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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Covariance</h1>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/covariance-examples.svg" alt="Covariance examples"></p>
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<p>Covariance is a single number we can calculate from a list of paired values.</p>
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<p>It tells us if the paired values tend to rise together, or if one tends to rise as the other falls.</p>
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<h2>The Calculations</h2>
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<p>Imagine we have pairs of values (x,y), ..., we do these calculations:</p>
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<ul>
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<li>Find the mean of the x values</li>
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<li>Find the mean of the y values</li></ul>
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<p>Then for each pair of values:</p>
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<ul>
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<li>subtract the mean of x from the x value</li>
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<li>subtract the mean of y from the y value</li>
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<li>multiply those together</li></ul>
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<p>And lastly:</p>
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<ul>
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<li>sum up all those multiplications</li>
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<li>divide by n−1 (where n is the total number of pairs)</li></ul>
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<p>And we get the covariance.</p>
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<h2>Example: Ice Cream Sales</h2>
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<p>The local ice cream shop keeps track of how much ice cream they sell
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versus the temperature on that day. Here are their figures for the last
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few days: </p>
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<table align="center" cellpadding="4" border="3">
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<tbody>
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<tr align="center">
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<td colspan="2"><b><em>Ice Cream Sales vs Temperature</em><em></em></b></td>
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</tr>
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<tr align="center">
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<th>Temperature °C</th>
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<th>Ice Cream Sales</th>
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</tr>
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<tr align="center">
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<td>14.2°</td>
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<td>$215 </td>
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</tr>
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<tr align="center">
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<td>16.4°</td>
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<td> $325 </td>
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</tr>
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<tr align="center">
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<td>15.2° </td>
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<td>$332 </td>
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</tr>
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<tr align="center">
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<td>22.6°</td>
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<td> $445 </td>
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</tr>
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<tr align="center">
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<td>17.2°</td>
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<td>$408</td>
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</tr>
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</tbody></table><p>Find the mean of the x values (temperature) by adding them up and dividing by how many:</p>
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<p class="center large">mean of x = <span class="intbl"><em>14.2 + 16.4 + 15.2 + 22.6 + 17.2</em><strong>5</strong></span> = 17.12</p>
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<p>Find the mean of the y values (sales in dollars):</p>
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<p class="center large">mean of y = <span class="intbl"><em>215 + 325 + 332 + 445 + 408</em><strong>5</strong></span> = 345</p>
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<p>Then for each pair of values subtract mean of x from x, mean of y from y and multiply:</p>
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<ul>
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<li>(14.2−17.12)(215−345) = −2.92 × −130 = 379.6</li>
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<li>(16.4−17.12)(325−345) = −0.72 × −20 = 14.4</li>
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<li>(15.2−17.12)(332−345) = −1.92 × −13 = 24.96</li>
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<li>(22.6−17.12)(445−345) = 5.48 × 100 = 548</li>
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<li>(17.2−17.12)(408−345) = 0.08 × 63 = 5.04</li></ul>
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<p>Add those results up and divide by n−1</p>
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<div class="center larger"><span class="intbl"><em>379.6 + 14.4 +24.96 + 548 + 5.04</em><strong>(5−1)</strong></span> = 243</div>
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<!-- 379.6+14.4+24.96+548+5.04/4 = 243 -->
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<p>The answer is <b>positive</b>: that tells us the x and y values tend to rise <b>together</b>.</p>
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<p>That is all it says. Not how strongly linked they are. Not how fast they rise or fall. Just that they tend to rise and fall together.</p>
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<p>A <b>negative </b>result would say that x <b>rises </b>as y <b>falls </b>(and vice versa).</p>
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<p>A <b>zero </b>result (rarely happens with statistical data) just means the covariance does not let us know if x and y rise or fall together.</p>
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<div class="def">
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<h3>Covariance:</h3>
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<ul>
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<li><b>positive </b>says they rise and fall together</li>
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<li><b>negative </b>says that one rises as the other falls</li></ul></div>
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<p><i>The size of the covariance is not important. Imagine we used cents instead of dollars: we would get much larger values but the data still has the same relationship.</i></p>
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<p>Note: the covariance can also be useful in other calculations.</p>
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<h2>Formula</h2>
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<p>As a formula covariance is:</p>
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<div class="center larger"><span class="intbl"><em>1</em><strong>n−1</strong></span>
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<div class="sig">
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<div class="to">n</div>
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<div class="symb"></div>
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<div class="from">i=1</div>
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</div> (x<sub>i</sub>−x̄)(y<sub>i</sub>−ȳ)</div>
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<!-- 1/n-1 SIG{i=1,n} (x-x)(y-y) -->
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<p>Where:</p>
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<ul>
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<li><b>n</b> = number of pairs</li>
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<li><b>Σ</b> means sum up (see <a href="../algebra/sigma-notation.html">Sigma Notation</a>)</li>
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<li><b>x<sub>i</sub></b> and <b>y<sub>i</sub></b> are each pair of x,y values</li>
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<li><b>x̄</b> and <b>ȳ</b> are the mean of x and mean of y</li></ul>
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<h2>Upgrade to Correlation</h2>
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<p><a href="correlation.html">Correlation</a> has a few more steps in its calculation but also gives the useful result of telling us how <b>well related</b> x and y are.</p>
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<h2>Why n−1 ?</h2>
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<p>We divide by <b>n−1</b> in the final step when our data is a sample (which it usually is) because we are unsure of the true population mean.</p>
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<p>But if our data is the entire population then we should divide by <b>n</b> in the final step.</p>
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<p><br></p>
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<div class="related">
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<a href="correlation.html">Correlation </a>
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<a href="standard-deviation.html">Standard Deviation and Variance</a>
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<a href="index.html">Data Index</a>
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