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<h4>Correlation Coefficient</h4>
<p>The correlation coefficient is a number between \(-1\) and \(+1\) (including \(-1\) and \(+1\)) that measures the strength and direction of a linear relationship. The correlation coefficient is denoted by the letter \(r\). Several scatter plots are shown below. The value of the correlation coefficient for the data displayed in each plot is also given. </p>
<p><img alt height="440" role="presentation" src="https://k12.openstax.org/contents/raise/resources/145c40c22a66d767a839006e27971719b1f5ed40" width="350"><br></p>
<p><img alt height="340" role="presentation" src="https://k12.openstax.org/contents/raise/resources/6d12efe2b4803c74d0e500d8ded9bdfdc0c8c8ca" width="350"><br></p>
<p><img alt height="340" role="presentation" src="https://k12.openstax.org/contents/raise/resources/05ab55b9fcb372cf1f3d26385a2e6be01c33b3c8" width="350"><br></p>
<p><img alt height="340" role="presentation" src="https://k12.openstax.org/contents/raise/resources/92fb7cb25185ddad4146c7fd8a5b804d3ca1ff7f" width="350"><br></p>
<p><img alt height="340" role="presentation" src="https://k12.openstax.org/contents/raise/resources/c05dae989827fb0b0fc7ed78d260b9033167c3e1" width="350"><br></p>
<p><img alt height="340" role="presentation" src="https://k12.openstax.org/contents/raise/resources/a7ae5b629d8807a43abc7696ded9e1912ca72b90" width="350"><br></p>
<p><img alt height="340" role="presentation" src="https://k12.openstax.org/contents/raise/resources/3319a8d06f502242dc15ee91d771e05e409d8d05" width="350"><br></p>
<p>When is the value of the correlation coefficient positive?</p>
<ul>
<li>The correlation coefficient is positive when as the \(x\)-values increase, the \(y\)-values also tend to increase. </li>
</ul>
<p>When is the value of the correlation coefficient negative?</p>
<ul>
<li>The correlation coefficient is negative when as the \(x\)-values increase, the \(y\)-values tend to decrease. </li>
</ul>
<p>Is the linear relationship stronger when the correlation coefficient is closer to 0 or to 1 (or \(-1\))?</p>
<ul>
<li>As the points form a stronger negative or positive linear relationship, the correlation coefficient gets farther from 0. Students note that when all of the points are on a line with a positive slope, the correlation coefficient is \(+1\). The correlation coefficient is \(-1\) if all of the points are on a line with a negative slope. </li>
</ul>
<p>The table below shows how you can informally interpret the value of a correlation coefficient.</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">If the value of the correlation coefficient is . . .</th>
<th scope="col">You can say that . . .</th>
</tr>
</thead>
<tbody>
<tr>
<td>
\(r = 1.0\)
</td>
<td>
There is a perfect positive linear relationship.
</td>
</tr>
<tr>
<td>
\( 0.7\;\leq\;r\;<\;1.0\)
</td>
<td>
There is a strong positive linear relationship.
</td>
</tr>
<tr>
<td>
\( 0.3\;\leq\;r\;<\;0.7\)
</td>
<td>
There is a moderate positive linear relationship.
</td>
</tr>
<tr>
<td>
\( 0\;<\;r\;<\;0.3\)
</td>
<td>
There is a weak positive linear relationship.
</td>
</tr>
<tr>
<td>
\(r = 0\)
</td>
<td>
There is no linear relationship.
</td>
</tr>
<tr>
<td>
\( -0.3\;<\;r\;<\;0\)
</td>
<td>
There is a weak negative linear relationship.
</td>
</tr>
<tr>
<td>
\( -0.7\;<\;r\;\leq\;-0.3\)
</td>
<td>
There is a moderate negative linear relationship.
</td>
</tr>
<tr>
<td>
\( -1.0\;<\;r\;\leq\;-0.7\)
</td>
<td>
There is a strong negative linear relationship.
</td>
</tr>
<tr>
<td>
\(r = -1.0\)
</td>
<td>
There is a perfect negative linear relationship.
</td>
</tr>
</tbody>
</table>
<br>
<h4>Try It: Correlation Coefficient</h4><br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<p><img src="https://k12.openstax.org/contents/raise/resources/278eaa2d070c2003bcb24f90dabb39f48a33436b" width="700"></p>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.
</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Compare your answer:</p>
<p>Your answer may vary, but here is a sample.</p>
<p>The strongest linear relationship would be Scatter Plot 3. The points are closest to having a linear model. The weakest linear relationship would be Scatter Plot 2 because the points are spread apart the most and are less likely to have a strong linear relationship.</p>
</div>