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14.5: r² and the Standard Error of the Estimate of y′ (2024)
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SartyUniversity of Saskatchewan\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}..."><meta property="article:published_time" content="2024-11-26T22:29:11+07:00"><meta property="article:modified_time" content="2024-11-26T22:29:11+07:00"><link rel="canonical" href="https://fantasygameday.net/article/14-5-r2-and-the-standard-error-of-the-estimate-of-y/1233"/><meta property="og:title" content="14.5: r² and the Standard Error of the Estimate of y′ (2024)" /><meta property="og:description" content="Last updated Save as PDF Page ID52061Gordon E. SartyUniversity of Saskatchewan\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}..." /><meta property="og:type" content="article" /><meta property="og:site_name" content="Fantasygameday" /><meta property="og:url" content="https://fantasygameday.net/article/14-5-r2-and-the-standard-error-of-the-estimate-of-y/1233" /><meta property="og:image" content="https://i0.wp.com/a.mtstatic.com/@public/production/site_4462/1474933326-social-share.png" /><meta name="twitter:title" content="14.5: r² and the Standard Error of the Estimate of y′ (2024)" /><meta name="twitter:description" content="Last updated Save as PDF Page ID52061Gordon E. SartyUniversity of Saskatchewan\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}..." /><meta name="twitter:image" content="https://i0.wp.com/a.mtstatic.com/@public/production/site_4462/1474933326-social-share.png" /><meta name="twitter:card" content="summary_large_image" /><meta name="twitter:label1" content="Written by" /><meta name="twitter:data1" content="Edwin Metz" /><meta name="twitter:label2" content="Est. reading time" /><meta name="twitter:data2" content="3 minutes" /><script type="application/ld+json">{"@context":"https://schema.org","@type":"AggregateRating","name":"14.5: r² and the Standard Error of the Estimate of y′ (2024)","description":"Last updated Save as PDF Page ID52061Gordon E. 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md:mt-3 ml-0 flex-col md:flex-row"> <div class="w-full"> <h1 class="entry-title mb-5 text-center font-semibold text-2xl xl:text-4xl !leading-common"> 14.5: r² and the Standard Error of the Estimate of y′ (2024) </h1> <div class="text-center my-2 sm:my-4 " > <ins class="adsbygoogle adsense h-ad3 " data-ad-client="ca-pub-3373908032421588" style="display:block" data-ad-format="auto" data-ad-slot="3352989513" data-full-width-responsive="true" data-adtest="off" > </ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script> </div> <article class="article-detail"> <section class="entry-content paragraph"><article tabindex="-1"> <header> <ol data-ga-action="Page actions"> <li><dl> <dt>Last updated</dt> <dd> <span data-timestamp="2024-11-22T09:28:55Z"></span> </dd></dl></li> <li><span> Save as PDF</span></li> </ol> </header> <header><li><dl><dt>Page ID</dt><dd>52061</dd></dl></li><div><ul><li><img alt="14.5: r² and the Standard Error of the Estimate of y′ (1)" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" title="14.5: r² and the Standard Error of the Estimate of y′ (1)" class="lazyload" referrerpolicy="no-referrer" data-src="https://i0.wp.com/biz.libretexts.org/@api/deki/files/5084/girl-160172__340.png" data-src2="https://biz.libretexts.org/@api/deki/files/5084/girl-160172__340.png" onerror="if(this.src==this.getAttribute('data-src2')){onImageError(this,true)}else{this.src=this.getAttribute('data-src2')}"></li><li><a href="https://artsandscience.usask.ca/profile/GSarty" rel="noopener nofollow" target="_blank">Gordon E. Sarty</a></li><li><a href="https://www.usask.ca/" rel="noopener nofollow" target="_blank">University of Saskatchewan</a></li></ul></div><div><p>\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)</p><p>\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)</p><p>\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)</p><p>( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)</p><p>\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)<div class="sm:flex sm:items-center sm:justify-center sm:flex-row sm:space-x-1 insert" style="margin-top:0; margin-bottom:0;"><ins class="adsbygoogle sm:w-1/2" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-full-width-responsive="true" data-ad-slot="3352989513" data-ad-client="ca-pub-3373908032421588" data-adtest="off"></ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script><ins class="adsbygoogle sm:w-1/2" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-full-width-responsive="true" data-ad-slot="3352989513" data-ad-client="ca-pub-3373908032421588" data-adtest="off"></ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script></div></p><p>\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)</p><p>\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)</p><p>\( \newcommand{\Span}{\mathrm{span}}\)</p><p>\( \newcommand{\id}{\mathrm{id}}\)</p><p>\( \newcommand{\Span}{\mathrm{span}}\)</p><p>\( \newcommand{\kernel}{\mathrm{null}\,}\)</p><p>\( \newcommand{\range}{\mathrm{range}\,}\)</p><p>\( \newcommand{\RealPart}{\mathrm{Re}}\)<div class="bg-primary-100 rounded-lg my-6 px-2 py-3 font-semibold insert" style="color: rgb(37, 99, 235)"><div class="text-common !leading-common ml-1 insert" style="margin-top: 0.2rem; margin-bottom: 0.2rem">See Also</div><a class="list-item toc-h2 py-1" style="border-bottom: none" href="https://fantasygameday.net/article/a1c-diabetes-treatment-options-humulin-r-u-500/1233" target="_blank" rel="noopener">A1C & Diabetes Treatment Options | Humulin R U-500</a><a class="list-item toc-h2 py-1" style="border-bottom: none" href="https://fantasygameday.net/article/r-squared-r2-formula-example-calculation-use-explanation/1233" target="_blank" rel="noopener">R-squared (R2) - Formula | Example | Calculation | Use Explanation</a><a class="list-item toc-h2 py-1" style="border-bottom: none" href="https://fantasygameday.net/article/testing-the-significance-of-the-correlation-coefficient-introductory-statistics/1233" target="_blank" rel="noopener">Testing the Significance of the Correlation Coefficient – Introductory Statistics</a><a class="list-item toc-h2 py-1" style="border-bottom: none" href="https://fantasygameday.net/article/what-is-insulation-r-value-get-the-facts-before-you-insulate/1233" target="_blank" rel="noopener">What Is Insulation R-Value? Get the Facts Before You Insulate</a></div></p><p>\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)</p><p>\( \newcommand{\Argument}{\mathrm{Arg}}\)<div class="sm:flex sm:items-center sm:justify-center sm:flex-row sm:space-x-1 insert" style="margin-top:0; margin-bottom:0;"><ins class="adsbygoogle sm:w-1/2" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-full-width-responsive="true" data-ad-slot="3352989513" data-ad-client="ca-pub-3373908032421588" data-adtest="off"></ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script><ins class="adsbygoogle sm:w-1/2" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-full-width-responsive="true" data-ad-slot="3352989513" data-ad-client="ca-pub-3373908032421588" data-adtest="off"></ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script></div></p><p>\( \newcommand{\norm}[1]{\| #1 \|}\)</p><p>\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)</p><p>\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)</p><p>\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)</p><p>\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)</p><p>\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)</p><p>\( \newcommand{\vectorC}[1]{\textbf{#1}}\)</p><p>\( \newcommand{\vectorD}[1]{\overrightarrow{#1}}\)</p><p>\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}\)</p><p>\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)<div class="sm:flex sm:items-center sm:justify-center sm:flex-row sm:space-x-1 insert" style="margin-top:0; margin-bottom:0;"><ins class="adsbygoogle sm:w-1/2" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-full-width-responsive="true" data-ad-slot="3352989513" data-ad-client="ca-pub-3373908032421588" data-adtest="off"></ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script><ins class="adsbygoogle sm:w-1/2" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-full-width-responsive="true" data-ad-slot="3352989513" data-ad-client="ca-pub-3373908032421588" data-adtest="off"></ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script></div></p><p>\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)</p><p>\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)</p><p>\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)</p></div></header> <section> <p>Consider the deviations :</p><p><img alt="14.5: r² and the Standard Error of the Estimate of y′ (2)" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" class="lazyload" referrerpolicy="no-referrer" title="14.5: r² and the Standard Error of the Estimate of y′ (2)" data-src="https://i0.wp.com/stats.libretexts.org/@api/deki/files/39800/devgraph-300x144.jpg?revision=1" data-src2="https://stats.libretexts.org/@api/deki/files/39800/devgraph-300x144.jpg?revision=1" onerror="if(this.src==this.getAttribute('data-src2')){onImageError(this,true)}else{this.src=this.getAttribute('data-src2')}"></p><p>Looking at the picture we see that</p>\[\begin{eqnarray*} \textrm{total deviation} & = & \textrm{explained deviation + unexplained deviation}\\ (y_{i} - \overline{y}_{i}) & = & (y^{\prime}_{i} - \overline{y}_{i}) + (y_{i} - y^{\prime}_{i}) \end{eqnarray*}\]<p><span>Remember that variance is the sum of the squared deviations (divided by degrees of freedom), so squaring the above and summing gives:</span></p>\[ \sum_{i=1}^{n} (y_{i} - \overline{y}_{i})^{2} = \sum_{i=1}^{n} (y^{\prime}_{i} - \overline{y}_{i})^{2} + \sum_{i=1}^{n} (y_{i} - y^{\prime}_{i})^{2} \]<p>(the cross terms all cancel because \(y^{\prime}\) is the least square solution and \(a = \overline{y} - b \overline{x}\), see Section 14.6.1, below, for details). This is also a sum of squares statement:</p>\[ \mbox{SS}_{T} = \mbox{SS}_{R} + \mbox{SS}_{E} \]<p>where SS\(_{E} = \sum (y_{i} - y^{\prime}_{i})^{2}\), SS\(_{T} = \sum (y_{i} - \overline{y})^{2}\) and SS\(_{R} = \sum (y^{\prime}_{i} - \overline{y})^{2}\) are the sum of squares — error, sum of squares — total and sum of squares — regression (explained) respectively.</p><p>Dividing by the degrees of freedom, which is \(n-2\) in this {\em bivariate} situation, we get:<div class="sm:flex sm:items-center sm:justify-center sm:flex-row sm:space-x-1 insert" style="margin-top:0; margin-bottom:0;"><ins class="adsbygoogle sm:w-1/2" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-full-width-responsive="true" data-ad-slot="3352989513" data-ad-client="ca-pub-3373908032421588" data-adtest="off"></ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script><ins class="adsbygoogle sm:w-1/2" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-full-width-responsive="true" data-ad-slot="3352989513" data-ad-client="ca-pub-3373908032421588" data-adtest="off"></ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script></div></p>\[\begin{eqnarray*} \frac{\sum (y_{i} - \overline{y}_{i})^{2}}{n-2} & = & \frac{\sum (y^{\prime}_{i} - \overline{y}_{i})^{2}}{n-2} + \frac{\sum (y_{i} - y^{\prime}_{i})^{2}}{n-2} \\ \mbox{total variance} & = & \mbox{explained variance} + \mbox{unexplained variance} \\ & = & \mbox{signal (or model)} + \mbox{noise} \end{eqnarray*}\]<p>It turns out that</p>\[ r^{2} = \frac{\mbox{explained variance}}{\mbox{total variance}} = \frac{ \mbox{SS}_{R} }{ \mbox{SS}_{T} } \]<p>The quantity \(r^{2}\) is called the <em>coefficient of determination</em> and gives the <strong><em>the fraction of variance explained by the model</em></strong> (here the model is the equation of a line). The quantity \(r^{2}\) appears with many statistical models. For example with ANOVA it turns out that the “effect size” eta-squared is the fraction of variance explained by the ANOVA model<span>[1]</span>, \(\eta^{2} = r^{2}\).</p><p>The <em>standard error of the estimate</em> is the standard deviation of the noise (the square root of the unexplained variance) and is given by</p>\[ s_{\mbox{est}} = \sqrt{\frac{\sum (y - y^{\prime})^{2} }{n-2} } = \sqrt{\frac{\sum y^{2} - a \sum y - b \sum xy}{n-2} } \]<p><strong>Example 14.4</strong>: Continuing with the data of Example 14.3, we had</p>\[ \sum y = 511 \;\;\; \sum y^{2} = 38993 \;\;\; \sum xy = 3745 \;\;\; a = 102.493 \;\;\; b = -3.622 \;\;\; n=7 \]<p>so</p>\[\begin{eqnarray*} s_{\mbox{est}} & = & \sqrt{\frac{(38993) - (102.493)(511) - (-3.622)(3745)}{5} } \\ s_{\mbox{est}} & = & \sqrt{\frac{38993 - 52373.923 + 13564.39}{5} } \\ s_{\mbox{est}} & = & 6.06 \end{eqnarray*}\]<p>▢</p><p>Here is a graphical interpretation of \(s_{\mbox{est}}\) :</p><img alt="14.5: r² and the Standard Error of the Estimate of y′ (3)" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" class="lazyload" referrerpolicy="no-referrer" title="14.5: r² and the Standard Error of the Estimate of y′ (3)" data-src="https://i0.wp.com/stats.libretexts.org/@api/deki/files/39818/sest-300x150.jpg?revision=1" data-src2="https://stats.libretexts.org/@api/deki/files/39818/sest-300x150.jpg?revision=1" onerror="if(this.src==this.getAttribute('data-src2')){onImageError(this,true)}else{this.src=this.getAttribute('data-src2')}"><p>The assumption for computing confidence intervals for is that \(s_{\mbox{est}}\) is independent of \(x\). This is the assumption of homoscedasticity. You can think of the regression situation as a generalized one-way ANOVA where instead of having a finite number of discrete populations for the IV, we have an infinite number of (continuous) populations. All the populations have the same variance \(\sigma^{2}\) (and they are assumed to be normal) and \(s_{\mbox{est}^{2}}\) is the pooled estimate of that variance.</p><div mt-section-origin="Bookshelves/Applied_Statistics/Introduction_to_Applied_Statistics_for_Psychology_Students_(Sarty)/14:_Correlation_and_Regression/14.05:_r_and_the_Standard_Error_of_the_Estimate_of_y"><p>Squaring both sides of<div class="sm:flex sm:items-center sm:justify-center sm:flex-row sm:space-x-1 insert" style="margin-top:0; margin-bottom:0;"><ins class="adsbygoogle sm:w-1/2" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-full-width-responsive="true" data-ad-slot="3352989513" data-ad-client="ca-pub-3373908032421588" data-adtest="off"></ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script><ins class="adsbygoogle sm:w-1/2" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-full-width-responsive="true" data-ad-slot="3352989513" data-ad-client="ca-pub-3373908032421588" data-adtest="off"></ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script></div></p><p>\[ (y_{i} - \overline{y}_{i}) = (y^{\prime}_{i} - \overline{y}_{i}) + (y_{i} - y^{\prime}_{i}) \]</p><p>and summing gives</p><p>\[ \sum (y_{i} - \overline{y}_{i})^{2} = \sum (y^{\prime}_{i} - \overline{y}_{i})^{2} + \sum (y_{i} - y^{\prime}_{i})^{2} + \sum 2 (y^{\prime}_{i} - \overline{y})(y_{i} - y^{\prime}_{i}) \]</p><p>Working on that cross term, using \(a = \overline{y} - b \overline{x}\), we get</p><p>\[\begin{eqnarray*} \sum 2 (y^{\prime}_{i} - \overline{y})(y_{i} - y^{\prime}_{i}) & = & \sum 2((\overline{y} - b \overline{x} + b x_{i}) - \overline{y})(y_{i} - y^{\prime}_{i}) \\ & = & \sum 2((\overline{y} + b(x_{i} - \overline{x}))-\overline{y})(y_{i} - y^{\prime}_{i}) \\ & = & \sum 2(b(x_{i}- \overline{x}))(y_{i} - y^{\prime}_{i}) \\ & = & \sum 2 b (x_{i} - \overline{x})(y_{i} - (\overline{y} + b(x_{i} - \overline{x}))) \\ & = & \sum 2 b ((y_{i} - \overline{y})(x_{i} - \overline{x}) - b(x_{i} - \overline{x})^{2}) \\ & = & 2 b \sum ((y_{i} - \overline{y})(x_{i} - \overline{x}) - (y_{i} - \overline{y})(x_{i} - \overline{x})) = 0 \end{eqnarray*}\]</p><p>where</p><p>\[ b = \frac{\sum (x_{i}-\overline{x})(y_{i}-\overline{y})}{\sum(x_{i}-\overline{x})^{2}} \]</p><p>was used in the last line.</p><div><ol> <li>In ANOVA the ``model'' is the difference of means between the groups. We will see more about this aspect of ANOVA in <a href="https://openpress.usask.ca/introtoappliedstatsforpsych/part/17-overview-of-the-general-linear-model" rel="noopener nofollow" target="_blank">Chapter 17</a>. <span>↵</span></li></ol></div> </div> </section> </article></section> <img class="lazyload" data-src="https://ts2.mm.bing.net/th?q=14.5: r² and the Standard Error of the Estimate of y′ (2024)" alt="14.5: r² and the Standard Error of the Estimate of y′ (2024)" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mP8/h8AAvMB+NzmkbcAAAAASUVORK5CYII="/> <section> </section> </article> <h2 id="references" class="text-xl font-semibold mt-3 mb-3">References</h2> <ul class="ml-5 list-disc"> <li><a target="_blank" rel="noopener nofollow" class="text-sm break-all" href="https://pressbooks-dev.oer.hawaii.edu/introductorystatistics/chapter/testing-the-significance-of-the-correlation-coefficient/">https://pressbooks-dev.oer.hawaii.edu/introductorystatistics/chapter/testing-the-significance-of-the-correlation-coefficient/</a> </li> <li><a target="_blank" rel="noopener nofollow" class="text-sm break-all" href="https://www.bobvila.com/interior/what-is-r-value/">https://www.bobvila.com/interior/what-is-r-value/</a> </li> <li><a target="_blank" rel="noopener nofollow" class="text-sm break-all" href="https://www.thefinanceintellect.com/the-finance-intellect/financial-ratios/r-squared.html">https://www.thefinanceintellect.com/the-finance-intellect/financial-ratios/r-squared.html</a> </li> <li><a target="_blank" rel="noopener nofollow" class="text-sm break-all" href="https://insulins.lilly.com/humulin">https://insulins.lilly.com/humulin</a> </li> <li><a target="_blank" rel="noopener nofollow" class="text-sm break-all" href="https://stats.libretexts.org/Bookshelves/Applied_Statistics/Introduction_to_Applied_Statistics_for_Psychology_Students_(Sarty)/14%3A_Correlation_and_Regression/14.05%3A_r_and_the_Standard_Error_of_the_Estimate_of_y">https://stats.libretexts.org/Bookshelves/Applied_Statistics/Introduction_to_Applied_Statistics_for_Psychology_Students_(Sarty)/14%3A_Correlation_and_Regression/14.05%3A_r_and_the_Standard_Error_of_the_Estimate_of_y</a> </li> </ul> <div class="text-2xl font-semibold mt-5 mb-5">Top Articles</div> <div class="container-doc"> <a href="https://fantasygameday.net/article/rise:-race-the-future-mailbox/1233" target="_blank" rel="noopener" class="group mb-20 flex flex-col items-center"> <img class="lazyload h-56 object-cover" data-src="https://ts2.mm.bing.net/th?q=Rise: Race The Future Mailbox" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mP8/h8AAvMB+NzmkbcAAAAASUVORK5CYII="/> <div class="group-hover:text-primary-500 block font-medium mt-5 text-2xl"> Rise: Race The Future Mailbox </div> </a> <a href="https://fantasygameday.net/article/14-free-fun-christmas-do-a-dot-printables-for-kids/1233" target="_blank" rel="noopener" class="group mb-20 flex flex-col items-center"> <img class="lazyload h-56 object-cover" data-src="https://ts2.mm.bing.net/th?q=14 Free & Fun Christmas Do-A-Dot Printables for Kids" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mP8/h8AAvMB+NzmkbcAAAAASUVORK5CYII="/> <div class="group-hover:text-primary-500 block font-medium mt-5 text-2xl"> 14 Free & Fun Christmas Do-A-Dot Printables for Kids </div> </a> <a href="https://fantasygameday.net/article/broken-spell-2-underground/1233" target="_blank" rel="noopener" class="group mb-20 flex flex-col items-center"> <img class="lazyload h-56 object-cover" data-src="https://ts2.mm.bing.net/th?q=Broken Spell 2 Underground" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mP8/h8AAvMB+NzmkbcAAAAASUVORK5CYII="/> <div class="group-hover:text-primary-500 block font-medium mt-5 text-2xl"> Broken Spell 2 Underground </div> </a> </div> <div class="text-2xl font-semibold mt-5 mb-5">Latest Posts</div> <div class="container-doc"> <a href="https://fantasygameday.net/article/ea-sports-fc-24-xbox-one-xbox-series-x-games-bol/1233" target="_blank" rel="noopener" class="group mb-20 flex flex-col items-center"> <img class="lazyload h-56 object-cover" data-src="https://ts2.mm.bing.net/th?q=EA Sports FC 24 - 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