<sect1 id="sect-stat-analysis-overview">
  <title>Overview</title>

  <para>
    All tools have the same output options (see <xref
    linkend="fig-outputoptions" />).  The results can be printed into a
    new sheet, into a new workbook, or into a given output range on a
    sheet of the current workbook.  To select the output method select
    one of the radio buttons inside the <guilabel>Output</guilabel>
    frame. If you have chosen <quote><guibutton>Output
    Range</guibutton></quote> you must also enter a single range in
    the entry field. 
  </para>

  <para>Select the <guilabel>Autofit
    Columns</guilabel> option to automatically adjust the widths of
    the columns in the output range.
  </para> 

  <para>You will normally want to select the <guilabel>Clear
    Output Range</guilabel> option, since otherwise some of the cells with 
    existing content will remain in the output range.
  </para> 

  <para> The <guilabel>Retain Output Range Formatting</guilabel> and 
    <guilabel>Retain Output Range Comments</guilabel> options are useful 
    if you have already preformatted the output range.
  </para>

  <para>All analysis tools also provide a choice whether
    they will enter formul&#xe6; or just values in the cells. By default 
    &gnum; will usually enter formul&#xe6;. These formul&#xe6; will automatically
    reevaluate when the data change. For some tools, the formul&#xe6; also
    permit modification of certain parameters. 
  </para> 
  

  <note>
    <para>
      If the chosen output range is too small, some of the results
      will be lost.
    </para>
  </note>

  <note>
    <para>
      The old data in the output range is deleted and cannot be
      recovered.
    </para>
  </note>

  <figure id="fig-outputoptions">
    <title>Common output options of the data analysis tools</title>
    <screenshot id="outputoptions-shot">
      <mediaobject>
        <imageobject>
          <imagedata fileref="figures/analysistools-outputoptions.png" 
              format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output options dialog used by
              the statistical analysis tools.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

  <para>
    To enter a range into an entry field, you can either type the
    range specification into the text field, or click in the text
    field and then select the range on the sheet (see <xref
    linkend="specifyingranges" />).
  </para>

  <figure id="specifyingranges">
    <title>Specifying Ranges</title>
    <screenshot>
      <mediaobject>
        <imageobject>
          <imagedata fileref="figures/analysistools-ranges.png" format="PNG" />
      </imageobject>
            <textobject>
              <phrase>An image of the input range text box used by the
              statistical analysis tools.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

  <para>
    Some entry fields accept lists of ranges. To enter these lists,
    select one range, type a comma, and then select the next range. At
    any time, you may switch to another sheet of the workbook.
  </para>

</sect1>

<sect1 id="descriptive_statistics">
  <title>Descriptive Statistics</title>
  <sect2 id="correlation-tool">
     <title>Correlation Tool</title>

  <figure id="correlation-tool-dialog">
    <title>Correlation Tool Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-correlation.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the correlation analysis dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <para>The correlation tool calculates the pairwise Pearson 
     correlation coefficients of the
     given variables.  Use this tool to calculate any number of
     correlation coefficients at the same time.  The variables for
     which the correlations are calculated are specified by the <quote><guilabel>Input
     Range:</guilabel></quote> entry. The input range can consist of either a single 
     range or a comma separated list of ranges. The given range or 
     ranges can be grouped by columns, by rows, or by areas.</para>

     <para>If the first row or column of the given ranges, or the 
     first field of each area contains labels,  the
     <quote><guibutton>Labels</guibutton></quote> option should be selected.
     </para>

  <figure id="correlation-example-1">
    <title>Some Example Data</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-correlation-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of an example data set for a
              correlation analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

 <example id="usingcorrelationtool">
 <title>Using the Correlation Tool</title>
    <para>For example, you want to calculate the correlation between
     three variables, one each in columns A, B, and C.
     Both variables have 10 values in rows 2 to 11 with labels in row 1
     (see <xref linkend="correlation-example-1" />).</para>
<orderedlist>
     <listitem><para>
     Enter A1:B11 in the <quote><guilabel>Input Range:</guilabel></quote> entry by typing 
     this directly into the entry or clicking in the entry field and 
     then selecting that range on the sheet. In the latter case the 
     entry will also contain the sheet name. </para></listitem>
     <listitem><para>
     Select the <quote><guibutton>Columns</guibutton></quote> radio button next to 
     <quote><guilabel>Grouped By:</guilabel></quote>, 
     since each variable is in its own column.</para></listitem>
     <listitem><para> Select the <quote><guibutton>Labels</guibutton></quote>
     option since the first row contains labels. (see 
     <xref linkend="correlation-example-2" />).</para></listitem>
     <listitem><para> Specify the output 
     options as described above.</para></listitem>
     <listitem><para> Press the OK button. </para></listitem>
</orderedlist>
     <para>The calculated correlations are given in a table with each column and
     row labeled with the names of the variables.  If the
     names are not given in the input range, &gnum; generates them.
     In our example, the 
     correlation between the variables in column A and B, can be found
     in the second column and third row of the results table (see 
     <xref linkend="correlation-example-3" />).</para>
 </example>
  <figure id="correlation-example-2">
    <title>Completed Correlation Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-correlation-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the completed correlation analysis
              dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  <figure id="correlation-example-3">
    <title>Correlation Tool Output</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-correlation-ex3.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output of the correlation
              analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect2>

  <sect2 id="covariance-tool">
     <title>Covariance Tool</title>

  <figure id="covariance-tool-dialog">
    <title>Covariance Tool Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-covariance.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the covariance analysis
              dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <para>The covariance tool calculates the pairwise 
     covariance coefficients of the
     given variables.  Use this tool to calculate any number of
     covariance coefficients at the same time.  The variables for
     which the covariances are calculated are specified by the <quote><guilabel>Input
     Range:</guilabel></quote> entry. The input range can consist of either a single 
     range or a comma separated list of ranges. The given range or 
     ranges can be grouped by columns, by rows, or by areas.</para>

     <para>If the first row or column of the given ranges, or the 
     first field of each area contains labels,  the
     <quote><guibutton>Labels</guibutton></quote> option should be selected.
     </para>

  <figure id="covariance-example-1">
    <title>Some Example Data</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-covariance-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image example data for a covariance
              analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

 <example id="usingcovariancetool">
 <title>Using The Covariance Tool</title>
    <para>For example, you want to calculate the covariance between
     three variables, one each in columns A, B, and C.
     Both variables have 10 values in rows 2 to 11 with labels in row 1
     (see <xref linkend="covariance-example-1" />).</para>
<orderedlist>
     <listitem><para>
     Enter A1:B11 in the <quote><guilabel>Input Range:</guilabel></quote> entry by typing 
     this directly into the entry or clicking in the entry field and 
     then selecting that range on the sheet. In the latter case the 
     entry will also contain the sheet name. </para></listitem>
     <listitem><para>
     Select the <quote><guibutton>Columns</guibutton></quote> radio button next to 
     <quote><guilabel>Grouped By:</guilabel></quote>, 
     since each variable is in its own column.</para></listitem>
     <listitem><para> Select the <quote><guibutton>Labels</guibutton></quote>
     option since the first row contains labels.
     </para></listitem>
     <listitem><para> Specify the output 
     options as described above.</para></listitem>
     <listitem><para> Press the OK button. </para></listitem>
</orderedlist>
     <para>The calculated covariances are given in a table with each column and
     row labeled with the names of the variables.  If the
     names are not given in the input range, &gnum; generates them.
     In our example, the 
     covariance between the variables in column A and B, can be found
     in the second column and third row of the results table (see 
     <xref linkend="covariance-example-2" />).</para>
 </example>
  <figure id="covariance-example-2">
    <title>Covariance Tool Output</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-covariance-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output of a covariance analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect2>


  <sect2 id="descriptive-statistics-tool">
     <title>Descriptive Statistics Tool</title>

  <figure id="descriptive-statistics-tool-dialog">
    <title>Descriptive Statistics Tool Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-descstats.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the descriptive statistics dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <para>The descriptive statistics tool calculates various statistics 
     for the given variables and a confidence interval for the population
     mean. The variables are specified via the <quote><guilabel>Input
     Range:</guilabel></quote> entry.  The given range or list of ranges can be grouped into 
     variables by columns, rows, or areas.</para>

     <para>This tool can produce four different kinds of statistical
     data.
     </para>
     <itemizedlist>
     <listitem><para>If the <quote><guibutton>Summary Statistics</guibutton></quote> option is selected,  
     this tool calculates the
     mean, standard error, median, mode, standard deviation, sample
     variance, kurtosis, skewness, range, minimum, maximum, sum, and
     count for each variable.</para>
     </listitem>
     <listitem><para>If the <quote><guibutton>Confidence Interval for the Mean</guibutton></quote> option is 
     selected, the tool calculates  confidence intervals for the population
     mean of each variable.
     Specify the confidence level in the entry box.  The default confidence 
     level is 95&#037;.</para> 

     <note><para>The interval given will usually be wider than the 
     interval obtained using the CONFIDENCE function. The CONFIDENCE function
     assumes that the population standard deviation is known. This tool
     estimates the population standard deviation using the sample standard
     deviation.</para></note></listitem>

     <listitem><para>If the <quote><guibutton>Kth Largest:</guibutton></quote> option is selected, the tool finds
     the <parameter>k</parameter>th largest value of each of the variables.  Specify 
     <parameter>k</parameter> in
     the entry box next to the option. The default is 1.
     </para></listitem>

     <listitem><para>If the <quote><guibutton>Kth Smallest:</guibutton></quote> option is selected, the tool finds
     the <parameter>k</parameter>th smallest value of each of the variables.  Specify 
     <parameter>k</parameter> in
     the entry box next to the option. The default is 1.
     </para></listitem>
     </itemizedlist>
     <para>If the first entry for each variable contains the label,
     select the <quote><guibutton>Labels</guibutton></quote> option.
     </para>
  <figure id="descstats-example-1">
    <title>Some Example Data</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-descstats-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of some example data for descriptive
              statistics.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
 <example id="usingdescstatstool"><title>Using the Descriptive Statistics Tool</title>
     <para><xref linkend="descstats-example-1" /> shows some example data, 
     <xref linkend="descstats-example-1-options" /> the selected options, and 
     <xref linkend="descstats-example-2" /> the corresponding output.
     </para>
</example>
  <figure id="descstats-example-1-options">
    <title>The Options Page For Descriptive Statistics</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-descstats-ex1-options.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of some example data for descriptive
              statistics.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  <figure id="descstats-example-2">
    <title>Descriptive Statistics Tool Output</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-descstats-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output of a descriptive
              statistics analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect2>

  <sect2 id="frequencies-tools">
      <title>Frequency Tables</title>
      <para>
	&gnum; provides two types of frequencies tables:
      </para>
     <itemizedlist>
     <listitem><para>The frequency table tools is primarily useful for non-numeric data 
	 (data of nominal and ordinal level of measurement). It allows to determine 
	 frequencies for given values.
     </para></listitem>
     <listitem><para> The histogram tool is useful for numeric data that is supposed to be 
	 classified into a certain number of intervals. These intervals can be either 
	 specified or calculated.
     </para></listitem>
     </itemizedlist>

    <sect3 id="frequency-tool">
      <title>Frequency Tables Tool</title>

      <sect4 id="frequency-tool-intro">
	<title>Introduction</title>
	
	<para> The frequency tool can be used to create frequency tables for 
	  non-numerical data.  It presents this table 
	  numerically as well as graphically.
	</para>
	
	<note><para>
	    If your data are numeric and you want to accumulate  whole intervals of values into
	    frequency counts then this tool is not appropriate. In that case you may 
	    want to use the histogram table tool described in section <xref linkend="histogram-tool" />.
	</para></note>
	
	<figure id="frequency-tool-dialog">
	  <title>Frequency Tool Dialog</title>
	  <screenshot>
	    <mediaobject>
              <imageobject>
		<imagedata fileref="figures/analysistools-frequency.png" 
			   format="PNG" />
              </imageobject>
              <textobject>
		<phrase>An image of the dialog to generate various
		  frequency tables open to the "Input" tab.</phrase>
              </textobject>
            </mediaobject>
	  </screenshot>
	</figure>
	
	<para>As shown in <xref linkend="frequency-tool-dialog" />, the
	  frequency table dialog has four tabs. We will introduce them in
	  sequence.
	</para>
      </sect4>
      
      <sect4 id="frequency-tool-inputtab">
	<title>The <quote><guilabel>Input</guilabel></quote> Tab</title>
	
	<para>The <quote><guilabel>Input</guilabel></quote> tab shown in 
	  <xref linkend="frequency-tool-dialog" /> contains
	  the field specifying the data to be used for the
	  histogram.</para>
	<para>
	  The <quote><guilabel>Input Range</guilabel></quote> entry
	  contains a single range or a list of ranges, that can be grouped
	  into variables by rows, columns, or areas.
	</para>
	<para>If the first row or column of the given input ranges, or
	  the first field of each area contains labels, the
	  <quote><guibutton>Labels</guibutton></quote> option should
	  be selected.
	  If the input is grouped by areas and the top left cell contains a label, the
	  other cells in the first row are being ignored.
	</para>
      </sect4>
      
      <sect4 id="frequency-tool-catstab">
	<title>The <quote><guilabel>Categories</guilabel></quote> Tab</title>
	
	<para>The <quote><guilabel>Categories</guilabel></quote> tab permits the specification
	  of a range that contains the possible values that are supposed to be counted in the 
	  input range.
	</para>
	
	<note><para>The <quote><guilabel>automatic categories</guilabel></quote> option is 
	    disabled since it is not yet implemented.
	</para></note>

	<figure id="frequency-tool-dialog-cats">
	  <title>Frequency Tool Dialog Categories Tab</title>
	  <screenshot>
	    <mediaobject>
              <imageobject>
		<imagedata fileref="figures/analysistools-frequency-cats.png" 
			   format="PNG" />
              </imageobject>
              <textobject>
		<phrase>An image of the dialog to generate various
		  frequency tables open to the "Categories" tab.</phrase>
              </textobject>
            </mediaobject>
	  </screenshot>
	</figure>
      </sect4>
      
      
      <sect4 id="frequency-tool-optionstab">
	<title>The <quote><guilabel>Graphs &amp;  Options</guilabel></quote> Tab</title>
	
	<para>The <quote><guilabel>Graphs &amp;  Options</guilabel></quote> tab allows various 
	  options to be set. In the top half of the tab you can choose whether you would like 
	  a graph to be created. If you choose to have a graph created you can specify whether 
	  you would like to see a bar chart or a column chart. 	
	</para>
	<para>In the bottom part of the tab you 
	  can select the  <quote><guilabel>percentages</guilabel></quote> option. This option 
	  replaces the frequency counts with percentages.
	</para>
	<note><para>If the categories range contains repeated values, then the percentages may
	    add up to more than 100%. If the the categories range does not contain all values that 
	    occur in the input range, the percentages may sum to less than 100%.
	</para></note>
	<para>The <quote><guilabel>Use exact comparisons</guilabel></quote> checkbox determines how 
	  category values and input range values are compared. If it is checked then the function 
	  <function>EXACT</function> is used for the comparison. If it isn't checked then simple
	  equality is used. In this latter case, empty cells and cells containing the numerical 
	  value 0 are considered equal. As a consequence you usually want that checkbox to be selected. 
	</para>
	

	<figure id="frequency-tool-dialog-graphs">
	  <title>Frequency Tool Dialog Graphs &amp;  Options Tab</title>
	  <screenshot>
	    <mediaobject>
              <imageobject>
		<imagedata fileref="figures/analysistools-frequency-graphs.png" 
			   format="PNG" />
              </imageobject>
              <textobject>
		<phrase>An image of the dialog to generate various
		  frequency tables open to the "Graphs &amp;  Options" tab.</phrase>
              </textobject>
            </mediaobject>
	  </screenshot>
	</figure>
      </sect4>
      
      
      <sect4 id="frequency-tool-results-sect">
	<title>Frequency Tool Results</title>
	<figure id="frequency-tool-results">
	  <title>Frequency Tool Results</title>
	  <screenshot>
	    <mediaobject>
	      <imageobject>
		<imagedata fileref="figures/analysistools-frequency-results.png" 
			   format="PNG" />
	      </imageobject>
	      <textobject>
		<phrase>Sample results of the frequencies tool.</phrase>
	      </textobject>
	    </mediaobject>
	  </screenshot>
	</figure>
      </sect4>
    </sect3>
    
    <sect3 id="histogram-tool">
      <title>Histogram Tool</title>
      
      <sect4 id="histogram-tool-intro">
	<title>Introduction</title>
	
	
	<para> The histogram tool can be used to create histograms or frequency tables for 
	  numerical data. Using this tool you can define intervals, or <quote>bins</quote>. 
	  The tool determines how many data points belong to each bin and presents this number 
	  numerically as well as graphically.
	</para>
	
	<note><para>
	    If your data are non-numeric this tool is not appropriate. In that case you may 
	    want to use the frequency table tool described in section <xref linkend="frequency-tool" />.
	</para></note>
	
	<figure id="histogram-tool-dialog">
	  <title>Histogram Tool Dialog</title>
	  <screenshot>
	    <mediaobject>
              <imageobject>
		<imagedata fileref="figures/analysistools-histogram.png" 
			   format="PNG" />
              </imageobject>
              <textobject>
		<phrase>An image of the dialog to generate various
		  histograms open to the "Input" tab.</phrase>
              </textobject>
            </mediaobject>
	  </screenshot>
	</figure>
	
	<para>As shown in <xref linkend="histogram-tool-dialog" />, the
	  histogram dialog has five tabs. We will introduce them in
	  sequence.
	</para>
      </sect4>

  <sect4 id="histogram-tool-inputtab">
     <title>The <quote><guilabel>Input</guilabel></quote> Tab</title>

     <para>The <quote><guilabel>Input</guilabel></quote> tab shown in 
       <xref linkend="histogram-tool-dialog" /> contains
       the field specifying the data to be used for the
       histogram.</para>
     <para>
       The <quote><guilabel>Input Range</guilabel></quote> entry
       contains a single range or a list of ranges, that can be grouped
       into variables by rows, columns, or areas.
     </para>
     <para>If the first row or column of the given input ranges, or
       the first field of each area contains labels, the
       <quote><guibutton>Labels</guibutton></quote> option should
       be selected.
       If the input is grouped by areas and the top left cell contains a label, the
       other cells in the first row are being ignored.
     </para>
  </sect4>
  
  <sect4 id="histogram-tool-cutoffsstab">
     <title>The <quote><guilabel>Cutoffs</guilabel></quote> Tab</title>
  <figure id="histogram-tool-dialog-cutoffs">
    <title>Histogram Tool Dialog <quote><guilabel>Cutoffs</guilabel></quote> Tab</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-histogram-cutoffs.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the dialog to generate various
              histograms open to the "Cutoffs" tab.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

  <para>The cutoffs for the histogram can either be predetermined by data 
    contained in your workbook or calculated by the histogram tool.  These cutoffs 
    determine bins as defined
    by the selection on the <quote><guilabel>Bins</guilabel></quote> tab.
  </para>

  <para>Select the <quote><guilabel>Predetermined Cutoffs</guilabel></quote> option to specify 
    data on your worksheet in the <quote><guilabel>Cutoff Range:</guilabel></quote> entry. The 
    values in this range will be used as cutoffs <parameter>c<subscript>1</subscript></parameter>,
    <parameter>c<subscript>2</subscript></parameter>, and so on 
    to  <parameter>c<subscript>n</subscript></parameter>.
  </para> 

  <para>Select the <quote><guilabel>Calculated Cutoffs</guilabel></quote> option to have the 
    cutoffs determined by the tool. Enter the desired number of cutoffs in the 
    <quote><guilabel>Number of Cutoffs</guilabel></quote> entry. It is strongly recommended 
    (but optional) that you 
    specify the minimum and maximum cutoffs in the <quote><guilabel>Minimum cutoff</guilabel></quote>
    and <quote><guilabel>Maximum cutoff</guilabel></quote> entries. If the minimum or maximum
    cutoff is not specified, the tool will use the minimum and/or maximum of the current data. 
  </para>
  </sect4>
  
  <sect4 id="histogram-tool-binstab">
     <title>The <quote><guilabel>Bins</guilabel></quote> Tab</title>
  <figure id="histogram-tool-dialog-bins">
    <title>Histogram Tool Dialog <quote><guilabel>Bins Tab</guilabel></quote></title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-histogram-bins.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the dialog to generate various
              histograms open to the "Bins" tab.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

  <para> The bins tab is used to determine how the cutoffs <parameter>c<subscript>1</subscript></parameter>,
    <parameter>c<subscript>2</subscript></parameter>, and so on 
    to  <parameter>c<subscript>n</subscript></parameter> are translated into bins. Specifically, 
    it has to be determined whether first and/or last bins reaching from &#x2212;&#x221e; to 
    <parameter>c<subscript>1</subscript></parameter> and from 
    <parameter>c<subscript>n</subscript></parameter> to &#x221e; are added and whether data points that much
    cutoffs exactly are included in the bin to the right or the left.
  </para>
  <para> For example the option  
    <quote><guilabel>[&#x2219;,&#x2219;),[&#x2219;,&#x2219;),&#x22ef;,
	[&#x2219;,&#x2219;),[&#x2219;,&#x221e;)
    </guilabel></quote>
    indicates that the first bin starts at the first cutoff while the last bin ends at &#x221e;. Moreover,
    each cutoff value belongs to the bin on its right.
  </para>
    </sect4>

  <sect4 id="histogram-tool-optionstab">
     <title>The <quote><guilabel>Graphs &amp; Options</guilabel></quote> Tab</title>
 <figure id="histogram-tool-dialog-options">
    <title>Histogram Tool Dialog <quote><guilabel>Graphs &amp; Options Tab</guilabel></quote></title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-histogram-graphs.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the dialog to generate various
              histograms open to the "Graphs &amp; Options" tab.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <para> The options in the graphs and options tab specify any graph to be created and modify 
       the appearance of the histogram:</para>

     <itemizedlist>
     <listitem>
     <para> The <quote><guibutton>No chart</guibutton></quote> option causes the chart to be omitted.
     </para>
     </listitem>
     <listitem>
     <para> The <quote><guibutton>Bar chart</guibutton></quote> option causes a bar chart to be 
       added to the histogram. For each bin, the bar chart shows a horizontal bar indicating the frequency.
     </para>
     <para> The <quote><guibutton>Column chart</guibutton></quote> option causes a column chart to be 
       added to the histogram. For each bin, the column chart shows a vertical bar indicating the frequency.
     </para>
     <para> The <quote><guibutton>Histogram chart</guibutton></quote> option causes a histogram chart to be 
       added to the histogram. For each bin, the histogram chart shows a vertical bar indicating the density 
       (that is the frequency divided by the width of the bin).
     </para>
     </listitem>
     <listitem>
     <para> The <quote><guibutton>Percentages</guibutton></quote> option causes the frequencies to be 
       expressed as percentages. 
     </para>
     </listitem>
     <listitem>
     <para> The <quote><guibutton>Cumulative answers</guibutton></quote> option causes a cumulative 
       frequency table (either with counts or with pecentages) to be created.  
     </para>
     </listitem>
     <listitem>
     <para> The <quote><guibutton>Count numbers only</guibutton></quote> option determines whether only numbers are counted. If also non-numbers are counted they are first converted into numbers, usually into 0.  
     </para>
     </listitem>
     </itemizedlist>

  </sect4>


  <sect4 id="histogram-tool-outputtab">
     <title>The <quote><guilabel>Output</guilabel></quote> Tab</title>

      <para>
        The Output tab contains the standard output options and fields
        described in <xref
        linkend="sect-stat-analysis-overview" />.
      </para>
  </sect4>


  <sect4 id="histogram-tool-example">
    <title>A Histogram Example</title>

    <figure id="histogram-example-1">
      <title>Some Example Data</title>
      <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-histogram-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of some example data for use with the
              histogram tool.</phrase>
            </textobject>
           </mediaobject>
      </screenshot>
    </figure>

    <example id="usinghistogramtool">
      <title>Using the Histogram Tool</title>

      <para>
        For example, you want to calculate a histogram for the number of
        successes in several sequences of trials. The numbers of
        successes are recorded in column A and the cutoffs of interest
        in column C (see <xref linkend="histogram-example-1" />).
      </para>

      <orderedlist>
        <listitem>
          <para>
            Enter A1:A31 in the <quote><guilabel>Input
            Range:</guilabel></quote> entry of the
            <quote><guilabel>Input</guilabel></quote> tab by typing
            this directly into the entry or clicking in the entry
            field and then selecting that range on the sheet. In the
            latter case the entry may also contain the sheet
            name. 
          </para>
        </listitem>
        <listitem>
          <para>
            Since you only have one variable select the
            <quote><guibutton>Areas</guibutton></quote> or
            <quote><guibutton>Columns</guibutton></quote> radio button
            next to <quote><guilabel>Grouped By:</guilabel></quote>.
            </para>
        </listitem> 
        <listitem><para> Select the
            <quote><guibutton>Labels</guibutton></quote> option
            since the first cell of the Input Range contains a
            label.</para>
        </listitem>
	
        <listitem><para> Enter C2:C5 in
            the <quote><guilabel>Cutoff Range:</guilabel></quote> entry
            of the <quote><guilabel>Cutoffs</guilabel></quote> tab. The
            <quote><guilabel>Predetermined Cutoffs</guilabel></quote>
            option will now also be selected (see <xref
            linkend="histogram-example-2" />). </para>
        </listitem>
        <listitem><para> In the  <quote><guilabel>Bins</guilabel></quote> tab 
	    select the second option since we want to add two bins reaching to &#x2213;&#x221e; and 
	    we want to count each cutoff value in the bin to its right (see <xref
            linkend="histogram-example-3" />).</para>
        </listitem>
        <listitem><para> Select the
            <quote><guibutton>Percentage</guibutton></quote> option of the
            <quote><guilabel>Graphs &amp;Options</guilabel></quote> tab to have
	    the frequencies expressed as percentages.
            </para>
        </listitem>
        <listitem><para> Select the
            <quote><guibutton>Column Chart</guibutton></quote> option of the
            <quote><guilabel>Graphs &amp;Options</guilabel></quote> tab to have
	    a column chart added to the histogram (see <xref
            linkend="histogram-example-4" />).
            </para>
        </listitem>
        <listitem>
          <para>
            In the <quote><guilabel>Output</guilabel></quote> tab,
            specify the output options as described in
            <xref linkend="sect-stat-analysis-overview" />.
          </para>
        </listitem>
        <listitem><para>
            Press the OK button. </para>
        </listitem>
      </orderedlist>
     <para> The results are shown in 
       <xref linkend="histogram-example-5" />. Note that the graph will by default appear on top 
       of the histogram table. It usually needs to be moved in to proper position. That has
       already been done here.
     </para>
 </example>
 

    <figure id="histogram-example-2">
      <title>Histogram Tool: Specifying Cutoffs</title>
      <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-histogram-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of selecting the cutoffs for the example
              data used with the histogram tool.</phrase>
            </textobject>
           </mediaobject>
      </screenshot>
    </figure>

 <figure id="histogram-example-3">
    <title>Histogram Tool: Specifying Bins</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-histogram-ex3.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of selecting a certain bins type.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

 <figure id="histogram-example-4">
    <title>Histogram Tool: Specifying Options</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-histogram-ex4.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of specifying the required options.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

 <figure id="histogram-example-5">
    <title>Histogram Tool Output</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-histogram-ex5.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output from the histogram
              analysis tool.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect4>
  </sect3>
  </sect2>

  <sect2 id="rank-and-percentile-tool">
     <title>Rank and Percentile Tool</title>

  <figure id="rank-and-percentile-tool-dialog">
    <title>Rank and Percentile Tool Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-rank.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the rank and percentile analysis
              tool.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <para>Use this tool to rank given data and to calculate the
     percentiles of each data point.</para>
     <para>Specify the datasets to use in the <quote><guilabel>Input  
     Range:</guilabel></quote> entry.  
     The given range can be grouped into datasets by columns, by rows, or by areas.</para>

     <para>For each dataset, the tool creates three columns in the 
     output table:</para>
     <orderedlist>
     <listitem><para>The first column gives the indices of the 
     ordered data from largest to smallest data value.</para></listitem>
     <listitem><para>The second column 
     gives data values corresponding to the indices in the first column.</para></listitem>
     <listitem><para>The  third column indicates
     the percentile of the  data value in the second column.</para></listitem>
     </orderedlist>

     <para>If you have labels
     in the first cell of each data set, select the
     <quote><guilabel>Labels</guilabel></quote> option.</para>

   <figure id="rank-example-1">
    <title>Some Example Data for the Rank and Percentile Tool</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-rank-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of example data for use with the rank
              and percentile analysis tool.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
 <example id="usingranktool"><title>Using the Rank and Percentile Tool</title>
     <para><xref linkend="rank-example-1" /> shows some example data and 
     <xref linkend="rank-example-2" /> the corresponding output.
     </para>
</example>
  <figure id="rank-example-2">
    <title>Rank and Percentile Tool Output</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-rank-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output from a rank and
              percentile analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  <note><para>In the case of ties, the rank calculated by this tool differs from the 
  value of the RANK function for the same data. This tool calculates the rank as it is 
  normally used in Statistics: If two values are tied, the assigned rank is the average
  rank for those entries. For example in <xref
  linkend="rank-example-1" /> the two values 10
  are the second and third largest values. Since they are equal each receives the rank of 
  2.5, the average of 2 and 3. The rank function on the other hand assigns the rank as it 
  is normally used to determine placements. The two values 10 would therefore each receive
  a rank of 2.   
  </para></note>
  </sect2>
</sect1>

<sect1 id="sampling-tool">
     <title>Sampling Tool</title>
     <figure>
        <title>Sampling Tool Dialog</title>
	<screenshot>
	   	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-sampling.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the sampling tool.</phrase>
            </textobject>
           </mediaobject>
	</screenshot>
      </figure>
     <para>Use the sampling tool to take a sample of a data set.  This
     tool can take both a random sample of a given size or a periodic
     sample:</para>
     <variablelist>
     <varlistentry><term>random sample</term>
     <listitem><para>A random sample is a subset of the population such that 
     every subset of that size has the same chance of being picked.</para></listitem>
     </varlistentry>
     <varlistentry><term>periodic sample</term>
     <listitem><para>In a periodic sample every <parameter>k</parameter>th element in 
     the population is selected.</para></listitem>
     </varlistentry>
     </variablelist>

     <para>To use this tool, first specify the data set or data sets by setting the
     <quote><guilabel>Input Range:</guilabel></quote> entry. The range or ranges 
     given can be grouped into datasets by rows, by columns, or by areas.</para>
     <para>If the first entry in each data set contains a variable, select the 
     <quote><guilabel>Labels</guilabel></quote> option.</para>

     <para>Select the sampling method which
     can be either periodic or random.</para>
     <variablelist>
     <varlistentry><term>random sample</term>
     <listitem><para>Specify the size of the random sample in the <quote><guilabel>Size 
     of Sample:</guilabel></quote> entry.</para></listitem>
     </varlistentry>
     <varlistentry><term>periodic sample</term>
     <listitem><para>Specify the period in the <quote><guilabel>Period:</guilabel></quote>
     entry.</para></listitem>
     </varlistentry>
     </variablelist>

     <para>Specify the number of samples you would like to obtain in the <quote><guilabel>
     Number of Samples:</guilabel></quote> entry.</para>
     <note><para> Since the period uniquely determines a periodic sample, if you specify 
     that you would like 2 samples you will be given the identical sample twice.</para></note>
     <note><para>If the dataset for a periodic sample is a two dimensional range, &gnum; 
     will enumerate the data points by row first.</para></note>

   <figure id="sampling-example-1">
    <title>Some Example Data for the Sampling Tool</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-sampling-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of example data for use with the
              sampling tool.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
 <example id="usingsamplingtool"><title>Using the Sampling Tool</title>
     <para><xref linkend="sampling-example-1" /> shows some example data and 
     <xref linkend="sampling-example-2" /> the corresponding output.
     </para>
</example>
  <figure id="sampling-example-2">
    <title>Sampling Tool Output</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-sampling-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output from the sampling
              tool.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect1>


<sect1 id="dependent_observations">
  <title>Dependent Observations</title>
 <sect2 id="forecast-tools">
     <title>Forecast Tools</title>
  <sect3 id="exp-smoothing-tool">
     <title>Exponential Smoothing Tool</title>

  <figure id="smoothing-tool-dialog">
    <title>Exponential Smoothing Tool Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-smoothing.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the exponential smoothing
              dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <para>The Exponential Smoothing tool performs the exponential
       smoothing for the given set or sets of values. It provides the choice of 5
       different exponential smoothing methods: 
     </para>
     <itemizedlist>
     <listitem>
       <para>Simple exponential smoothing according to (Hunter, 1968).</para>
     </listitem>
     <listitem>
       <para>Simple exponential smoothing according to (Roberts, 1959).</para>
     </listitem>
     <listitem>
       <para>Holt's trend corrected exponential smoothing (occasionally also 
	 referred to as double exponential smoothing)</para>
     </listitem>
     <listitem>
       <para>Additive Holt-Winters exponential smoothing</para>
     </listitem>
     <listitem>
       <para>Multiplicative Holt-Winters exponential smoothing (occasionally also 
	 referred to as triple exponential smoothing)</para>
     </listitem>
     </itemizedlist>

     <para>Since the kind of options available depend on the type of exponential 
       smoothing desired, you can choose the type on the <quote><guilabel>Input
       </guilabel></quote>
       page.
     </para>

     <sect4 id="exp-smoothing-tool-common">
 <title>Common Options of the Exponential Smoothing Tool</title>

     <para>Specify the cells containing the datasets in the <quote><guilabel>Input
     Range</guilabel></quote> entry. The entered range or ranges are grouped into 
     datasets either by rows or by columns.</para> 

     <para>If you have labels
     in the first cell of each data set, select the
     <quote><guilabel>Labels</guilabel></quote> option.</para>

     <para> If you select the <quote><guilabel>Include chart</guilabel></quote> 
       option, &gnum;
       will also create a chart showing both the data and corresponding smoothed 
       values.
     </para>
     </sect4>

  <sect4 id="exp-smoothing-tool-hunter">
 <title>Exponential Smoothing According to Hunter</title>

    <para>  Each value in the
     smoothed set is predicted based on the forecast for the prior
     period.  The formula is given in <xref linkend="exp-smoothing-tool-formula-hunter" />. 
     &#x03b1; is the value given as <quote><guilabel>Damping factor</guilabel></quote>.
     <parameter>y<subscript>t</subscript></parameter> is the <parameter>t</parameter>th
     value in the original data set and <parameter>l<subscript>t</subscript></parameter>
     the corresponding smoothed value.</para>

  <figure id="exp-smoothing-tool-formula-hunter">
    <title>Exponential Smoothing Formula According To Hunter</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-exp-smoothing-hunter-formula.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>The formula used in exponential smoothing according to Hunter.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <para>For example, a value for &#x03b1;  between 0.2 and 0.3 represents 20 to 30 percent error
       adjustment in the prior forecast.
     </para>

     <note><para>
	 If you choose to have the tool enter formul&#xe6; rather than values into the output region, 
	 then you can modify the damping factor &#x03b1; even after you executed the tool. 
     </para></note>

     <para>To have the standard errors output as well, check the 
       <quote><guilabel>Standard error</guilabel></quote> check box. The formula 
       used is given in  <xref linkend="exp-smoothing-tool-formula-hunter-stderr" />.  
       The denominator can be adjusted by selecting the appropriate radio button. Since 
       there are <parameter>t&#x2212;1</parameter> terms in the sum of the denominator, 
       selecting <quote><guilabel>n&#x2212;1</guilabel></quote> means that the denominator 
       will be <parameter>t&#x2212;2</parameter>.
     </para>


  <figure id="exp-smoothing-tool-formula-hunter-stderr">
    <title>The Standard Error Formula For Exponential Smoothing According To Hunter</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-exp-smoothing-hunter-stderr.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>The formula used to calculate the standard error of exponential 
		smoothing according to Hunter
	      </phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
 
   <para>If you check the <quote><guilabel>Include chart</guilabel></quote> check box, a line
    graph showing the observations <parameter>y<subscript>t</subscript></parameter> and the 
    predicted values <parameter>l<subscript>t</subscript></parameter> will also be created.
  </para>

 <example id="usingsmoothingtool"><title>Using the Exponential Smoothing Tool</title>
     <para><xref linkend="smoothing-example-1" /> shows some example data, <xref linkend="smoothing-example-2" /> the selected options and 
     <xref linkend="smoothing-example-3" /> the corresponding output.
     </para>
</example>
  <figure id="smoothing-example-1">
    <title>Some Example Data for the Exponential Smoothing Tool</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-smoothing-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of example data for exponential
              smoothing.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
   <figure id="smoothing-example-2">
    <title>The Options for the Exponential Smoothing Tool</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-smoothing-ex3.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the options tab of the exponential smoothing tool.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  <figure id="smoothing-example-3">
    <title>Exponential Smoothing Tool Output (Hunter)</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-smoothing-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output of an exponential
              smoothing analysis according to Hunter.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

  </sect4>

  <sect4 id="exp-smoothing-tool-roberts">
<title>Exponential Smoothing According to Roberts</title>

  <para>The simple exponential smoothing method according to Roberts is used for 
    forecasting a time series without a trend or seasonal pattern, but for which 
    the level is nevertheless slowly changing over time. The predicted values are 
    calculated according to the formula given in 
    <xref linkend="exp-smoothing-tool-formula-roberts" />. &#x03b1; is the value 
    given as <quote><guilabel>Damping factor</guilabel></quote>.
    <parameter>y<subscript>t</subscript></parameter> is the <parameter>t</parameter>th
    value in the original data set and <parameter>l<subscript>t</subscript></parameter>
    the predicted value. <parameter>l<subscript>0</subscript></parameter> is the 
    predicted value at time 0 and must be estimated. This tool uses the average 
    value of the first 5 observations as estimate.  
  </para>

     <note><para>
	 If you choose to have the tool enter formul&#xe6; rather than values into 
	 the output region, 
	 then you can modify the damping factor &#x03b1; and the estimated value
	 at time 0 after executing the tool. 
     </para></note>

  <figure id="exp-smoothing-tool-formula-roberts">
    <title>Exponential Smoothing Formula According To Roberts</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-exp-smoothing-roberts-formula.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>The formula used in exponential smoothing according to Roberts.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <para>To have the standard errors output as well, check the 
       <quote><guilabel>Standard error</guilabel></quote> check box. The formula used is 
       given in  <xref linkend="exp-smoothing-tool-formula-roberts-stderr" />.  The 
       denominator can be adjusted by selecting the appropriate radio button.
     </para>

  <figure id="exp-smoothing-tool-formula-roberts-stderr">
    <title>The Standard Error Formula For Exponential Smoothing According To Roberts</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-exp-smoothing-roberts-stderr.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>The formula used to calculate the standard error of exponential 
		smoothing according to Roberts
	      </phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

  <para>If you check the <quote><guilabel>Include chart</guilabel></quote> check box, a line
    graph showing the observations <parameter>y<subscript>t</subscript></parameter> and the 
    predicted values <parameter>l<subscript>t</subscript></parameter> will also be created.
  </para>

  <example id="usingsmoothingtool-roberts"><title>Using the Exponential Smoothing Tool</title>
     <para> 
     <xref linkend="smoothing-example-4" /> shows example output for the exponential smoothing
     tool using the formula according to Roberts. Cell A4 contains the estimated level at time 0.
     If you requested to have formul&#xe6; rather than values entered into the sheet, then changing 
     the estimate in A4 and/or the value for &#x03b1; in A2 will result in an immediate change to 
     the predicted values.
     </para>
</example>


  <figure id="smoothing-example-4">
    <title>Exponential Smoothing Tool Output (Roberts)</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-smoothing-ex4.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output of an exponential
              smoothing analysis according to Roberts.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect4>

  <sect4 id="exp-smoothing-tool-holt">
<title>Holt's Trend Corrected Exponential Smoothing</title>

  <para>Holt's trend corrected exponential smoothing is appropriate when both the level and the growth rate of a time series are changing. (If the time series has a fixed growth rate and therefore exhibits a linear trend, a linear regression model is more appropriate.) 
  </para>

  <para><parameter>y<subscript>t</subscript></parameter> is the true value at time 
    <parameter>t</parameter>, <parameter>l<subscript>t</subscript></parameter>
    is the estimated level at time <parameter>t</parameter> and <parameter>b<subscript>t
    </subscript></parameter>
    is the estimated growth rate at time <parameter>t</parameter>. We use the two smoothing equations
    given in <xref linkend="exp-smoothing-tool-formula-holt" /> to update our estimates.
    &#x03b1; is the value 
    given as <quote><guilabel>Damping factor</guilabel></quote> and &#x03b3; is the value 
    given as <quote><guilabel>Growth damping factor</guilabel></quote>. 
  </para>

  <para>This tool obtains initial (time 0) estimates for the level and growth rate by performing
    a linear regression using the first 5 data values.
  </para>

  <figure id="exp-smoothing-tool-formula-holt">
    <title>Formulae Of Holt's Trend Corrected Exponential Smoothing</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-exp-smoothing-holt-formula.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>The formulae used in Holt's trend corrected exponential smoothing.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <note><para>
	 If you choose to have the tool enter formul&#xe6; rather than values into 
	 the output region, 
	 then you can modify the damping factors &#x03b1; and &#x03b3; as well as the estimated level and growth rate
	 at time 0 after executing the tool. 
     </para></note>


     <para>To have the standard errors output as well, check the 
       <quote><guilabel>Standard error</guilabel></quote> check box. The formula used is 
       given in  <xref linkend="exp-smoothing-tool-formula-holt-stderr" />.  The 
       denominator can be adjusted by selecting the appropriate radio button.
     </para>

  <figure id="exp-smoothing-tool-formula-holt-stderr">
    <title>The Standard Error Formula For Holt's Trend Corrected Exponential Smoothing</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-exp-smoothing-holt-stderr.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>The formula used to calculate the standard error for Holt's trend
		corrected exponential smoothing.
	      </phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

  <para>If you check the <quote><guilabel>Include chart</guilabel></quote> check box, a line
    graph showing the observations <parameter>y<subscript>t</subscript></parameter> and the 
    estimated level values <parameter>l<subscript>t</subscript></parameter> will also be created.
  </para>

  <example id="usingsmoothingtool-holt"><title>Using the Exponential Smoothing Tool</title>
     <para> 
     <xref linkend="smoothing-example-5" /> shows example output for Holt's trend corrected 
     exponential smoothing. Cell A4 contains the estimated level at time 0 and B4 the estimated 
     growth rate at time 0.
     If you requested to have formul&#xe6; rather than values entered into the sheet, then changing 
     the estimates in A4, B4, the values for &#x03b1; in A2 and/or for &#x03b3; in B2 will result 
     in an immediate change to 
     the predicted values.
     </para>
</example>


  <figure id="smoothing-example-5">
    <title>Exponential Smoothing Tool Output (Holt's)</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-smoothing-ex5.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output of Holt's trend corrected exponential
              smoothing.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect4>


  <sect4 id="exp-smoothing-tool-additive-holt-winters">
<title>Additive Holt-Winters Method</title>

  <para>The additive Holt-Winters method of exponential smoothing is appropriate when a time 
    series with a linear trend has an additive seasonal pattern for which the level, the growth 
    rate and the seasonal pattern may be changing. An additive seasonal pattern is a pattern in 
    which the seasonal variation can be explained by the addition of a seasonal constant 
    (although we allow for this constant to change slowly.) 
  </para>

   <para><parameter>y<subscript>t</subscript></parameter> is the true value at time 
    <parameter>t</parameter>, <parameter>l<subscript>t</subscript></parameter>
    is the estimated level at time <parameter>t</parameter>, <parameter>b<subscript>t
    </subscript></parameter>
    is the estimated growth rate at time <parameter>t</parameter> and <parameter>s<subscript>t
    </subscript></parameter>
    is the estimated seasonal adjustment for time <parameter>t</parameter>.
    We use the three smoothing equations
    given in <xref linkend="exp-smoothing-tool-formula-a-holt-winters" /> to update our estimates.
    &#x03b1; is the value 
    given as <quote><guilabel>Damping factor</guilabel></quote>, &#x03b3; is the value 
    given as <quote><guilabel>Growth damping factor</guilabel></quote> and &#x03b4; is the value 
    given as <quote><guilabel>Seasonal damping factor</guilabel></quote>. <parameter>L</parameter>
    is the value 
    given as <quote><guilabel>Seasonal period</guilabel></quote>. If your data consist of monthly 
    values, then  <parameter>L</parameter> should be 12, if it consist of quarterly values then 
    <parameter>L</parameter> should be 4.
  </para>

  <para>This tool obtains initial (time 0) estimates for the level and growth rate by performing
    a linear regression using all data values. It obtains estimates 
    for the seasonal adjustments by averaging the appropriate seasonal differences from values 
    predicted by linear regression alone.  
  </para>


 <figure id="exp-smoothing-tool-formula-a-holt-winters">
    <title>Exponential Smoothing Formulae Of The Additive Holt-Winters Method</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-exp-smoothing-a-holt-winters-formula.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>The formulae used in the additive Holt-Winters Method.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <note><para>
	 If you choose to have the tool enter formul&#xe6; rather than values into 
	 the output region, 
	 then you can modify the damping factors &#x03b1;, &#x03b3;  and  &#x03b4; as well as all
	 estimates after executing the tool. 
     </para></note>

     <para>To have the standard errors output as well, check the 
       <quote><guilabel>Standard error</guilabel></quote> check box. The formula used is 
       given in  <xref linkend="exp-smoothing-tool-formula-a-holt-winters-stderr" />.  
       The denominator can be adjusted by selecting the appropriate radio button.
     </para>

  <figure id="exp-smoothing-tool-formula-a-holt-winters-stderr">
    <title>The Standard Error Formula Of The Additive Holt-Winters Method</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-exp-smoothing-a-holt-winters-stderr.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>The formula used to calculate the standard error in the additive 
		Holt-Winters Method
	      </phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

   <para>If you check the <quote><guilabel>Include chart</guilabel></quote> check box, a line
    graph showing the observations <parameter>y<subscript>t</subscript></parameter> and the 
    estimated level values <parameter>l<subscript>t</subscript></parameter> will also be created.
  </para>

 <example id="usingsmoothingtool-ahw"><title>Using the Exponential Smoothing Tool</title>
     <para> 
     <xref linkend="smoothing-example-6" /> shows the options' tab of the exponential smoothing 
     tool for the additive Holt-Winters method. The data is expected to have a seasonal period 
     of 4 (this would for example happen if we have a data value for each quarter of a year). 
     <xref linkend="smoothing-example-7" /> shows the corresponding example output for the
     additive Holt-Winters method. Cell C7 contains the estimated level at time 0, D7 the 
     estimated growth rate at time 0, and E4 to E7 the initial seasonal adjustments for each 
     of the 4 seasons preceding our data time period.
     If you requested to have formul&#xe6; rather than values entered into the sheet, then changing
     any of these estimates, the values for &#x03b1; in A2, for &#x03b3; in B2 and/or for &#x03b4; 
     in C2 will result in an immediate change to the estimated values.
     </para>
</example>

  <figure id="smoothing-example-6">
    <title>Exponential Smoothing Tool Options (Additive Holt-Winters))</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-smoothing-ex6.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the options' tab for the additive Holt-Winters method.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

  <figure id="smoothing-example-7">
    <title>Exponential Smoothing Tool Output (Additive Holt-Winters)</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-smoothing-ex7.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output of the additive Holt-Winters method.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

    </sect4>

  <sect4 id="exp-smoothing-tool-multiplicative-holt-winters">
<title>Multiplicative Holt-Winters Method</title>

  <para>The multiplicative Holt-Winters method of exponential smoothing is appropriate when a time 
    series with a linear trend has a multiplicative seasonal pattern for which the level, the growth 
    rate and the seasonal pattern may be changing. A multiplicative seasonal pattern is a pattern in 
    which the seasonal variation can be explained by the multiplication of a seasonal constant 
    (although we allow for this constant to change slowly.) 
  </para>

   <para><parameter>y<subscript>t</subscript></parameter> is the true value at time 
    <parameter>t</parameter>, <parameter>l<subscript>t</subscript></parameter>
    is the estimated level at time <parameter>t</parameter>, <parameter>b<subscript>t
    </subscript></parameter>
    is the estimated growth rate at time <parameter>t</parameter> and <parameter>s<subscript>t
    </subscript></parameter>
    is the estimated seasonal adjustment for time <parameter>t</parameter>.
    We use the three smoothing equations
    given in <xref linkend="exp-smoothing-tool-formula-m-holt-winters" /> to update our estimates.
    &#x03b1; is the value 
    given as <quote><guilabel>Damping factor</guilabel></quote>, &#x03b3; is the value 
    given as <quote><guilabel>Growth damping factor</guilabel></quote> and &#x03b4; is the value 
    given as <quote><guilabel>Seasonal damping factor</guilabel></quote>. <parameter>L</parameter>
    is the value 
    given as <quote><guilabel>Seasonal period</guilabel></quote>. If your data consist of monthly 
    values, then  <parameter>L</parameter> should be 12, if it consist of quarterly values then 
    <parameter>L</parameter> should be 4.
  </para>

  <para>This tool obtains initial (time 0) estimates for the level and growth rate by performing
    a linear regression using the data values of the first 4 seasonal periods. It obtains estimates 
    for the seasonal adjustments by averaging the appropriate seasonal differences from values 
    predicted by linear regression alone during the first 4 seasonal periods.  
  </para>

  <figure id="exp-smoothing-tool-formula-m-holt-winters">
    <title>Exponential Smoothing Formulae Of The Multiplicative Holt-Winters Method</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-exp-smoothing-m-holt-winters-formula.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>The formulae used in the multiplicative Holt-Winters Method</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <note><para>
	 If you choose to have the tool enter formul&#xe6; rather than values into 
	 the output region, 
	 then you can modify the damping factors &#x03b1;, &#x03b3;  and  &#x03b4; as well as all
	 estimates after executing the tool. 
     </para></note>

  <para>To have the standard errors output as well, check the 
    <quote><guilabel>Standard error</guilabel></quote> check box. The formula used is given in  
    <xref linkend="exp-smoothing-tool-formula-m-holt-winters-stderr" />.  The denominator 
    can be adjusted by selecting the appropriate radio button.
  </para>

  <figure id="exp-smoothing-tool-formula-m-holt-winters-stderr">
    <title>The Standard Error Formula Of The Multiplicative Holt-Winters Method</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-exp-smoothing-m-holt-winters-stderr.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>The formula used to calculate the standard error in the multiplicative 
		Holt-Winters Method
	      </phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

  <para>If you check the <quote><guilabel>Include chart</guilabel></quote> check box, a line
    graph showing the observations <parameter>y<subscript>t</subscript></parameter> and the 
    estimated level values <parameter>l<subscript>t</subscript></parameter> will also be created.
  </para>

  <example id="usingsmoothingtool-mhw"><title>Using the Exponential Smoothing Tool</title>
     <para> 
     <xref linkend="smoothing-example-8" /> shows the example output for the
     multiplicative Holt-Winters method, assuming 4 seasons. Cell C7 contains the estimated level 
     at time 0, D7 the estimated growth rate at time 0, and E4 to E7 the initial seasonal 
     adjustments for each of the 4 seasons preceding our data time period.
     If you requested to have formul&#xe6; rather than values entered into the sheet, then changing
     any of these estimates, the values for &#x03b1; in A2, for &#x03b3; in B2 and/or for &#x03b4; 
     in C2 will result in an immediate change to the estimated values.
     </para>
</example>

 <figure id="smoothing-example-8">
    <title>Exponential Smoothing Tool Output (Multiplicative Holt-Winters)</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-smoothing-ex8.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output of the multiplicative Holt-Winters method.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

  </sect4>
  </sect3>

  <sect3 id="moving-average-tool">
     <title>Moving Average Tool</title>

  <figure id="moving-tool-dialog">
    <title>Moving Average Tool Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-moving-average.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the dialog for the moving average
              analysis tool.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <para>Use the moving average tool to calculate moving averages of
     one or more data sets.  A moving average provides useful trend
     information of the data that is lost in a simple average.  In
     addition, moving averages can be used to eliminate random
     variance.  For example, use this tool to create a smoother curve
     of a stock prize.</para>

     <para>Specify the cells containing the datasets in the
     <quote><guilabel>Input Range</guilabel></quote> entry. The
     entered range or ranges are grouped into datasets either by rows
     or by columns.</para>

     <para>If you have labels in the first cell of each data set,
     select the <quote><guilabel>Labels</guilabel></quote>
     option.</para>

     <para>Choose the type of moving average you would like to calculate. The tool can
       determine 4 types of moving averages:
     </para>
     <orderedlist spacing="compact">
       <listitem><para>
	   Simple moving average
       </para></listitem>
       <listitem><para>
	   Cumulative moving average
       </para></listitem>
       <listitem><para>
	   Weighted moving average
       </para></listitem>
       <listitem><para>
	   Spencer's 15 point moving average
       </para></listitem>
     </orderedlist>

  <figure id="moving-tool-dialog-options">
    <title>
      Moving Average Tool Dialog: The 
      <quote><guilabel>Options</guilabel></quote> Tab
    </title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-moving-average-options.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the <quote><guilabel>Options</guilabel></quote> 
		tab of the moving average
		analysis tool.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

  <para>Specify the <quote><guilabel>Interval</guilabel></quote>
    for the moving average.  The interval <parameter>i</parameter> is
    the number of consecutive values to be included in each moving
    average. This options is only available for the simple and weighted 
    moving averages.
  </para>

  <para>Check the <quote><guilabel>Standard errors</guilabel></quote>
    checkbox if you would also like the standard error to be calculated.  
    Since there is no general agreement on the denominator for the standard 
    error you can choose the appropriate radio button.
  </para>

  <para>In the case of the simple moving average, you can also choose between 
    a prior moving average and a central moving average, or you may even specify 
    any other desired offset.
  </para>
     <orderedlist>
       <listitem><para>
	   <quote><guilabel>Prior moving average</guilabel></quote>: Each average 
	   takes into account the current observation and the most recent prior 
	   observations for a total of <parameter>i</parameter> observations.
       </para></listitem>
       <listitem><para>
	   <quote><guilabel>Central moving average</guilabel></quote>
	   with <parameter>i</parameter> being odd: Each average 
	   takes into account the current observation and the same number of most recent prior 
	   observations and closest future observations for a total of 
	   <parameter>i</parameter> observations.
       </para></listitem>
       <listitem><para>
	   <quote><guilabel>Central moving average</guilabel></quote>
	   with <parameter>i</parameter> being even:
	   This is calculated according to the formula given in 
	   <xref linkend="moving-formula-central" />. 
	   <parameter>a<subscript>t</subscript></parameter> is the moving average
	   at time <parameter>t</parameter> and 
	   <parameter>y<subscript>t</subscript></parameter> is the observation at
	   time <parameter>t</parameter>.
       </para></listitem>
       <listitem><para>
	   <quote><guilabel>Other offset</guilabel></quote>: If the offset is 0,
	   this is just the prior moving average. Otherwise the offset indicates 
	   the number of closest future observations to include in the average. 
	   Correspondingly, the number of most recent past observations is decreased.
       </para></listitem>
     </orderedlist>

       <figure id="moving-formula-central">
	 <title>Formula For The Central Moving Average With Even Interval</title>
	 <screenshot>
	   <mediaobject>
             <imageobject>
               <imagedata fileref="figures/analysistools-moving-average-formula-central.png" 
			  format="PNG" />
             </imageobject>
             <textobject>
               <phrase>The formula for the central moving average if the interval 
		 length is even.</phrase>
             </textobject>
           </mediaobject>
	 </screenshot>
       </figure>


  <para>The results are given in one column for each dataset (with a second 
    column added if you have chosen standard errors to be calculated). Each
    row represents the moving average of the corresponding row or
    column in the input range.  Depending on the type of average and 
    the offset, the moving average cannot be
    calculated for the first rows in the
    input range.
  </para>

     <sect4 id="moving-averages-simple">
       <title>Simple Moving Average</title>
       <para>
	 A simple moving average is the unweighted average of a collection of 
	 observations. Exactly which observations are included depends on whether 
	 a prior or central moving average is calculated.  
       </para>
     </sect4>
     <sect4 id="moving-averages-cumulative">
       <title>Cumulative Moving Average</title>
       <para>A cumulative moving average is a prior moving average in which the current 
       and all prior observations are included.</para>
     </sect4>
     <sect4 id="moving-averages-weighted">
       <title>Weighted Moving Average</title>
       <para>A weighted moving average with an interval <parameter>i</parameter> is a prior 
	 moving average calculated according to formula 
	 <xref linkend="moving-formula-central" />. 
	 <parameter>a<subscript>t</subscript></parameter> is the moving average
	 at time <parameter>t</parameter> and 
	 <parameter>y<subscript>t</subscript></parameter> is the observation at
	 time <parameter>t</parameter>.
       </para>

       <figure id="moving-averages-weighted-formula">
	 <title>Formula For The Weighted Moving Average With Interval 
	   <parameter>i</parameter></title>
	 <screenshot>
	   <mediaobject>
             <imageobject>
               <imagedata fileref="figures/analysistools-moving-average-formula-weighted.png" 
			  format="PNG" />
             </imageobject>
             <textobject>
               <phrase>The formula for the weighted moving average if the interval 
		 length is <parameter>i</parameter>.</phrase>
             </textobject>
           </mediaobject>
	 </screenshot>
       </figure>
     </sect4>
   

     <sect4 id="moving-averages-spencer">
       <title>Spencer's 15 Point Moving Average</title>
       <para>Spencer's 15 point moving average is a central moving average calculated 
	 according to formula 
       <xref linkend="moving-formula-spencer" />. 
       <parameter>a<subscript>t</subscript></parameter> is the moving average
       at time <parameter>t</parameter> and 
       <parameter>y<subscript>t</subscript></parameter> is the observation at
       time <parameter>t</parameter>.
       </para>

       <figure id="moving-formula-spencer">
	 <title>Formula For Spencer's 15 Point Moving Average</title>
	 <screenshot>
	   <mediaobject>
             <imageobject>
               <imagedata fileref="figures/analysistools-moving-average-formula-spencer.png" 
			  format="PNG" />
             </imageobject>
             <textobject>
               <phrase>The formula for the Spencer's 15 point moving average.</phrase>
             </textobject>
           </mediaobject>
	 </screenshot>
       </figure>
     </sect4>

     <sect4 id="moving-averages-example">
       <title>A Moving Average Example</title>
       <figure id="moving-example-1">
	 <title>Some Example Data for the Moving Average Tool</title>
	 <screenshot>
	   <mediaobject>
             <imageobject>
               <imagedata fileref="figures/analysistools-moving-average-ex1.png" 
			  format="PNG" />
             </imageobject>
             <textobject>
               <phrase>An image of some example data for use with the
		 moving average analysis tool.</phrase>
             </textobject>
           </mediaobject>
	 </screenshot>
       </figure>
       <example id="usingmovingtool"><title>Using the Moving Average Tool</title>
	 <para><xref linkend="moving-example-1" /> shows some example data,  
	   <xref linkend="moving-example-2" /> shows the option settings, and 
	   <xref linkend="moving-example-3" /> the corresponding output.
	 </para>
       </example>
       <figure id="moving-example-2">
	 <title>Moving Averages Tool Options</title>
	 <screenshot>
	   <mediaobject>
             <imageobject>
               <imagedata fileref="figures/analysistools-moving-average-ex2.png" 
			  format="PNG" />
             </imageobject>
             <textobject>
               <phrase>An image of the option settings of the moving averages 
		 example.</phrase>
             </textobject>
           </mediaobject>
	 </screenshot>
       </figure>
       <figure id="moving-example-3">
	 <title>Moving Averages Tool Output</title>
	 <screenshot>
	   <mediaobject>
             <imageobject>
               <imagedata fileref="figures/analysistools-moving-average-ex3.png" 
			  format="PNG" />
             </imageobject>
             <textobject>
               <phrase>An image of the output from the moving average
		 analysis tool.</phrase>
             </textobject>
           </mediaobject>
	 </screenshot>
       </figure>
     </sect4>
  </sect3>
  </sect2>

  <sect2 id="fourier-analysis-tool">
     <title>Fourier Analysis Tool</title>

  <figure id="fourier-tool-dialog">
    <title>Fourier Analysis Tool Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-fourier.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the fourier analysis
              dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <para>
       The Fourier Analysis tool normally performs a Fast Fourier
       Transform to obtain the discrete fourier transform
       F<subscript>s</subscript> of the given sequence
       f<subscript>t</subscript> of real numbers according to the
       formula given in <xref linkend="fourier-tool-formula"
       />.</para> <para>Select the
       <quote><guilabel>Inverse</guilabel></quote> option to calculate
       the inverse discrete fourier transform
       f<subscript>t</subscript> of the given sequence
       F<subscript>s</subscript> of real numbers</para> <note><para>If
       the number of terms in the given sequence is not
       a power of 2 (i.e.  2, 4, 8, 16, 32, 64, 128, etc.), this tool
       will append zeros to reach such a power of 2!</para></note>
     
     <para>Specify the cells containing the datasets in the
     <quote><guilabel>Input Range</guilabel></quote> entry. The
     entered range or ranges are grouped into sequences either by rows
     or by columns.</para>

     <para>If you have labels
     in the first cell of each data set, select the
     <quote><guilabel>Labels</guilabel></quote> option.</para>

  <figure id="fourier-tool-formula">
    <title>Fourier Analysis Formulae</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-fourier-formula.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>The formulae used in a fourier analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <note>
       <para>Before using the numbers obtained by this tool, ensure
       that these are in fact the correct formulae for your
       discipline. In the physical sciences this fourier transform
       tends to be called the inverse fourier transform and vice
       versa. Moreover, frequently the scaling factor varies.</para>
       <para>For example <application>Mathematica</application> uses
       the terms fourier transform and inverse fourier transform with
       the reversed meaning than &gnum;
       and it uses a scaling factor of
       <parameter>1/SQRT(N)</parameter> rather than
       <parameter>1/N</parameter>.</para></note>
  </sect2>

  <sect2 id="kaplan-meier-tool">
      <title>Kaplan Meier Estimates Tool</title>
      <para/>

  <sect3 id="kaplan-meier-tool-inputtab">
     <title>The <quote><guilabel>Input</guilabel></quote> Tab</title>

     <para>The <quote><guilabel>Input</guilabel></quote> tab shown in 
        <xref linkend="kaplan-meier-tool-dialog" /> contains
        the fields specifying the data to be used for the
        Kaplan Meier Estimates. The time column contains the times or dates 
        at which the subjects died or were censored. If any of the subjects
        were censored, the <guilabel>Permit censorship</guilabel> checkbox is
        checked and the Censor column contained the censorship marks. Censorship
        marks are typically 0s or 1s. The range of censor marks or labels can be 
        set using the the 
        remaining two spinboxes.</para>

 <figure id="kaplan-meier-tool-dialog">
    <title>Kaplan-Meier Tool Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-kaplan.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the .Kaplan-Meier tool dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect3>

  <sect3 id="kaplan-meier-tool-group-tab">
     <title>The <quote><guilabel>Groups</guilabel></quote> Tab</title>

    <para>
        If the subjects belong to several groups and the groups are supposed to be
        analyzed separately, the groups tab can be used. 
    </para>

 <figure id="kaplan-meier-tool-dialog-groups">
    <title>Kaplan-Meier Tool Dialog Groups Tab</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-kaplan-groups.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the Kaplan-Meier tool dialog groups tab.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

    <para>
        The groups tab can be enabled via the <guilabel>Define multiple groups
        </guilabel> checkbox. The groups column entry contains the address of 
        the column specifying the group membership. Groups can then be defined 
        or deleted via the <guilabel>Add</guilabel> and <guilabel>Remove
        </guilabel> buttons.
    </para>

  </sect3>

  <sect3 id="kaplan-meier-tool-optionstab">
     <title>The <quote><guilabel>Options</guilabel></quote> Tab</title>

    <para>The options tab of the Kaplan-Meier tools dialog is used to set 
        various options of the Kaplan-Meier tool.
    </para>

 <figure id="kaplan-meier-tool-dialog-options">
    <title>Kaplan-Meier Tool Dialog Options Tab</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-kaplan-options.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the Kaplan-Meier tool dialog options tab.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect3>

  <sect3 id="kaplan-meier-tool-outputtab">
     <title>The <quote><guilabel>Output</guilabel></quote> Tab</title>

      <para>
        The Output tab contains the standard output options and fields
        described in <xref linkend="sect-stat-analysis-overview" />.
      </para>
  </sect3>

  <sect3 id="kaplan-meier-tool-example">
    <title>A Kaplan-Meier Example</title>

      <figure id="kaplan-meier-tool-example-1">
	<title>Kaplan-Meier Tool Example Input</title>
	<screenshot>
	    <mediaobject>
		<imageobject>
		  <imagedata fileref="figures/analysistools-kaplan-ex1.png" 
		      format="PNG" />
		</imageobject>
		<textobject>
		  <phrase>An image of the input to the Kaplan-Meier estimate example 
            and of the input 
		  tab of the Kaplan-Meier analysis tool.</phrase>
		</textobject>
	       </mediaobject>
	</screenshot>
      </figure>

    <example id="usingkaplan-meiertool">
      <title>Using the Kaplan-Meier Tool</title>
      <para>
        Suppose you want to calculate Kaplan-Meier Estimates 
        for the as given in <xref linkend="kaplan-meier-tool-example-1" />. Each 
        row contains the data for one subject. Column A contains the survival time,
        i.e. the time until death or censure. Column B contains the group number, 
        we are considering two groups of subjects. Column C indicates whether the 
        subject died (0) or was censured (1).
        </para>
        <para>
        We complete the fields of the <guilabel>Input</guilabel> tab as shown in 
        <xref linkend="kaplan-meier-tool-example-1" />. The time column is A2:A21
        and the censure column is C2:C21.
        </para>
        <para>
        Since we have two groups of subjects, on the <guilabel>Groups</guilabel> 
        tab we check the  <guilabel>Define multiple groups</guilabel> check box and 
        set up two groups with identifiers 1 and 2 in column B2:B21:
      <figure id="kaplan-meier-tool-tool-example-3">
	 <title>Kaplan-Meier Tool Example Group Tab</title>
	 <screenshot>
	     <mediaobject>
		 <imageobject>
		   <imagedata fileref="figures/analysistools-kaplan-ex3.png" 
		       format="PNG" />
		 </imageobject>
		 <textobject>
		   <phrase>An image of the group tab of the Kaplan-Meier
		   analysis tool.</phrase>
		 </textobject>
		</mediaobject>
	 </screenshot>
       </figure>
      </para>
    <para>On the <guilabel>Options</guilabel> tab all checkboxes are pre-checked
     and we leave them that way to obtain a maximum amount of information.
    </para>
    <para>On the output tab we choose where we would like the output to be placed. For 
    the purposes of this example we retain the <guilabel>New Sheet</guilabel> target. 
    After clicking <guilabel>OK</guilabel> we get the output shown in 
    <xref linkend="kaplan-meier-tool-example-2" />. Note that the graph initially 
        always appears on top of the numerical result and was moved for the 
        screen shot.
    </para>
    <para>
        B1:F17 shows the results of the first group, G1 to K17 the results of the 
        second group. The graph shows the Kaplan-Meier survival curves for both 
        groups.
    </para>
    <para>
        M4:N7 shows the result of the Mantel-Haenszel Log-Rank Test. In this case 
        the p-value is larger than 0.3 and we would fail to reject the Null 
        hypothesis. There is no evidence that the survival times differ.
    </para>
    </example>

      <figure id="kaplan-meier-tool-example-2">
	 <title>Kaplan-Meier Tool Example Output</title>
	 <screenshot>
	     <mediaobject>
		 <imageobject>
		   <imagedata fileref="figures/analysistools-kaplan-ex2.png" 
		       format="PNG" />
		 </imageobject>
		 <textobject>
		   <phrase>An image of the output of the Kaplan-Meier
		   analysis tool.</phrase>
		 </textobject>
		</mediaobject>
	 </screenshot>
       </figure>
  </sect3>
  </sect2>

  <sect2 id="principal-component-tool">
      <title>Principal Component Analysis</title>
   <figure id="pcanalysis-tool-dialog">
    <title>Principal Component Analysis Tool Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-pcanalysis.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the principal component analysis tool dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
     <para>Principal Component Analysis Tool performs a principal component analysis (PCA). 
           PCA is a useful statistical technique with application in
	   fields such as face recognition and image compression. It is a common technique for
           finding patterns in data of high dimension.
     </para>

     <para>Specify the cells containing the datasets in the
     <quote><guilabel>Input Range</guilabel></quote> entry. The
     entered range or ranges are grouped into the factors either by rows
     or by columns.</para>

     <para>If you have labels
     in the first cell of each factor, select the
     <quote><guilabel>Labels</guilabel></quote> option.</para>

 <figure id="pcanalysis-example-1">
    <title>Principal Component Analysis Example Data</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-pcanalysis-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of example data for use with the
               principal component analysis tool.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

 <example id="usingpcanalysistool">
 <title>Using the Principal Component Analysis Tool Tool</title>
 <para>
    Suppose you want to perform a principal component analysis on the data
    given in <xref linkend="pcanalysis-example-1" /> having the two dimensions (factors) 
    <inlineequation><mathphrase>x</mathphrase></inlineequation> and 
    <inlineequation><mathphrase>y</mathphrase></inlineequation>.</para>
<orderedlist>
     <listitem><para>
     Enter Sheet1!$A$1:$B$11 (or just A1:B11) in the <quote><guilabel>Input Range:</guilabel></quote> 
     entry by typing 
     this directly into the entry or clicking in the entry field and 
     then selecting the range on the sheet.</para></listitem>
     <listitem><para> Select the <quote><guibutton>Labels</guibutton></quote>
     option since the first row contains labels. (see 
     <xref linkend="pcanalysis-tool-dialog" />).</para></listitem>
     <listitem><para> Specify the output 
     options as described above.</para></listitem>
     <listitem><para> Press the OK button. </para></listitem>
</orderedlist>
     <para> The output of this principal component analysis is shown in
     <xref linkend="regression-example-3" />. The output shows the covariance matrix,
     the eigenvalues and corresponding eigenvectors. The principal component is the 
     constructed factor with the highest percent of trace, 
     <inlineequation><mathphrase>&#x03be;1</mathphrase></inlineequation>.</para>
 </example>

   <figure id="pcanalysis-example-2">
    <title>Principal Component Analysis Tool Output</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-pcanalysis-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output from a principal component
              analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect2>



  <sect2 id="regression-tool">
     <title>Regression Tool</title>
  <figure id="regression-tool-dialog">
    <title>Regression Tool Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-regression.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the regression tool dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  <para>The regression tool performs a multiple regression analysis.</para>
  <para>Enter a range or list of ranges containing the independent variables 
  into the <quote><guilabel>X Variables:</guilabel></quote> entry.</para>
  <para>Enter a single range containing the dependent variable into the
   <quote><guilabel>Y Variable:</guilabel></quote> entry.</para>
  <para>If the ranges for the independent and dependent variables also contains 
  labels in the first field of each row, column or area, select the <quote>
  <guilabel>Labels</guilabel></quote> option.</para>
  <para> Specify the confidence level in the <quote><guilabel>Confidence
  Level:</guilabel></quote> entry. The default is 95&#037;.</para>
  <para>To force the regression line or plane to pass through the origin, select the
  <quote><guilabel>Force Intercept To Be Zero</guilabel></quote> option.</para>
  <para>Specify the output options as described above. If the output is directed 
  into a specific output range, that
  range should contain at least seven columns and 17 rows more than there are 
  independent variables.</para>

  <figure id="regression-example-1">
    <title>Regression Example Data</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-regression-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of example data for use with the
              regression tool.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

 <example id="usingregressiontool">
 <title>Using the Regression Tool</title>

  <para>
    Suppose you want to perform a regression analysis on the data
    given in <xref linkend="regression-example-1" /> using
    <parameter>v</parameter> and <parameter>y</parameter> as
    independent variables and <parameter>u</parameter> as dependent
    variable.</para>
<orderedlist>
     <listitem><para>
     Enter B1:C11 in the <quote><guilabel>X Variables:</guilabel></quote> 
     entry by typing 
     this directly into the entry or clicking in the entry field and 
     then selecting the range on the sheet.</para></listitem>
     <listitem><para>
     Enter A1:A11  in the <quote><guilabel>Y Variable:</guilabel></quote> 
     entry. </para></listitem>
     <listitem><para> Select the <quote><guibutton>Labels</guibutton></quote>
     option since the first row contains labels. (see 
     <xref linkend="regression-example-2" />).</para></listitem>
     <listitem><para> Specify the output 
     options as described above.</para></listitem>
     <listitem><para> Press the OK button. </para></listitem>
</orderedlist>
     <para> The output of this regression analysis is shown in
     <xref linkend="regression-example-3" />.</para>
 </example>
  <figure id="regression-example-2">
    <title>Completed Regression Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-regression-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the regression tool dialog with the
              required fields completed.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  <figure id="regression-example-3">
    <title>Regression Tool Output</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-regression-ex3.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output from a regression
              analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect2>
</sect1>

<sect1 id="one_sample_tests">
  <title>One Sample Tests</title>

  <sect2 id="normality-tool">
      <title>Normality Tests</title>
<para>The normality test tool provides for four tests of normality.</para>
<orderedlist spacing="compact">
     <listitem><para>Anderson Darling Test</para></listitem>
     <listitem><para>Cram&#xe9;r-von Mises Test</para></listitem>
     <listitem><para>Lilliefors (Kolmogorov-Smirnov) Test</para></listitem>
     <listitem><para>Shapiro-Francia Test</para></listitem>
</orderedlist>
   <figure id="normality-tool-dialog">
    <title>Normality Test Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-normality.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the normality test dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
      <para>The data range is specified via the <quote><guilabel>Input
      Range:</guilabel></quote> entry 
      (see <xref linkend="normality-tool-dialog" />).  The given range 
      or list of ranges can be grouped into 
      separate data sets by columns, rows, or areas. The tool performs a
      separate test for each data set.</para>
   <figure id="normality-tool-testspec-dialog">
    <title>Test Tab of the Normality Test Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-normality-testspec.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the test tab of the normality 
	      test dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
      <para>On the test tab one specifies which of the four tests to
      perform, the significance level for the test and whether to include
      a normal probability plot of the data 
      (see <xref linkend="normality-tool-testspec-dialog" />).</para>
   <figure id="normality-example-1">
    <title>Normality Test Example Data</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-normality-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of example data for a normality test.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

 <example id="usingnormalitytesttool">
 <title>Using the Normality Test Tool</title>

  <para>
    Suppose you want to perform a Lilliefors (Kolmogorov-Smirnov) Test
    for Normality on the data
    given in <xref linkend="normality-example-1" />.</para>
<orderedlist>
     <listitem><para>
     Enter A1:A50 (or Sheet1!$A$1:$A$50) in the 
     <quote><guilabel>Input Range:</guilabel></quote> 
     entry by typing 
     this directly into the entry or clicking in the entry field and 
     then selecting the range on the sheet.</para></listitem>
     <listitem><para> Select the <quote><guibutton>Labels</guibutton></quote>
     option since the first row contains a label. (see 
     <xref linkend="normality-example-2" />).</para></listitem>
     <listitem><para> On the test tab of the dialog
     (see <xref linkend="normality-example-3" />) select the
     Lilliefors (Kolmogorov-Smirnov) Test.</para></listitem>
     <listitem><para> Specify an appropriate significance level
     Alpha, say 0.05.</para></listitem>
     <listitem><para> Select the <quote><guibutton>Create Normal 
     Probability Plot</guibutton></quote>
     option to include a normal 
     probability plot in the output.</para></listitem>
     <listitem><para> Specify the output 
     options as described above.</para></listitem>
     <listitem><para> Press the OK button. </para></listitem>
</orderedlist>
     <para> The output of this normality test is shown in
     <xref linkend="normality-example-4" />. Note that the graph appears 
     initially on top of the output data and needs to be moved to make 
     the data visible.</para>
 </example>

   <figure id="normality-example-2">
    <title>Completed Input Tab of the Normality Test Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-normality-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the completed input tab of the normality 
	      test dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
   <figure id="normality-example-3">
    <title>Completed Test Tab of the Normality Test Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-normality-ex3.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the completed test tab of the normality 
	      test dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
   <figure id="normality-example-4">
    <title>Normality Test Output</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-normality-ex4.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output from a normality test.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
 </sect2>


  <sect2 id="one-median-tool">
      <title>One Median</title>
  
      <para>The One Median test tool provides two non-parametric
      tests that test the null hypothesis that the sample comes from a 
      population with a given median:</para>
      <orderedlist spacing="compact">
	<listitem><para>Sign Test</para></listitem>
	<listitem><para>Wilcoxon Signed Rank Test</para></listitem>
      </orderedlist>
      <para>Selecting the appropriate submenu item opens the dialog with
      the respective test preselected.</para>

  <sect3 id="sign-test-tool">
      <title>Sign Test</title>
  <note>
    <para>
      This section describes the one sample sign test to test the 
      null hypothesis that the sample comes from a 
      population with the given median. The tool to perform a sign test to
      test the null hypothesis that two paired samples come from populations 
      with the same median is in section  
      <xref linkend="two-median-sign-test-tool" />.
    </para>
  </note>
   <figure id="one-median-tool-dialog">
    <title>One-Median Test Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-signtest.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the one-median test dialog used by 
	      the Sign Test and the Wilcoxon Signed Rank Test.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
      <para>The Sign Test tool performs a one-sample sign test whether
      the sample comes from a  population with a given median.</para>
      <para>The sample data range is specified via the <quote><guilabel>Input
      Range:</guilabel></quote> entry 
      (see <xref linkend="one-median-tool-dialog" />).  The given range 
      or list of ranges can be grouped into 
      separate data sets by columns, rows, or areas. The tool performs a
      separate test for each data set.</para>
      <para>On the <quote><guilabel>Test</guilabel></quote>tab of the dialog
      (see <xref linkend="one-median-tool-dialog-test-tab" />) the predicted 
      median as well as the significance level are specified.</para>
   <figure id="one-median-tool-dialog-test-tab">
    <title>The Test Tab of the One-Median Test Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-signtest-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the test tab of the one-median test 
	      dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

<example id="usingsigntesttool">
 <title>Using the Sign Test Tool</title>

  <para>
    Suppose you want to perform a Sign Test
    on the data
    given in <xref linkend="one-median-tool-dialog" /> to determine whether
    the sample comes from a population of mean 3.</para>
<orderedlist>
     <listitem><para>
     Enter A1:A19 (or Sheet1!$A$1:$A$19) in the 
     <quote><guilabel>Input Range:</guilabel></quote> 
     entry by typing 
     this directly into the entry or clicking in the entry field and 
     then selecting the range on the sheet.</para></listitem>
     <listitem><para> Select the <quote><guibutton>Labels</guibutton></quote>
     option since the first row contains a label. (see 
     <xref linkend="one-median-tool-dialog" />).</para></listitem>
     <listitem><para> On the <quote><guibutton>Test</guibutton></quote> tab 
     of the dialog
     (see <xref linkend="one-median-tool-dialog-test-tab" />) select the
     Sign Test.</para></listitem>
     <listitem><para> Specify an appropriate significance level
     Alpha, say 0.05.</para></listitem>
     <listitem><para> Select thepecify the median of the null hypothesis (3)
     in the <quote><guibutton>Predicted Median</guibutton></quote> entry.
     </para></listitem>
     <listitem><para> Specify the output 
     options as described above.</para></listitem>
     <listitem><para> Press the OK button. </para></listitem>
</orderedlist>
     <para> The output of this sign test is shown in
     <xref linkend="sign-test-dialog-output" />.</para>
 </example>

   <figure id="sign-test-dialog-output">
    <title>Output of a Sign Test</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-signtest-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output of a Sign Test.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect3>

  <sect3 id="wilcoxon-sign-test-tool">
      <title>Wilcoxon Signed Rank Test</title>
  <note>
    <para>
      This section describes the one sample Wilcoxon signed rank test to 
      test the 
      null hypothesis that the sample comes from a 
      population with the given median. The tool to perform a Wilcoxon
      signed rank test to
      test the null hypothesis that two paired samples come from populations 
      with the same median is in section  
      <xref linkend="two-median-wilcoxon-sign-test-tool" />.
    </para>
  </note>
       <para>The  Wilcoxon Signed Rank TTest tool performs a one-sample 
      sign test whether
      the sample comes from a  population with a given median.</para>
      <para>The sample data range is specified via the <quote><guilabel>Input
      Range:</guilabel></quote> entry 
      (see <xref linkend="one-median-tool-dialog" />).  The given range 
      or list of ranges can be grouped into 
      separate data sets by columns, rows, or areas. The tool performs a
      separate test for each data set.</para>
      <para>On the <quote><guilabel>Test</guilabel></quote>tab of the dialog
      (see <xref linkend="one-median-tool-dialog-test-tab" />) the predicted 
      median as well as the significance level are specified.</para>
 
  <note>
    <para>
      The p-values given by this tool are determined using a normal
      approximation. This approximation is only valid if the sample 
      size is at least 12.
    </para>
  </note>

<example id="usingwilcoxonsignedranktesttool">
 <title>Using the Wilcoxon Signed Rank Test Tool</title>
  <para>
    Suppose you want to perform a Wilcoxon Signed Rank Test
    on the data
    given in <xref linkend="one-median-tool-dialog" /> to determine whether
    the sample comes from a population of mean 3.</para>
<orderedlist>
     <listitem><para>
     Enter A1:A19 (or Sheet1!$A$1:$A$19) in the 
     <quote><guilabel>Input Range:</guilabel></quote> 
     entry by typing 
     this directly into the entry or clicking in the entry field and 
     then selecting the range on the sheet.</para></listitem>
     <listitem><para> Select the <quote><guibutton>Labels</guibutton></quote>
     option since the first row contains a label. (see 
     <xref linkend="one-median-tool-dialog" />).</para></listitem>
     <listitem><para> On the <quote><guibutton>Test</guibutton></quote> tab 
     of the dialog
     (see <xref linkend="one-median-tool-dialog-test-tab" />) select the
     Wilcoxon Signed Rank Test.</para></listitem>
     <listitem><para> Specify an appropriate significance level
     Alpha, say 0.05.</para></listitem>
     <listitem><para> Select thepecify the median of the null hypothesis (3)
     in the <quote><guibutton>Predicted Median</guibutton></quote> entry.
     </para></listitem>
     <listitem><para> Specify the output 
     options as described above.</para></listitem>
     <listitem><para> Press the OK button. </para></listitem>
</orderedlist>
     <para> The output of this sign test is shown in
     <xref linkend="wilcoxon-sign-test-dialog-output" />.</para>
 </example>


   <figure id="wilcoxon-sign-test-dialog-output">
    <title>Output of a Wilcoxon Signed Rank Test</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-signtest-ex3.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output of a Wilcoxon Signed Rank 
	      Test.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect3>

  </sect2>

</sect1>

<sect1 id="two_sample_tests">
  <title>Two Sample Tests</title>

<sect2 id="t-test-tool">
     <title>Comparing Means of Two Populations</title>
     <para>&gnum; provides four similar
     tools to test whether the difference of two population means is
     equal to a hypothesized value. These four tools use the same
     dialog (see <xref linkend="ttest-dialog" />).</para>

  <figure id="ttest-dialog">
    <title><parameter>t</parameter>- and <parameter>z</parameter>-Test
    Tool Dialog</title>

    <screenshot>
      <mediaobject>
        <imageobject>
          <imagedata fileref="figures/analysistools-ttest.png" format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the t-test and z-test dialog.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

     <para>Depending on the options settings, the appropriate test
     will be performed. The entries in the
     <quote><guilabel>Input</guilabel></quote>,
     <quote><guilabel>Test</guilabel></quote>, and
     <quote><guilabel>Output</guilabel></quote> frames are independent
     from the specific test.</para>

     <para>Enter the first variable in the <quote><guilabel>Variable 1
     Range</guilabel></quote> entry and the second variable in the
     <quote><guilabel>Variable 2 Range</guilabel></quote>
     entry.</para> <para>Enter the hypothesized difference between the
     population means in the <quote><guilabel>Hypothesized Mean
     Difference</guilabel></quote> entry, which has a default of 0.
     Enter the significance level in the
     <quote><guilabel>Alpha</guilabel></quote> entry, which has a
     default of 5 &#037;.</para> <para> Specify the output options as
     described above. If the output is printed into a range, it should
     have at least three columns and ten rows.</para>

     <para>There are up to three possible options that can be selected:</para>
     <variablelist>
     <varlistentry><term><quote><guilabel>Paired</guilabel></quote> versus <quote><guilabel>Unpaired</guilabel></quote>
     </term><listitem><para>
     If the variables are dependent (or paired) select the <quote><guilabel>Paired</guilabel></quote>
     option.
     </para></listitem>
     </varlistentry>
     <varlistentry><term><quote><guilabel>Known</guilabel></quote> versus <quote><guilabel>Unknown</guilabel></quote>
     </term><listitem><para>
     For unpaired or independent variables, the population variances may be known 
     or unknown. In the latter case they will be estimated using the sample variances.
     Select the <quote><guilabel>Known</guilabel></quote> option if you in fact know the population 
     variances prior to collecting the sample.
     </para></listitem>
     </varlistentry>
     <varlistentry><term><quote><guilabel>Equal</guilabel></quote> versus <quote><guilabel>Unequal</guilabel></quote>
     </term><listitem><para>
     For paired variables with unknown population variances, we may either assume 
     that the population variances are equal or not. If the population variances are
     assumed to be equal, &gnum; will estimate the common variance by pooling the 
     sample variances. Select the <quote><guilabel>Equal</guilabel></quote> option to assume that
     the population variances are equal.
     </para></listitem>
     </varlistentry>
     </variablelist>

  <sect3 id="t-test-paired-two-samples-for-means-tool">
     <title><parameter>t</parameter>-Test: Paired Two Sample for Means Tool</title>
  <figure id="ttest-dialog-paired">
    <title><parameter>t</parameter>-Test (Paired) Tool Dialog Options</title>
    <screenshot>
      <mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ttest-paired.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the options for the t-test.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
     <para>For paired variables, when you click on 
     <quote><guibutton>OK</guibutton></quote>, &gnum; will test whether the 
     mean of the difference between the paired variables is equal to 
     the given hypothesized mean difference.</para>

 <example id="usingttestpairedtool">
 <title>Using the <parameter>t</parameter>-Test (Paired) Tool</title>
     <para>See <xref linkend="ttest-paired-tool-ex1" /> for an example 
     of a completed dialog and <xref linkend="ttest-paired-tool-ex2" />
     for the corresponding output.
     </para>
 </example>
  <figure id="ttest-paired-tool-ex1">
    <title><parameter>t</parameter>-Test (Paired) Example Data</title>
    <screenshot>
      <mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ttest-paired-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the example for a t-test.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  <figure id="ttest-paired-tool-ex2">
    <title>Output from the <parameter>t</parameter>-Test (Paired) Tool</title>
    <screenshot>
      <mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ttest-paired-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output results from a t-test.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>

  </sect3>

  <sect3 id="t-test-two-sample-equal-variances-tool">
     <title><parameter>t</parameter>-Test: Two-Sample Assuming Equal Variances Tool</title>
  <figure id="ttest-dialog-equal">
    <title><parameter>t</parameter>-Test (Equal Variances) Tool Dialog
    Options</title>
    <screenshot>
      <mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ttest-equal.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the options for a t-test
              analysis of two samples with equal variances.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
     <para>For unpaired variables with unknown but assumed equal population variances,
     when you click on <quote><guibutton>OK</guibutton></quote>, &gnum; will test whether the 
     mean of the difference between the paired variables is equal to the given hypothesized
     mean difference.</para>

 <example id="usingttestequaltool">
 <title>Using the <parameter>t</parameter>-Test (Unknown but Equal Variances) Tool</title>
     <para>See <xref linkend="ttest-equal-tool-ex1" /> for an example 
     of a completed dialog and <xref linkend="ttest-equal-tool-ex2" />
     for the corresponding output.
     </para>
 </example>
  <figure id="ttest-equal-tool-ex1">
    <title><parameter>t</parameter>-Test (Unknown but Equal Variances) Example Data</title>
    <screenshot>
      <mediaobject>
        <imageobject>
          <imagedata fileref="figures/analysistools-ttest-equal-ex1.png" 
              format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of example data for use with a t-test
              with unknown but equal variances.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  <figure id="ttest-equal-tool-ex2">
    <title>Output from the <parameter>t</parameter>-Test (Unknown but Equal Variances) Tool</title>
    <screenshot>
      <mediaobject>
        <imageobject>
          <imagedata fileref="figures/analysistools-ttest-equal-ex2.png" 
              format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output from a t-test
              with unknown but equal variances.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect3>

  <sect3 id="t-test-two-samples-unequal-variances">
     <title><parameter>t</parameter>-Test: Two-Sample Assuming Unequal Variances Tool</title>
  <figure id="ttest-dialog-unequal">
    <title><parameter>t</parameter>-Test (Unknown and Unequal Variances) Tool 
    Dialog Options</title>
    <screenshot>
      <mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ttest-unequal.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the options in a t-test of two
              samples with unknown and possibly unequal
              variances.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
     <para>For unpaired  variables with unknown and assumed unequal population variances,
     when you click on <quote><guibutton>OK</guibutton></quote>, &gnum; will test whether the 
     mean of the difference between the paired variables is equal to the given hypothesized
     mean difference.</para> 

 <example id="usingttestunwqualtool">
 <title>Using the <parameter>t</parameter>-Test (Unknown and Unequal Variances) Tool</title>
     <para>See <xref linkend="ttest-unequal-tool-ex1" /> for an example 
     of a completed dialog and <xref linkend="ttest-unequal-tool-ex2" />
     for the corresponding output.
     </para>
 </example>
  <figure id="ttest-unequal-tool-ex1">
    <title><parameter>t</parameter>-Test (Unknown and Unequal Variances) Example Data</title>
    <screenshot>
      <mediaobject>
        <imageobject>
          <imagedata fileref="figures/analysistools-ttest-unequal-ex1.png" 
              format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of example data for use in a t-test of two
              samples with unknown and possibly unequal
              variances.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  <figure id="ttest-unequal-tool-ex2">
    <title>Output from the <parameter>t</parameter>-Test (Unknown and Unequal Variances) 
    Tool</title>
    <screenshot>
      <mediaobject>
        <imageobject>
          <imagedata fileref="figures/analysistools-ttest-unequal-ex2.png" 
              format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output of a t-test of two
              samples with unknown and possibly unequal
              variances.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect3>

  <sect3 id="ztest-two-samples-for-means-tool">
     <title><parameter>z</parameter>-Test: Two Samples for Means Tool</title>
  <figure id="ztest-dialog">
    <title><parameter>z</parameter>-Test Tool Dialog Options</title>
    <screenshot>
      <mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ztest.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the options in a z-test of two
              samples.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
     <para>For unpaired  variables with known population variances, enter those variances 
     in the <quote><guilabel>Variable 1 Pop. Variance</guilabel></quote> and 
     <quote><guilabel>Variable 2 Pop. Variance</guilabel></quote> entries.
     When you click on <quote><guibutton>OK</guibutton></quote>, &gnum; will test whether the 
     mean of the difference between the paired variables is equal to the given hypothesized
     mean difference.</para> 

 <example id="usingztesttool">
 <title>Using the <parameter>z</parameter>-Test Tool</title>
     <para>See <xref linkend="ztest-tool-ex1" /> for an example 
     of a completed dialog and <xref linkend="ztest-tool-ex2" />
     for the corresponding output.
     </para>
 </example>
  <figure id="ztest-tool-ex1">
    <title><parameter>z</parameter>-Test Example Data</title>
    <screenshot>
      <mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ztest-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of example data for use in a z-test of two
              samples.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  <figure id="ztest-tool-ex2">
    <title>Output from the <parameter>z</parameter>-Test Tool</title>
    <screenshot>
      <screeninfo>Output from the <parameter>z</parameter>-Test
              Tools
      </screeninfo>
      <mediaobject>
        <imageobject>
          <imagedata fileref="figures/analysistools-ztest-ex2.png" 
              format="PNG" />
          </imageobject>
          <textobject>
            <phrase>An image of the output from a z-test of two
            samples.</phrase>
          </textobject>
      </mediaobject>
    </screenshot>
  </figure>
  </sect3>
</sect2>

  <sect2 id="two-medians-tool">
      <title>Comparing Medians of Two Populations</title>

      <para>&gnum; provides three non-parametric tests to test the null 
      hypothesis that the two samples come from 
      populations with the same median. Two tests, performed through the same
      tool, apply in the case of paired samples:</para>
      <itemizedlist spacing="compact">
 	<listitem><para>Sign Test</para></listitem>
	<listitem><para>Wilcoxon Signed Rank Test</para></listitem>
      </itemizedlist>
      <para>One test applies in the case of unpaired samples:</para>
      <itemizedlist spacing="compact">
 	<listitem><para>Wilcoxon-Mann-Whitney Test</para></listitem>
      </itemizedlist>

      <para></para>
  <sect3 id="two-median-sign-test-tool">
      <title>Sign Test</title>
  <note>
    <para>
      This section describes the two sample (paired) sign test to test the 
      null hypothesis that the two samples come from 
      populations with the same median. The tool to perform a sign test to
      test the null hypothesis that the single sample comes from a  population 
      with a given median is in section  <xref linkend="sign-test-tool" />.
    </para>
  </note>
      <para>This section has not yet been written</para>
  </sect3>
  <sect3 id="two-median-wilcoxon-sign-test-tool">
      <title>Wilcoxon Signed Rank Test</title>
  <note>
    <para>
      This section describes the two sample (paired) Wilcoxon signed rank
      test to test the 
      null hypothesis that the two samples come from 
      populations with the same median. The tool to perform a Wilcoxon 
      signed rank test to
      test the null hypothesis that the single sample comes from a  population 
      with a given median is in section  
      <xref linkend="wilcoxon-sign-test-tool" />.
    </para>
  </note>
      <para>This section has not yet been written</para>
  </sect3>
  <sect3 id="two-median-wilcoxon-mann-whitney-test-tool">
      <title>Wilcoxon-Mann-Whitney Test</title>
      <para>This section has not yet been written</para>
  </sect3>

  </sect2>

  <sect2 id="ftest-two-sample-for-variances-tool">
     <title>F-Test: Two-Sample for Variances Tool</title>

  <figure id="ftest-tool-dialog">
    <title>F-Test Tool Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ftest.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the dialog for an F-test analysis of
              the equality of two variances.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
     <para>Use the F-Test tool to test whether two population  
     variances are different against the null hypothesis that
     they are not.</para>

     <para>Specify the variables in the <quote><guilabel>Variable 1 Range:</guilabel></quote>
     and <quote><guilabel>Variable 2 Range:</guilabel></quote> entries. The <quote><guilabel>Alpha:</guilabel></quote> 
     entry contains the 
     significance level which is by default 5&#037;.</para>

     <para>If the first field of each range contains labels, 
     select the <quote><guibutton>Labels</guibutton></quote> option. The names of 
     the variables will be included in the  output table.</para>

     <para>The results are given in a table.  This table contains
     the mean, variance, count of observations and the degree
     of freedom for both variables. The output table also includes the F-value,
     the one-tailed probability for the F-value, and the F Critical
     value for one-tailed test and the corresponding values for a two 
     tailed test. The one-tailed probability for the
     F-value (<quote><inlineequation><mathphrase>P(F≤f)</mathphrase></inlineequation> one-tail</quote> row) is the probability of making a
     Type I error in the one-tailed test. Similarly, the two-tailed 
     probability for the F-value (<quote><guilabel>P two-tail</guilabel></quote> row)
     is the probability of making a Type I error in the two-tailed test.
     Since in the two-tailed F-Test both critical values are positive, the
     <quote><guilabel>F Critical two-tail</guilabel></quote> row contains two numbers.</para>

     <para>If the output is directed into a specific output range, that
     range should contain at least three columns and eight rows.</para>

   <figure id="ftest-example-1">
    <title>Some Example Data</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ftest-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of some example data for an F-test of
              the equality of two variances.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
 <example id="usingftesttool"><title>Using the F-Test Tool</title>
     <para><xref linkend="ftest-example-1" /> shows some example data and 
     <xref linkend="ftest-example-2" /> the corresponding output.
     </para>
</example>
  <figure id="ftest-example-2">
    <title>F-Test Tool Output</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ftest-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output of an F-test analysis of
              the equality of two variances.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
    
  </sect2>

</sect1>

<sect1 id="multiple_sample_tests">
  <title>Multiple Sample Tests</title>
<sect2 id="anova">
  <title>Analysis of Variance</title>

  <sect3 id="anova-single-factor-tool">
     <title>ANOVA: Single Factor Tool</title>

     <para>
       Use this tool to perform a single factor analysis of the
       variances of given variables. The variables are specified by
       the <quote><guilabel>Input Range:</guilabel></quote> entry.
       The given range can be grouped into the variables either by
       columns, by rows or by areas.  The
       <quote><guilabel>Alpha:</guilabel></quote> entry specifies the
       significance level which is by default 5&#037;.
     </para>

     <para>If the first row or first column of the given range, or the 
     first field of each area contains labels, select the <quote><guibutton>Labels
     </guibutton></quote> option. The names of 
     the variables will be included in the  output table.</para>

     <para>The results of this analysis of variance are presented in 
     a standard ANOVA table. The <quote><guilabel>F critical</guilabel></quote>
     value is the largest value of F that is statistically significant
     using the given significance level (<quote><guilabel>Alpha</guilabel></quote>).</para>

     <para>This tool also calculates the count, sum, average,
     and the variance of each variable.</para>

   <figure id="anova-one-factor-tool-ex1">
    <title>1-factor ANOVA Dialog and Example Data</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ANOVA1-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of a multilevel single factor ANOVA
              analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
<example id="usinganovaonefactortool">
 <title>Using the single factor ANOVA</title>
     <para>See <xref linkend="anova-one-factor-tool-ex1" /> for an example 
     of a completed dialog and <xref
     linkend="anova-one-factor-tool-ex2" />
     for the corresponding output.
     </para>
 </example>
  <figure id="anova-one-factor-tool-ex2">
    <title>Output From a 1-factor ANOVA</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ANOVA1-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output from a multilevel single
              factor ANOVA analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
     

  </sect3>

  <sect3 id="anova-two-factor-tool">
     <title>ANOVA: Two-Factor Tool</title>

     <para>&gnum; can perform two factor fixed effects ANOVAs with and 
     without replication. The same dialog is used and the 
     appropriate tool is selected depending on whether the number of rows 
     per sample is 1 or larger than 1.</para> 

  <sect4 id="anova-two-factor-without-tool">
     <title>ANOVA: Two-Factor Without Replication Tool</title>

     <para>If the number of rows per sample is given as 1, &gnum; 
     performs a two factor fixed effects ANOVA without replication. Each
     column of the input range is interpreted as a level of the first 
     factor while each row is interpreted as a level of the second factor.
     </para>
     <para>The first row and column of the range may contain labels for 
     these levels. In this case the <quote><guibutton>Labels</guibutton></quote> option should be selected.
     </para>
     <para> The <quote><guilabel>Alpha:</guilabel></quote> entry specifies the 
     significance level which is by default 5&#037;.</para>
 <example id="usinganovatwofactorwotool">
 <title>Using the 2-factor ANOVA Without Replication Tool</title>
     <para>See <xref linkend="anova-two-factor-without-tool-ex1" /> for an example 
     of a completed dialog and <xref
     linkend="anova-two-factor-without-tool-ex2" />
     for the corresponding output.
     </para>
 </example>
  <figure id="anova-two-factor-without-tool-ex1">
    <title>2-factor ANOVA Without Replication Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ANOVA2wo-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of a two factor ANOVA without
              replication analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  <figure id="anova-two-factor-without-tool-ex2">
    <title>Output From a 2-factor ANOVA Without Replication</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ANOVA2wo-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output from a two factor ANOVA without
              replication analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  </sect4>

  <sect4 id="anova-two-factor-with-tool">
     <title>ANOVA: Two-Factor With Replication Tool</title>
     <para>If the number of rows per sample is larger than 1, &gnum; 
     performs a two factor fixed effects ANOVA with replication. Each
     column of the input range is interpreted as a level of the first 
     factor while groups of rows (the number of rows in each group given 
     by the <quote><guilabel>number of rows per sample</guilabel></quote> value) are interpreted as levels 
     of the second factor.
     </para>
     <para>The first row and column of the range may contain labels for 
     these levels. In this case the <quote><guibutton>Labels</guibutton></quote> option should be selected.
     </para>
     <para> The <quote><guilabel>Alpha:</guilabel></quote> entry specifies the 
     significance level which is by default 5&#037;.</para>
     <para>See <xref linkend="anova-two-factor-with-tool-ex1" /> for an example 
     of a completed dialog and <xref
     linkend="anova-two-factor-with-tool-ex2" />
     for the corresponding output.
     </para>
  <figure id="anova-two-factor-with-tool-ex1">
    <title>2-factor ANOVA With Replication Dialog</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ANOVA2w-ex1.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of a two factor ANOVA with replication
              analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
  <figure id="anova-two-factor-with-tool-ex2">
    <title>Output From a 2-factor ANOVA With Replication</title>
    <screenshot>
	<mediaobject>
            <imageobject>
              <imagedata fileref="figures/analysistools-ANOVA2w-ex2.png" 
                  format="PNG" />
            </imageobject>
            <textobject>
              <phrase>An image of the output from a two factor ANOVA
              with replication analysis.</phrase>
            </textobject>
           </mediaobject>
    </screenshot>
  </figure>
     
     <para>&gnum; will estimate missing
     values for each level combination as the mean of the existing
     values in that combination. The degrees of freedom are adjusted
     appropriately. </para>

  </sect4>
  </sect3>

</sect2>

  <sect2 id="chi-square-tool">
      <title>Tests for a Contingency Table</title>

  <sect3 id="homogeneity-tool">
      <title>Test of Homogeneity</title>
      <para></para>
  </sect3>
  <sect3 id="independence-tool">
      <title>Test of Independence</title>
      <para></para>
  </sect3>

  </sect2>
</sect1>








  




