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<h1>
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Pearson correlation coefficient
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</h1>
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<time class="published" datetime="2018-11-14T21:21:00+01:00">
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14 novembre 2018
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</li>
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<li>Mathematics</li>
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<li>Statistics</li>
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<h2>Covariance</h2>
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<p>This is a measure of how two random variables change together, or the strength of their correlation.</p>
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<p>Consider two random variables, <span class="math">\(X\)</span> and <span class="math">\(Y\)</span>, each with <span class="math">\(n\)</span> values (i.e., <span class="math">\(x_1\)</span>, <span class="math">\(x_2\)</span>, <span class="math">\(...\)</span>, <span class="math">\(x_n\)</span> and <span class="math">\(y_1\)</span>, <span class="math">\(y_2\)</span>, <span class="math">\(...\)</span>, <span class="math">\(y_n\)</span>). The covariance of <span class="math">\(X\)</span> and <span class="math">\(Y\)</span> can be found using either of the following equivalent formulas: </p>
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<div class="math">$$cov(X,Y)=\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})\cdot(y_i-\bar{y})$$</div>
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<p>or </p>
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<div class="math">$$cov(X,Y)=\frac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{1}{2}(x_i-x_j)\cdot(y_i-y_j))$$</div>
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<div class="math">$$cov(X,Y)=\frac{1}{n^2}\sum_{i}\sum_{j\gt i}^{n}(x_i-x_j)\cdot(y_i-y_j)$$</div>
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<p>where, <span class="math">\(\bar{x}\)</span> is the mean of <span class="math">\(X\)</span> (or <span class="math">\(\mu_X\)</span>) and <span class="math">\(\bar{y}\)</span> is the mean of <span class="math">\(Y\)</span> (or <span class="math">\(\mu_Y\)</span>)</p>
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<h2>Pearson correlation coefficient</h2>
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<p>The pearson correlation coefficient, <span class="math">\(\rho_{X,Y}\)</span>, is given by : </p>
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<div class="math">$$\rho_{X,Y}=\frac{cov(X,Y)}{\sigma_X\sigma_Y}=\frac{\sum_{i}(x_i-\bar{x})(y_i-\bar{y})}{n\sigma_X\sigma_Y}$$</div>
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<p>Here, <span class="math">\(\sigma_X\)</span> is the standard deviation of <span class="math">\(X\)</span> and <span class="math">\(\sigma_Y\)</span> is the standard deviation of <span class="math">\(Y\)</span>. You may also see <span class="math">\(\rho_{X,Y}\)</span> written as <span class="math">\(r_{X,Y}\)</span>.</p>
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<p>The pearson correlation coefficient is a measure of the linear correlation between two variables X and Y.</p>
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