link href="#analyzing the quality" title="Links to section" popup="false">To Analyze the Quality of the Fit</link>
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<link href="#data and function" title="Links to section" popup="false">To Plot the Data and the Curve on the Same 2D Graph</link>
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<link href="#correlation" title="Links to section" popup="false">To Calculate Correlation</link>
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<b>
<c val="#008080">Note:</c>
</b> If your data comes from an outside source please read the <b>
<u>
<link href="./feat_06.xmcd" title="Links to section" popup="false">Handling Data Tutorial</link>
</u>
</b> for more information.</p>
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<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Plot Your Data</p>
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<phrase>graphing data</phrase>
<phrase>data, graphing</phrase>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Before you analyze your data, you can plot it on a 2D graph in order to make a reasonable guess about a model function that fits the data. Here are several methods to plot your data depending on how you have entered or imported it into the worksheet.</p>
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<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Plot a Single Data Variable from One Data Table or Vector</p>
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<b>
<c val="#008080">1.</c>
</b>
<sp count="2"></sp>Anywhere below or to the right of your data variable definition, define a range variable to index the data variable. For example, if your data variable is <b>data</b>, type <f family="Verdana" charset="0" size="10">
<b>i:0;length(data)</b>
</f>.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">2.</c>
</b>
<sp count="2"></sp>Click below these definitions and choose <b>Graph &gt; X-Y Plot</b> from the <b>Insert</b> menu or type <b>@</b> to insert a 2D Graph.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">3.</c>
</b>
<sp count="2"></sp>In the x-axis placeholder, enter the name of the range variable (for example, <b>i</b>).</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">4.</c>
</b>
<sp count="2"></sp>In the y-axis placeholder, type <f family="Verdana" charset="0" size="10">
<b>data[i</b>
</f> to enter the name of the variable that contains your data subscripted with the range variable. Then, press <f family="Verdana" charset="0" size="10">
<b>[Enter]</b>
</f>.</p>
</text>
</region>
<region region-id="3740" left="6" top="750.75" width="306.75" height="14.25" align-x="17.25" align-y="762" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<c val="#008080">Note:</c>
</b> Arrays and tables have a starting index of 0 (zero) by default.</p>
</text>
</region>
<region region-id="3801" left="6" top="776.25" width="47.25" height="12" align-x="6" align-y="786" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Example</p>
</text>
</region>
<region region-id="3398" left="6" top="798.75" width="341.25" height="28.5" align-x="16.5" align-y="810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here is a small set of data entered in a single column Data Table. The data represent the completion times for various speed trials of a car:</p>
</text>
</region>
<region region-id="3735" left="18" top="840.75" width="117" height="147" align-x="50.25" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<component hide-arguments="false" clsid-buddy="01350081-1122-11DB-9380-000D56C6051A" item-idref="4" disable-calc="false">
<inputs></inputs>
<outputs>
<ml:id xml:space="preserve">times</ml:id>
</outputs>
</component>
<rendering item-idref="5"></rendering>
</region>
<region region-id="3736" left="168" top="840.75" width="96" height="15.75" align-x="178.5" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">i</ml:id>
<ml:range>
<ml:real>0</ml:real>
<ml:apply>
<ml:id xml:space="preserve">length</ml:id>
<ml:id xml:space="preserve">times</ml:id>
</ml:apply>
</ml:range>
</ml:define>
</math>
<rendering item-idref="6"></rendering>
</region>
<region region-id="3529" left="0" top="1008" width="6000" height="6" align-x="0" align-y="1008" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3415" left="6" top="1014.75" width="342" height="28.5" align-x="14.25" align-y="1026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We have formatted this graph to show the data as points and have adjusted the axis limits.</p>
</text>
</region>
<region region-id="3478" left="6" top="1056" width="220.5" height="214.5" align-x="6" align-y="1056" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<plot disable-calc="false" item-idref="7"></plot>
<rendering item-idref="8"></rendering>
</region>
<region region-id="3390" left="6" top="1298.25" width="328.5" height="12" align-x="22.5" align-y="1308" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="two input tables">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Plot Two Data Variables from Two Data Tables or Vectors</p>
</text>
</region>
<region region-id="3418" left="6" top="1320.75" width="341.25" height="99.75" align-x="6" align-y="1332" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">1.</c>
</b>
<sp count="2"></sp>Click anywhere below or to the right of your data tables and choose <b>Graph &gt; X-Y Plot</b> from the <b>Insert</b> menu or type <b>@</b> to insert a 2D Graph.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">2.</c>
</b>
<sp count="2"></sp>In the x-axis placeholder, enter the name of the variable containing your independent data (usually time).</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">3.</c>
</b>
<sp count="2"></sp>In the y-axis placeholder, enter the name of the variable containing your dependent data (usually what you measured) and press <f family="Verdana" charset="0" size="10">
<b>[Enter]</b>
</f>.</p>
</text>
</region>
<region region-id="3534" left="0" top="1434" width="6000" height="6" align-x="0" align-y="1434" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3803" left="6" top="1442.25" width="47.25" height="12" align-x="6" align-y="1452" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Example</p>
</text>
</region>
<region region-id="3425" left="6" top="1458.75" width="341.25" height="28.5" align-x="16.5" align-y="1470" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here are two vectors containing time and population data and a plot of time versus population, below.</p>
</text>
</region>
<region region-id="3445" left="18" top="1499.25" width="66" height="162.75" align-x="45" align-y="1584" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define warning="WarnRedefinedBIFunction">
<ml:id xml:space="preserve">time</ml:id>
<ml:matrix rows="8" cols="1">
<ml:real>0</ml:real>
<ml:real>5</ml:real>
<ml:real>10</ml:real>
<ml:real>15</ml:real>
<ml:real>20</ml:real>
<ml:real>25</ml:real>
<ml:real>30</ml:real>
<ml:real>35</ml:real>
</ml:matrix>
</ml:define>
</math>
<rendering item-idref="9"></rendering>
</region>
<region region-id="3446" left="120" top="1499.25" width="100.5" height="162.75" align-x="175.5" align-y="1584" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">population</ml:id>
<ml:matrix rows="8" cols="1">
<ml:real>0</ml:real>
<ml:real>6</ml:real>
<ml:real>11</ml:real>
<ml:real>25</ml:real>
<ml:real>50</ml:real>
<ml:real>90</ml:real>
<ml:real>153</ml:real>
<ml:real>230</ml:real>
</ml:matrix>
</ml:define>
</math>
<rendering item-idref="10"></rendering>
</region>
<region region-id="3448" left="6" top="1674" width="248.25" height="214.5" align-x="6" align-y="1674" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<plot disable-calc="false" item-idref="11"></plot>
<rendering item-idref="12"></rendering>
</region>
<region region-id="3535" left="0" top="1908" width="6000" height="6" align-x="0" align-y="1908" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3391" left="6" top="1916.25" width="341.25" height="24" align-x="22.5" align-y="1926" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="one input table">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Plot Two Data Variables from One Data Table, a File Input Component, or a Matrix</p>
</text>
</region>
<region region-id="3450" left="6" top="1950.75" width="341.25" height="142.5" align-x="6" align-y="1962" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">1.</c>
</b>
<sp count="2"></sp>Below your data table, define two variables to contain individual columns from your data table. For example, if time values are in the first column of the data table, <b>data</b>, type <f family="Verdana" charset="0" size="10">
<b>time:data[Ctrl]6 0</b>
</f>.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">2.</c>
</b>
<sp count="2"></sp>Click anywhere below or to the right of your data table and choose <b>Graph &gt; X-Y Plot</b> from the <b>Insert</b> menu or type <b>@</b> to insert a 2D Graph.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<c val="#008080">
<b>3.</b>
</c>
<sp count="2"></sp>In the x-axis placeholder, enter the name of the variable containing your independent data (usually time).</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">4.</c>
</b>
<sp count="2"></sp>In the y-axis placeholder, enter the name of the variable containing your dependent data (usually what you measured) and press <f family="Verdana" charset="0" size="10">
<b>[Enter]</b>
</f>.</p>
</text>
</region>
<region region-id="3796" left="6" top="2114.25" width="47.25" height="12" align-x="6" align-y="2124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Example</p>
</text>
</region>
<region region-id="3457" left="6" top="2130.75" width="341.25" height="42.75" align-x="16.5" align-y="2142" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here is a Data Table containing time and population data, two variables assigned to individual columns from the table, and a plot of time versus population, below.</p>
</text>
</region>
<region region-id="3475" left="6" top="2190.75" width="172.5" height="147.75" align-x="34.5" align-y="2202" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<component hide-arguments="false" clsid-buddy="01350081-1122-11DB-9380-000D56C6051A" item-idref="13" disable-calc="false">
<inputs></inputs>
<outputs>
<ml:id xml:space="preserve">data</ml:id>
</outputs>
</component>
<rendering item-idref="14"></rendering>
</region>
<region region-id="3476" left="216" top="2186.25" width="70.5" height="20.25" align-x="243" align-y="2202" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define warning="WarnRedefinedUDScalar">
<ml:id xml:space="preserve">time</ml:id>
<ml:apply>
<ml:matcol></ml:matcol>
<ml:id xml:space="preserve">data</ml:id>
<ml:real>0</ml:real>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="15"></rendering>
</region>
<region region-id="3477" left="216" top="2216.25" width="99" height="20.25" align-x="271.5" align-y="2232" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define warning="WarnRedefinedUDScalar">
<ml:id xml:space="preserve">population</ml:id>
<ml:apply>
<ml:matcol></ml:matcol>
<ml:id xml:space="preserve">data</ml:id>
<ml:real>1</ml:real>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="16"></rendering>
</region>
<region region-id="3536" left="0" top="2352" width="6000" height="6" align-x="0" align-y="2352" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3460" left="6" top="2364" width="248.25" height="214.5" align-x="6" align-y="2364" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<plot disable-calc="false" item-idref="17"></plot>
<rendering item-idref="18"></rendering>
</region>
<region region-id="3039" left="6" top="2606.25" width="197.25" height="12" align-x="6" align-y="2616" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="lsfit">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Perform a Least Squares Fit</p>
</text>
<indexes>
<phrase>least squares fit</phrase>
<phrase>linear regression</phrase>
</indexes>
</region>
<region region-id="3160" left="6" top="2622.75" width="341.25" height="71.25" align-x="6" align-y="2634" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">A least squares fit is the method most commonly used to find the line that best fits a set of data. "Least squares" refers to the sum of the squares of the distances from the individual data points to the line itself. A least squares fit finds the smallest possible sum. There are three built-in functions in Mathcad that can be used to fit data to a straight line.</p>
</text>
</region>
<region region-id="3169" left="6" top="2700.75" width="341.25" height="57" align-x="6" align-y="2712" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">•</c>
</b>
<sp count="3"></sp>The <b>line</b> function returns both the slope and the y-intercept of the least squares line in a vector.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">•</c>
</b>
<sp count="3"></sp>The <b>slope</b> function returns only the slope; the <b>intercept</b> function returns only the y-intercept.</p>
</text>
</region>
<region region-id="3479" left="6" top="2766.75" width="341.25" height="28.5" align-x="6" align-y="2778" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here are instructions for using the <b>line</b> function to get the slope and intercept of a line of best fit.</p>
</text>
</region>
<region region-id="3537" left="0" top="2802" width="6000" height="6" align-x="0" align-y="2802" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3171" left="6" top="2810.25" width="142.5" height="12" align-x="24.75" align-y="2820" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="line function">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The line Function</p>
</text>
</region>
<region region-id="3162" left="6" top="2832.75" width="341.25" height="171" align-x="6" align-y="2844" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<c val="#008080">1.</c>
</b>
<sp count="2"></sp>Click in a blank spot below your data tables.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">2.</c>
</b>
<sp count="2"></sp>To define the fit, enter the name of the variable that will contain the results of the least squares fit followed by the definition equals; for example, type <f family="Verdana" charset="0" size="10">
<b>fit:</b>
</f>
</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">3.</c>
</b>
<sp count="2"></sp>For the right side of the definition, choose <b>Function</b> from the <b>Insert</b> menu to get the Insert Function dialog box. In the list of all functions, click on "line" and then click OK.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">4.</c>
</b>
<sp count="2"></sp>In the two placeholders provided, enter the names of the independent data variable and the dependent data variable, in that order, and press <f family="Verdana" charset="0" size="10">
<b>[Enter]</b>
</f>.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">5.</c>
</b>
<sp count="2"></sp>To see the results of the fit, type <f family="Verdana" charset="0" size="10">
<b>fit=</b>
</f>. The first value is the y-intercept; the second value is the slope.</p>
</text>
</region>
<region region-id="3762" left="6" top="3026.25" width="326.25" height="12" align-x="26.25" align-y="3036" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="data and function">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Plot the Data and a Curve on the Same 2D Graph</p>
</text>
<indexes>
<phrase>plotting, data and curve</phrase>
</indexes>
</region>
<region region-id="3763" left="6" top="3042.75" width="340.5" height="42.75" align-x="18" align-y="3054" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Once you have performed your curve fit and plotted your data on a graph, you can superimpose the curve you found on the plot of your data to see whether the curve fits the data well or not.</p>
</text>
</region>
<region region-id="3772" left="0" top="3102" width="6000" height="6" align-x="0" align-y="3102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3770" left="6" top="3108.75" width="158.25" height="14.25" align-x="6" align-y="3120" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Start with a plot of your data:</p>
</text>
</region>
<region region-id="3758" left="24" top="3132" width="248.25" height="214.5" align-x="24" align-y="3132" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<plot disable-calc="false" item-idref="19"></plot>
<rendering item-idref="20"></rendering>
</region>
<region region-id="3808" left="6" top="3360.75" width="265.5" height="14.25" align-x="6" align-y="3372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Follow these instructions to add a curve plot to your graph:</p>
</text>
</region>
<region region-id="3809" left="6" top="3384.75" width="336.75" height="144.75" align-x="6" align-y="3396" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">1.</c>
</b>
<sp count="2"></sp>Click in the y-axis placeholder of your plot and press <f family="Verdana" charset="0" size="10">
<b>[Spacebar]</b>
</f> until the blue editing lines enclose the argument already there.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">2.</c>
</b>
<sp count="2"></sp>Type <f family="Verdana" charset="0" size="10">
<b>[</b>
</f>
<f family="Verdana" charset="0" size="10">
<b>,]</b>
</f> for a new placeholder, and enter the name of the function representing the curve that fits your data, for example, <region region-id="3668" left="265.5" top="3428.25" width="21" height="15.75" align-x="270.75" align-y="3439.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:id xml:space="preserve">g</ml:id>
<ml:id xml:space="preserve">x</ml:id>
</ml:apply>
</math>
<rendering item-idref="21"></rendering>
</region>.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">3.</c>
</b>
<sp count="2"></sp>Click in the x-axis placeholder and follow the same steps, but this time enter the name of the independent variable of your function, for example, x.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">4.</c>
</b>
<sp count="2"></sp>You may notice that the default range of the independent variable does not sufficiently cover the data. In which case, define an independent range variable above the graph such as <f family="Verdana" charset="0" size="10">
<b>x:0;40</b>
</f>.</p>
</text>
</region>
<region region-id="3782" left="6" top="3540.75" width="219" height="14.25" align-x="13.5" align-y="3552" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try to follow these steps using the graph, above.</p>
</text>
</region>
<region region-id="3787" left="0" top="3570" width="6000" height="6" align-x="0" align-y="3570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3793" left="6" top="3578.25" width="47.25" height="12" align-x="24.75" align-y="3588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Example</p>
</text>
</region>
<region region-id="3786" left="6" top="3594.75" width="341.25" height="28.5" align-x="6" align-y="3606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">In the previous example, <b>time</b> and <b>population</b> were defined. Use them again here and perform a least squares fit on the data.</p>
</text>
</region>
<region region-id="3482" left="18" top="3636.75" width="123.75" height="15.75" align-x="34.5" align-y="3648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">fit</ml:id>
<ml:apply>
<ml:id xml:space="preserve">line</ml:id>
<ml:sequence>
<ml:id xml:space="preserve">time</ml:id>
<ml:id xml:space="preserve">population</ml:id>
</ml:sequence>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="22"></rendering>
</region>
<region region-id="3486" left="18" top="3662.25" width="75" height="36.75" align-x="33.75" align-y="3684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">fit</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="2" cols="1">
<ml:real>-38</ml:real>
<ml:real>6.2071428571428573</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="23"></rendering>
</region>
<region region-id="3488" left="6" top="3708.75" width="342" height="57" align-x="17.25" align-y="3720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now, these results can be used in other definitions or referenced directly in calculations or a graph. Be sure to use an array subscript to refer to these values. For example, here we define <b>b</b> and <b>m</b> as the first and second values in the vector.</p>
</text>
</region>
<region region-id="3501" left="18" top="3780.75" width="38.25" height="17.25" align-x="32.25" align-y="3792" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">b</ml:id>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">fit</ml:id>
<ml:real>0</ml:real>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="24"></rendering>
</region>
<region region-id="3502" left="84" top="3780.75" width="48.75" height="14.25" align-x="86.25" align-y="3792" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">y-intercept</p>
</text>
</region>
<region region-id="3503" left="168" top="3780.75" width="40.5" height="17.25" align-x="184.5" align-y="3792" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define warning="WarnRedefinedBIUnit">
<ml:id xml:space="preserve">m</ml:id>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">fit</ml:id>
<ml:real>1</ml:real>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="25"></rendering>
</region>
<region region-id="3504" left="234" top="3780.75" width="24" height="14.25" align-x="245.25" align-y="3792" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">slope</p>
</text>
</region>
<region region-id="3507" left="6" top="3810.75" width="276" height="14.25" align-x="16.5" align-y="3822" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Then, you can define a function that uses these values:</p>
</text>
</region>
<region region-id="3508" left="18" top="3834.75" width="78.75" height="15.75" align-x="45.75" align-y="3846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:function>
<ml:id xml:space="preserve">f</ml:id>
<ml:boundVars>
<ml:id xml:space="preserve">x</ml:id>
</ml:boundVars>
</ml:function>
<ml:apply>
<ml:plus></ml:plus>
<ml:apply>
<ml:mult style="dot"></ml:mult>
<ml:id xml:space="preserve">m</ml:id>
<ml:id xml:space="preserve">x</ml:id>
</ml:apply>
<ml:id xml:space="preserve">b</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="26"></rendering>
</region>
<region region-id="3512" left="6" top="3864.75" width="282" height="14.25" align-x="15" align-y="3876" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">And, view that function and the original data in the same graph:</p>
</text>
</region>
<region region-id="3513" left="18" top="3894" width="255" height="214.5" align-x="18" align-y="3894" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<plot disable-calc="false" item-idref="27"></plot>
<rendering item-idref="28"></rendering>
</region>
<region region-id="3517" left="6" top="4122.75" width="341.25" height="28.5" align-x="9.75" align-y="4134" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Is a straight line a good fit for this data? Not really. In the next section, we fit this data to an exponential function to see if that does a better job.</p>
</text>
</region>
<region region-id="3788" left="0" top="4164" width="6000" height="6" align-x="0" align-y="4164" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3703" left="6" top="4172.25" width="207" height="12" align-x="6" align-y="4182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="special fit">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Perform a Specialized Curve Fit</p>
</text>
<indexes>
<phrase>curve fit, specialized</phrase>
<phrase>nonlinear least squares regression</phrase>
<phrase>least squares regression, nonlinear</phrase>
</indexes>
</region>
<region region-id="3545" left="6" top="4188.75" width="341.25" height="57" align-x="6" align-y="4200" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">There are built-in curve fitting functions for the most common model functions: exponential, logistic, logarithmic, sinusoidal, and power. In addition, you can use one of the generalized curve fitting functions to specify your own model function.</p>
</text>
</region>
<region region-id="3550" left="6" top="4260.75" width="341.25" height="93" align-x="6" align-y="4272" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here are instructions for using the <b>expfit</b> function to get the coefficients for the exponential model function, <region region-id="3552" left="148.5" top="4275.75" width="79.5" height="21" align-x="176.25" align-y="4292.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:equal style="equal"></ml:equal>
<ml:apply>
<ml:id xml:space="preserve">g</ml:id>
<ml:id xml:space="preserve">x</ml:id>
</ml:apply>
<ml:apply>
<ml:plus></ml:plus>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:id xml:space="preserve">a</ml:id>
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">e</ml:id>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:id xml:space="preserve">b</ml:id>
<ml:id xml:space="preserve">x</ml:id>
</ml:apply>
</ml:apply>
</ml:apply>
<ml:id xml:space="preserve">c</ml:id>
</ml:apply>
</ml:apply>
</math>
<rendering item-idref="29"></rendering>
</region>. Most of the other specialized curve fitting functions work the same way. For more information about how to use the other curve fitting functions, see the <b>
<u>
<link href=".\..\..\Data_Analysis\fittoc.xmcd" title="Links to section" popup="false">Data Analysis QuickSheets</link>
</u>
</b>. In particular, see <b>
<u>
<link href=".\..\..\Data_Analysis\linfitlo.xmcd" title="Links to section" popup="false">Using Linfit: Logarithmic Regression</link>
</u>
</b> and <b>
<u>
<link href=".\..\..\Data_Analysis\genfitpo.xmcd" title="Links to section" popup="false">Using Genfit: Power Regression</link>
</u>
</b>.</p>
</text>
</region>
<region region-id="3548" left="6" top="4376.25" width="169.5" height="12" align-x="24.75" align-y="4386" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="expfit function">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The expfit Function</p>
</text>
<indexes>
<phrase>expfit function</phrase>
</indexes>
</region>
<region region-id="3549" left="6" top="4398.75" width="340.5" height="242.25" align-x="6" align-y="4410" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">1.</c>
</b>
<sp count="2"></sp>In a blank spot below your data tables, first, define a vector containing your guesses for the three coefficients. To define this vector, type <f family="Verdana" charset="0" size="10">
<b>guess:</b>
</f>.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">2.</c>
</b>
<sp count="2"></sp>For the right side of the definition, type <f family="Verdana" charset="0" size="10">
<b>[Ctrl]m</b>
</f> to get the Insert Matrix dialog box and specify 3 rows and 1 column. In the vector placeholders, enter guesses for the three coefficients. Enter any values to start; refine them later. <b>
<c val="#008080">Note:</c>
</b> This argument is optional for <b>expfit</b>.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<c val="#008080">
<b>3.</b>
</c>
<sp count="2"></sp>To define the fit, enter the name of the variable that will contain the results of the fit followed by the definition equals; for example, type <f family="Verdana" charset="0" size="10">
<b>efit:</b>
</f>
</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">4.</c>
</b>
<sp count="2"></sp>For the right side of the definition, choose <b>Function</b> from the <b>Insert</b> menu to get the Insert Function dialog box. In the list of all functions, click on "expfit" and then click OK.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">5.</c>
</b>
<sp count="2"></sp>In the three placeholders provided, enter the names of the independent data variable, the dependent data variable, and the guess vector, in that order, and press <f family="Verdana" charset="0" size="10">
<b>[Enter]</b>
</f>.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">6.</c>
</b>
<sp count="2"></sp>To see the results of the fit, type <f family="Verdana" charset="0" size="10">
<b>efit=</b>
</f>. The three coefficients are given in the order they are specified in the model function.</p>
</text>
</region>
<region region-id="3789" left="0" top="4668" width="6000" height="6" align-x="0" align-y="4668" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3799" left="6" top="4676.25" width="47.25" height="12" align-x="6" align-y="4686" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Example</p>
</text>
</region>
<region region-id="3570" left="6" top="4692.75" width="342" height="28.5" align-x="6" align-y="4704" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">In the previous example, <b>time</b> and <b>population</b> were defined. Use them again here and perform a fit to the exponential model function.</p>
</text>
</region>
<region region-id="3571" left="18" top="4740.75" width="138.75" height="15.75" align-x="39.75" align-y="4752" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">efit</ml:id>
<ml:apply>
<ml:id xml:space="preserve">expfit</ml:id>
<ml:sequence>
<ml:id xml:space="preserve">time</ml:id>
<ml:id xml:space="preserve">population</ml:id>
</ml:sequence>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="30"></rendering>
</region>
<region region-id="3572" left="18" top="4773.75" width="87" height="57.75" align-x="39" align-y="4806" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">efit</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="1">
<ml:real>12.195070616037935</ml:real>
<ml:real>0.086105517104660412</ml:real>
<ml:real>-15.270321087400125</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="31"></rendering>
</region>
<region region-id="3573" left="6" top="4848.75" width="368.25" height="42.75" align-x="17.25" align-y="4860" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Again, these results can be used in other definitions or referenced directly in calculations or a graph. The definition you most likely need to create is the model function using your results. First, we define the three coefficients:</p>
</text>
</region>
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<pageBreak></pageBreak>
</region>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Then, define the function:</p>
</text>
</region>
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<ml:function>
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<ml:id xml:space="preserve">x</ml:id>
</ml:boundVars>
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<ml:apply>
<ml:plus></ml:plus>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:id xml:space="preserve">a</ml:id>
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">e</ml:id>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:id xml:space="preserve">b</ml:id>
<ml:id xml:space="preserve">x</ml:id>
</ml:apply>
</ml:apply>
</ml:apply>
<ml:id xml:space="preserve">c</ml:id>
</ml:apply>
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<rendering item-idref="35"></rendering>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now, view that function and the original data in the same graph:</p>
</text>
</region>
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<plot disable-calc="false" item-idref="36"></plot>
<rendering item-idref="37"></rendering>
</region>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This is a much better function to fit to the data, as you can see from the graph. It is typical, however, to do some sort of error analysis on the data and the fit to determine exactly how well the curve fits the data. </p>
</text>
</region>
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<pageBreak></pageBreak>
</region>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Analyze the Quality of the Fit</p>
</text>
</region>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Plot the Data and the Curve on the Same 2D Graph</p>
</text>
<indexes>
<phrase>plotting, data and curve</phrase>
</indexes>
</region>
<region region-id="3624" left="6" top="5358.75" width="340.5" height="14.25" align-x="18" align-y="5370" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now check to see if this function curve fits the data well or not.</p>
</text>
</region>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">With the results of your analysis:</p>
</text>
</region>
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<ml:id xml:space="preserve">efit</ml:id>
<ml:apply>
<ml:id xml:space="preserve">expfit</ml:id>
<ml:sequence>
<ml:id xml:space="preserve">time</ml:id>
<ml:id xml:space="preserve">population</ml:id>
</ml:sequence>
</ml:apply>
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<rendering item-idref="38"></rendering>
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<ml:real>0.086105517104660412</ml:real>
<ml:real>-15.270321087400125</ml:real>
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<math optimize="false" disable-calc="false">
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<ml:id xml:space="preserve">b</ml:id>
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<ml:indexer></ml:indexer>
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<ml:real>1</ml:real>
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<rendering item-idref="41"></rendering>
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<math optimize="false" disable-calc="false">
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<ml:id xml:space="preserve">c</ml:id>
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<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">efit</ml:id>
<ml:real>2</ml:real>
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<rendering item-idref="42"></rendering>
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<math optimize="false" disable-calc="false">
<ml:define warning="WarnRedefinedUDFunction">
<ml:function>
<ml:id xml:space="preserve">g</ml:id>
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<ml:id xml:space="preserve">x</ml:id>
</ml:boundVars>
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<ml:plus></ml:plus>
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<ml:id xml:space="preserve">a</ml:id>
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">e</ml:id>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:id xml:space="preserve">b</ml:id>
<ml:id xml:space="preserve">x</ml:id>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">And a plot of your data:</p>
</text>
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<plot disable-calc="false" item-idref="44"></plot>
<rendering item-idref="45"></rendering>
</region>
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<pageBreak></pageBreak>
</region>
<region region-id="3631" left="6" top="5802.75" width="265.5" height="14.25" align-x="6" align-y="5814" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Follow these instructions to add a curve plot to your graph:</p>
</text>
</region>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">1.</c>
</b>
<sp count="2"></sp>Click in the y-axis placeholder of your plot and press <f family="Verdana" charset="0" size="10">
<b>[Spacebar]</b>
</f> until the blue editing lines enclose the argument already there.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">2.</c>
</b>
<sp count="2"></sp>Type <f family="Verdana" charset="0" size="10">
<inlineAttr font-weight="normal">[</inlineAttr>
</f>
<f family="Verdana" charset="0" size="10">
<b>,</b>
</f>
<f family="Verdana" charset="0" size="10">
<inlineAttr font-weight="normal">]</inlineAttr>
</f> for a new placeholder and enter the name of the function representing the curve that fits your data, say, <region region-id="3771" left="227.25" top="5870.25" width="21" height="15.75" align-x="232.5" align-y="5881.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:id xml:space="preserve">g</ml:id>
<ml:id xml:space="preserve">x</ml:id>
</ml:apply>
</math>
<rendering item-idref="46"></rendering>
</region>.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">3.</c>
</b>
<sp count="2"></sp>Click in the x-axis placeholder and follow the same steps, this time entering the name of the independent variable of your function, say, x.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">4.</c>
</b>
<sp count="2"></sp>You may notice that the default range of the independent variable does not sufficiently cover the data. In which case, define an independent range variable above the graph such as <f family="Verdana" charset="0" size="10">
<b>x:0;40</b>
</f>.</p>
</text>
</region>
<region region-id="3694" left="6" top="5970.75" width="219" height="14.25" align-x="13.5" align-y="5982" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try following these steps using the graph, above.</p>
</text>
</region>
<region region-id="3610" left="6" top="6002.25" width="157.5" height="12" align-x="17.25" align-y="6012" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="correlation">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Calculate Correlation</p>
</text>
<indexes>
<phrase>correlation, calculating</phrase>
</indexes>
</region>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">One way to decide whether the function is a good fit for the data is by calculating a correlation. To calculate a correlation:</p>
</text>
</region>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">1.</c>
</b>
<sp count="2"></sp>In a blank spot below the results of your analysis, enter the built-in correlation function by choosing <b>Function</b> from the <b>Insert</b> menu and selecting "corr" from the function list.</p>
<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">2.</c>
</b>
<sp count="2"></sp>The <b>corr</b> function takes two arguments, the function evaluated over the set of x-coordinate data and the y-coordinate data. So, for example, enter your corr function as <f family="Verdana" charset="0" size="10">
<b>corr(time,population)=</b>
</f> for the example above.</p>
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<c val="#008080">3.</c>
</b>
<sp count="2"></sp>The closer this result is to 1, the better the fit.</p>
</text>
</region>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try it, below:</p>
</text>
</region>
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<picture>
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