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<meta itemprop="keywords" content="Analysis toolpack,analysis of variance">
<meta itemprop="keywords" content="Analysis toolpack,ANOVA">
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<a name="anova"></a>
<h1 id="hd_id1000060" dir="auto">
<a name="anovah1"></a>Analysis of Variance (ANOVA)</h1>
<p id="par_id1001240" class="paragraph" dir="auto"><span class="avis">Produces the analysis of variance (ANOVA) of a given data set</span></p>
<a name="howtoget"></a><div class="howtoget">
<div><p class="howtogetheader" dir="auto"><a name="wie"></a>To access this command...</p></div>
<div class="howtogetbody">
<p id="par_id1001250" class="paragraph" dir="auto"><a name="sam01"></a>Choose <span class="menuitem">Data - Statistics - Analysis of Variance (ANOVA)</span></p>
</div>
</div>
<br>
<p id="par_id1001260" class="paragraph" dir="auto">ANOVA is the acronym for <span class="emph">AN</span>alysis <span class="emph">O</span>f <span class="emph">VA</span>riance. This tool produces the analysis of variance of a given data set</p>
<div class="note">
<div class="noteicon" dir="auto"><img src="media/icon-themes/res/helpimg/note.svg" alt="note" style="width:40px;height:40px;"></div>
<div class="notetext"><p id="par_id1001270" dir="auto">For more information on ANOVA, refer to the <a target="_blank" href="https://en.wikipedia.org/wiki/ANOVA">corresponding Wikipedia article</a>.</p></div>
</div>
<br>
<div class="embedded">
<a name="data"></a>
<h3 id="hd_id1000010" dir="auto">Data</h3>
<p id="par_id1000010" class="paragraph" dir="auto"><span class="emph">Input Range</span>: The reference of the range of the data to analyze.</p>
<p id="par_id1000020" class="paragraph" dir="auto"><span class="emph">Results to</span>: The reference of the top left cell of the range where the results will be displayed.</p>
</div>
<div class="embedded">
<a name="grouped"></a>
<h4 id="hd_id1000020" dir="auto">Grouped By</h4>
<p id="par_id1000030" class="paragraph" dir="auto">Select whether the input data has <span class="emph">columns</span> or <span class="emph">rows</span> layout.</p>
</div>
<h2 id="hd_id1000070" dir="auto">Type</h2>
<p id="par_id1001280" class="paragraph" dir="auto">Select if the analysis is for a <span class="emph">single factor</span> or for <span class="emph">two factor</span> ANOVA.</p>
<h2 id="hd_id1000080" dir="auto">Parameters</h2>
<p id="par_id1001290" class="paragraph" dir="auto"><span class="emph">Alpha</span>: the level of significance of the test.</p>
<p id="par_id1001300" class="paragraph" dir="auto"><span class="emph">Rows per sample</span>: Define how many rows a sample has.</p>
<div class="embedded">
<a name="sampledata01"></a>
<div class="embedded">
<a name="sectionexample"></a>
<h4 id="hd_id281171" dir="auto">Example</h4>
</div>
<p id="par_id1000040" class="paragraph" dir="auto">The following data will be used as example </p>
<table border="1" class="" cellpadding="0" cellspacing="0">
<tr>
<td rowspan="" colspan="" dir="auto">
</td>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000050" class="tableheadintable" dir="auto">A</p>
</th>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000060" class="tableheadintable" dir="auto">B</p>
</th>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000070" class="tableheadintable" dir="auto">C</p>
</th>
</tr>
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000080" class="tableheadintable" dir="auto">1</p>
</th>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000090" class="tableheadintable" dir="auto">Maths</p>
</th>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000100" class="tableheadintable" dir="auto">Physics</p>
</th>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000110" class="tableheadintable" dir="auto">Biology</p>
</th>
</tr>
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000120" class="tableheadintable" dir="auto">2</p>
</th>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000130" class="tablecontentintable" dir="auto">47</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000140" class="tablecontentintable" dir="auto">67</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000150" class="tablecontentintable" dir="auto">33</p>
</td>
</tr>
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000160" class="tableheadintable" dir="auto">3</p>
</th>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000170" class="tablecontentintable" dir="auto">36</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000180" class="tablecontentintable" dir="auto">68</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000190" class="tablecontentintable" dir="auto">42</p>
</td>
</tr>
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000200" class="tableheadintable" dir="auto">4</p>
</th>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000210" class="tablecontentintable" dir="auto">40</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000220" class="tablecontentintable" dir="auto">65</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000230" class="tablecontentintable" dir="auto">44</p>
</td>
</tr>
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000240" class="tableheadintable" dir="auto">5</p>
</th>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000250" class="tablecontentintable" dir="auto">39</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000260" class="tablecontentintable" dir="auto">64</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000270" class="tablecontentintable" dir="auto">60</p>
</td>
</tr>
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000280" class="tableheadintable" dir="auto">6</p>
</th>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000290" class="tablecontentintable" dir="auto">38</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000300" class="tablecontentintable" dir="auto">43</p>
</td>
</tr>
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000310" class="tableheadintable" dir="auto">7</p>
</th>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000320" class="tablecontentintable" dir="auto">47</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000330" class="tablecontentintable" dir="auto">84</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000340" class="tablecontentintable" dir="auto">62</p>
</td>
</tr>
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000350" class="tableheadintable" dir="auto">8</p>
</th>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000360" class="tablecontentintable" dir="auto">29</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000370" class="tablecontentintable" dir="auto">80</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000380" class="tablecontentintable" dir="auto">51</p>
</td>
</tr>
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000390" class="tableheadintable" dir="auto">9</p>
</th>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000400" class="tablecontentintable" dir="auto">27</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000410" class="tablecontentintable" dir="auto">49</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000420" class="tablecontentintable" dir="auto">40</p>
</td>
</tr>
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000430" class="tableheadintable" dir="auto">10</p>
</th>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000440" class="tablecontentintable" dir="auto">57</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000450" class="tablecontentintable" dir="auto">49</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000460" class="tablecontentintable" dir="auto">12</p>
</td>
</tr>
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000470" class="tableheadintable" dir="auto">11</p>
</th>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000480" class="tablecontentintable" dir="auto">56</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000490" class="tablecontentintable" dir="auto">33</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000500" class="tablecontentintable" dir="auto">60</p>
</td>
</tr>
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000510" class="tableheadintable" dir="auto">12</p>
</th>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000520" class="tablecontentintable" dir="auto">57</p>
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
</tr>
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1000530" class="tableheadintable" dir="auto">13</p>
</th>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1000540" class="tablecontentintable" dir="auto">26</p>
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
</tr>
</table>
<br>
</div>
<p id="par_id1001310" class="paragraph" dir="auto">The following table displays the results of the <span class="emph">analysis of variance (ANOVA)</span> of the sample data above.</p>
<table border="1" class="" cellpadding="0" cellspacing="0">
<tr>
<th rowspan="" colspan="" class="tableheadcell" dir="auto">
<p id="par_id1001320" class="tableheadintable" dir="auto">ANOVA - Single Factor</p>
</th>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
</tr>
<tr>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001330" class="tablecontentintable" dir="auto">Alpha</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001340" class="tablecontentintable" dir="auto">0.05</p>
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
</tr>
<tr>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
</tr>
<tr>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001350" class="tablecontentintable" dir="auto">Groups</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001360" class="tablecontentintable" dir="auto">Count</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001370" class="tablecontentintable" dir="auto">Sum</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001380" class="tablecontentintable" dir="auto">Mean</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001390" class="tablecontentintable" dir="auto">Variance</p>
</td>
<td rowspan="" colspan="" dir="auto">
</td>
</tr>
<tr>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001400" class="tablecontentintable" dir="auto">Column 1</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001410" class="tablecontentintable" dir="auto">11</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001420" class="tablecontentintable" dir="auto">461</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001430" class="tablecontentintable" dir="auto">41.9090909091</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001440" class="tablecontentintable" dir="auto">139.4909090909</p>
</td>
<td rowspan="" colspan="" dir="auto">
</td>
</tr>
<tr>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001450" class="tablecontentintable" dir="auto">Column 2</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001460" class="tablecontentintable" dir="auto">10</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001470" class="tablecontentintable" dir="auto">597</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001480" class="tablecontentintable" dir="auto">59.7</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001490" class="tablecontentintable" dir="auto">287.1222222222</p>
</td>
<td rowspan="" colspan="" dir="auto">
</td>
</tr>
<tr>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001500" class="tablecontentintable" dir="auto">Column 3</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001510" class="tablecontentintable" dir="auto">10</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001520" class="tablecontentintable" dir="auto">447</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001530" class="tablecontentintable" dir="auto">44.7</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001540" class="tablecontentintable" dir="auto">227.3444444444</p>
</td>
<td rowspan="" colspan="" dir="auto">
</td>
</tr>
<tr>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
</tr>
<tr>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001550" class="tablecontentintable" dir="auto">Source of Variation</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001560" class="tablecontentintable" dir="auto">SS</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001570" class="tablecontentintable" dir="auto">df</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001580" class="tablecontentintable" dir="auto">MS</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001590" class="tablecontentintable" dir="auto">F</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001600" class="tablecontentintable" dir="auto">P-value</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id841554943563747" class="paragraphintable" dir="auto">F-critical</p>
</td>
</tr>
<tr>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001610" class="tablecontentintable" dir="auto">Between Groups</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001620" class="tablecontentintable" dir="auto">1876.5683284457</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001630" class="tablecontentintable" dir="auto">2</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001640" class="tablecontentintable" dir="auto">938.2841642229</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001650" class="tablecontentintable" dir="auto">4.3604117704</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001660" class="tablecontentintable" dir="auto">0.0224614952</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id411554944014360" class="paragraphintable" dir="auto">3.340385558</p>
</td>
</tr>
<tr>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001670" class="tablecontentintable" dir="auto">Within Groups</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001680" class="tablecontentintable" dir="auto">6025.1090909091</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001690" class="tablecontentintable" dir="auto">28</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001700" class="tablecontentintable" dir="auto">215.1824675325</p>
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
</tr>
<tr>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001710" class="tablecontentintable" dir="auto">Total</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001720" class="tablecontentintable" dir="auto">7901.6774193548</p>
</td>
<td rowspan="" colspan="" dir="auto">
<p id="par_id1001730" class="tablecontentintable" dir="auto">30</p>
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
<td rowspan="" colspan="" dir="auto">
</td>
</tr>
</table>
<br>
<a name="relatedtopics"></a><div class="relatedtopics">
<p class="related" itemprop="mentions" dir="auto"><a name="related"></a><span class="emph">Related Topics</span>
</p>
<div class="relatedbody" itemprop="mentions">
<div class="embedded">
<a name="otherstat"></a>
<div class="embedded"><p class="embedded" dir="auto"><a target="_top" href="en-US/text/scalc/01/statistics_sampling.html">Sampling</a></p></div>
<div class="embedded"><p class="embedded" dir="auto"><a target="_top" href="en-US/text/scalc/01/statistics_descriptive.html">Descriptive Statistics</a></p></div>
<div class="embedded"><p class="embedded" dir="auto"><a target="_top" href="en-US/text/scalc/01/statistics_anova.html">Analysis of Variance (ANOVA)</a></p></div>
<div class="embedded"><p class="embedded" dir="auto"><a target="_top" href="en-US/text/scalc/01/statistics_correlation.html">Correlation</a></p></div>
<div class="embedded"><p class="embedded" dir="auto"><a target="_top" href="en-US/text/scalc/01/statistics_covariance.html">Covariance</a></p></div>
<div class="embedded"><p class="embedded" dir="auto"><a target="_top" href="en-US/text/scalc/01/statistics_exposmooth.html">Exponential Smoothing</a></p></div>
<div class="embedded"><p class="embedded" dir="auto"><a target="_top" href="en-US/text/scalc/01/statistics_movingavg.html">Moving Average</a></p></div>
<div class="embedded"><p class="embedded" dir="auto"><a target="_top" href="en-US/text/scalc/01/statistics_regression.html">Regression Analysis</a></p></div>
<div class="embedded"><p class="embedded" dir="auto"><a target="_top" href="en-US/text/scalc/01/statistics_test_t.html">Paired t-test</a></p></div>
<div class="embedded"><p class="embedded" dir="auto"><a target="_top" href="en-US/text/scalc/01/statistics_test_f.html">F-test</a></p></div>
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