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docs: update education/statistics/5

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2026-02-11 14:43:25 +00:00
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@@ -2,7 +2,7 @@
title: 5 title: 5
description: description:
published: true published: true
date: 2026-02-11T14:42:59.270Z date: 2026-02-11T14:43:25.309Z
tags: tags:
editor: ckeditor editor: ckeditor
dateCreated: 2026-02-11T14:42:59.270Z dateCreated: 2026-02-11T14:42:59.270Z
@@ -65,18 +65,18 @@ dateCreated: 2026-02-11T14:42:59.270Z
<figure class="image"><img src="data:image/png;base64,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"></figure> <figure class="image"><img src="data:image/png;base64,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"></figure>
<p><span style="font-family:Arial, Helvetica, sans-serif;">R<sub>a</sub><sup>2</sup> will always be smaller than&nbsp;R<sup>2</sup>.</span></p> <p><span style="font-family:Arial, Helvetica, sans-serif;">R<sub>a</sub><sup>2</sup> will always be smaller than&nbsp;R<sup>2</sup>.</span></p>
<p>&nbsp;</p> <p>&nbsp;</p>
<h3><span style="font-family:Arial, Helvetica, sans-serif;"><strong>Assumptions</strong></span></h3> <h3><span style="font-family:Arial, Helvetica, sans-serif;">Assumptions</span></h3>
<h4><span style="font-family:Arial, Helvetica, sans-serif;"><strong>Linearity</strong></span></h4> <h4><span style="font-family:Arial, Helvetica, sans-serif;">Linearity</span></h4>
<p><span style="font-family:Arial, Helvetica, sans-serif;">The relationship between the explanatory X and the response variable Y should be linear.</span></p> <p><span style="font-family:Arial, Helvetica, sans-serif;">The relationship between the explanatory X and the response variable Y should be linear.</span></p>
<p><span style="font-family:Arial, Helvetica, sans-serif;">Methods for fitting a model to non-linear relationships exist but are beyond the scope of this course.</span></p> <p><span style="font-family:Arial, Helvetica, sans-serif;">Methods for fitting a model to non-linear relationships exist but are beyond the scope of this course.</span></p>
<p><span style="font-family:Arial, Helvetica, sans-serif;">Check using a scatterplot of the data, or a residuals plot.</span></p> <p><span style="font-family:Arial, Helvetica, sans-serif;">Check using a scatterplot of the data, or a residuals plot.</span></p>
<p>&nbsp;</p> <p>&nbsp;</p>
<h4><span style="font-family:Arial, Helvetica, sans-serif;"><strong>Nearly Normal Residuals</strong></span></h4> <h4><span style="font-family:Arial, Helvetica, sans-serif;">Nearly Normal Residuals</span></h4>
<p><span style="font-family:Arial, Helvetica, sans-serif;">The residuals should be nearly normal.</span></p> <p><span style="font-family:Arial, Helvetica, sans-serif;">The residuals should be nearly normal.</span></p>
<p><span style="font-family:Arial, Helvetica, sans-serif;">This condition may not be satisfied when there are unusual observations that do not follow the trend of the rest of the data.</span></p> <p><span style="font-family:Arial, Helvetica, sans-serif;">This condition may not be satisfied when there are unusual observations that do not follow the trend of the rest of the data.</span></p>
<p><span style="font-family:Arial, Helvetica, sans-serif;">Check using histogram or normal probability plot of residuals.</span></p> <p><span style="font-family:Arial, Helvetica, sans-serif;">Check using histogram or normal probability plot of residuals.</span></p>
<p>&nbsp;</p> <p>&nbsp;</p>
<h4><span style="font-family:Arial, Helvetica, sans-serif;"><strong>Constant variability</strong></span></h4> <h4><span style="font-family:Arial, Helvetica, sans-serif;">Constant variability</span></h4>
<p><span style="font-family:Arial, Helvetica, sans-serif;">The variability of points around the least squares line should be roughly constant.</span></p> <p><span style="font-family:Arial, Helvetica, sans-serif;">The variability of points around the least squares line should be roughly constant.</span></p>
<p><span style="font-family:Arial, Helvetica, sans-serif;">This implies that the variability of residuals around the 0 line should be roughly constant as well.</span></p> <p><span style="font-family:Arial, Helvetica, sans-serif;">This implies that the variability of residuals around the 0 line should be roughly constant as well.</span></p>
<p><span style="font-family:Arial, Helvetica, sans-serif;">It is also called homoscedasticity.</span></p> <p><span style="font-family:Arial, Helvetica, sans-serif;">It is also called homoscedasticity.</span></p>