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<h1 class="center">Correlation</h1>
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<p>When two sets of data are strongly linked together we say they have a <b>High Correlation</b>.</p>
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<div class="words">
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<p>The word Correlation is made of <b>Co-</b> (meaning "together"), and <b>Relation</b></p>
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</div>
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<ul>
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<li>Correlation is <b>Positive</b> when the values <b>increase</b> together, and </li>
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<li>Correlation is <b>Negative</b> when one value <b>decreases</b> as the other increases</li>
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</ul>
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<p>A correlation is assumed to be <b> linear </b> (following a line).</p>
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<img src="images/correlation-examples.svg" style="max-width:100%" alt="correlation examples">
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<p>Correlation can have a value: </p>
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<ul>
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<li><span class="large">1</span> is a perfect positive correlation</li>
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<li><span class="large">0</span> is no correlation (the values don't seem linked at all)</li>
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<li><span class="large">-1</span> is a perfect negative correlation</li>
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</ul>
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<p>The value shows <b>how good the correlation is</b> (not how steep the line is), and if it is positive or negative.</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 versus the temperature on that day. Here are their figures for the last 12 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>11.9° </td>
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<td>$185 </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>18.5° </td>
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<td>$406 </td>
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</tr>
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<tr align="center">
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<td>22.1° </td>
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<td>$522 </td>
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</tr>
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<tr align="center">
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<td>19.4° </td>
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<td>$412 </td>
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</tr>
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<tr align="center">
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<td>25.1° </td>
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<td>$614 </td>
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</tr>
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<tr align="center">
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<td>23.4° </td>
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<td>$544 </td>
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</tr>
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<tr align="center">
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<td>18.1°</td>
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<td> $421 </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>
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<p>And here is the same data as a <span class="center"><a href="scatter-xy-plots.html">Scatter Plot</a></span>:</p>
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<p class="center"><img src="images/scatter-ice-cream1.svg" style="max-width:100%" alt="scatter plot ice cream 1"></p>
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<p>We can easily see that warmer weather and higher sales go together. The relationship is good but not perfect.</p>
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<p class="center larger">In fact the correlation is <b>0.9575</b> ... see at the end how I calculated it.</p>
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Also try the <a href="correlation-calculator.html">Correlation Calculator</a>.
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<h2>Correlation Is Not Good at Curves</h2>
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<p>The correlation calculation only works properly for straight line relationships.</p>
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<div class="example">
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<h3>Our Ice Cream Example: <b>there has been a heat wave!</b> </h3>
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<p>It gets so hot that people aren't going near the shop, and <b>sales start dropping</b>.</p>
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<p>Here is the latest graph:</p>
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<p class="center"><img src="images/scatter-ice-cream2.svg" style="max-width:100%" alt="scatter ice cream plot 2"></p>
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<p class="center larger">The correlation value is now <b>0</b>: "No Correlation" ... !</p>
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<p>The <b>calculated</b> correlation value is <b>0</b> (I worked it out), which means "no correlation". </p>
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<p>But we can see the data follows a nice <b>curve</b> that reaches a peak around 25° C. </p>
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<p>But the correlation calculation is not "smart" enough to see this.</p>
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</div>
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<p class="center larger">Moral of the story: make a <a href="scatter-xy-plots.html">Scatter Plot</a>, and look at it! <br>
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You may see a relationship that the calculation does not.</p>
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<h2>"Correlation Is Not Causation"</h2>
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<p>A common saying is "Correlation Is Not Causation". </p>
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<p>What it <b>really</b> means is that a correlation does <b>not prove</b> one thing causes the other:</p>
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<ul>
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<li>One thing <b>might</b> cause the other </li>
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<li>The other <b>might</b> cause the first to happen</li>
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<li>They may be linked by a different thing</li>
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<li>Or it could be random chance! </li>
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</ul>
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<p>There can be many reasons the data has a good correlation. </p>
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<div class="example">
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<h3>Example: Sunglasses vs Ice Cream</h3>
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<p>Our Ice Cream shop finds how many sunglasses were sold by a big store for each day and compares them to their ice cream sales:</p>
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<p class="center"><img src="images/scatter-ice-cream3.svg" style="max-width:100%" alt="scatter ice cream plot 3"></p>
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<p class="center larger">The correlation between Sunglasses and Ice Cream sales is high</p>
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<p>Does this mean that sunglasses make people want ice cream?</p>
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</div>
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<div class="example">
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<h3>Example: Poor suburbs are more likely to have high pollution. </h3>
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<p>Why? </p>
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<ul>
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<li>Do poor people make pollution? </li>
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<li>Are polluted suburbs the only place poor people can afford? </li>
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<li>Is it a common link, such as factories with low paying jobs and lots of pollution? </li>
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</ul>
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</div>
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<div class="example">
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<h3>Example: A Real Case! </h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/study-sick.gif" alt="study sick" height="150" width="207"></p>
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<p>A few years ago a survey of employees found a strong<b></b> positive correlation between <b>"Studying an external course"</b> and <b>Sick Days</b>.</p>
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<p>Does this mean:</p>
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<ul>
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<li>Studying makes them sick?</li>
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<li>Sick people study a lot?</li>
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<li>Or did they lie about being sick so they can study more? </li>
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</ul>
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<p>Without further research we can't be sure why.</p>
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</div>
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<h2>How To Calculate</h2>
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<p>How did I calculate the value <b>0.9575</b> at the top?</p>
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<p>I used "Pearson's Correlation". There is software that can calculate it, such as the CORREL() function in Excel or LibreOffice Calc ...</p>
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<h3>... but here is how to calculate it yourself: </h3>
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<p>Let us call the two sets of data "x" and "y" (in our case Temperature is <b>x</b> and Ice Cream Sales is <b>y</b>):</p>
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<ul>
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<li>Step 1: Find the mean of <b>x</b>, and the mean of <b>y</b></li>
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<li>Step 2: Subtract the mean of x from every x value (call them "<b>a</b>"), and subtract the mean of y from every y value (call them "<b>b</b>")</li>
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<li>Step 3: Calculate: <b>ab</b>, <b>a<sup>2</sup></b> and <b>b<sup>2</sup></b> for every value</li>
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<li>Step 4: Sum up <b>ab</b>, sum up <b>a<sup>2</sup></b> and sum up <b>b<sup>2</sup></b></li>
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<li>Step 5: Divide the sum of ab by the square root of [(sum of a<sup>2</sup>) × (sum of b<sup>2</sup>)]</li>
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</ul>
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<p>Here is how I calculated the first Ice Cream example (values rounded to 1 or 0 decimal places):</p>
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<p class="center"><img src="images/correlation-calc1.svg" style="max-width:100%" alt="correlation calculations"></p>
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<p>As a formula it is:</p>
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<p class="center"><img src="images/correlation-formula.gif" alt="correlation formula" height="101" width="274"></p>
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<p>Where: </p>
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<ul>
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<li><span class="times">Σ</span> is <a href="../algebra/sigma-notation.html">Sigma</a>, the symbol for "sum up"</li>
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<li><span class="times"><i><img src="images/correlation-formula-x.gif" alt="correlation formula (xi - xbar)" height="24" width="68"></i></span> is each x-value minus the mean of x (called "a" above)</li>
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<li><img src="images/correlation-formula-y.gif" alt="correlation formula (yi - ybar)" height="22" width="63"> is each y-value minus the mean of y (called "b" above) </li>
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</ul>
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<p>You probably won't have to calculate it like that, but at least you know it is not "magic", but simply a routine set of calculations.</p>
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<div class="center80">
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<h3>Note for Programmers</h3>
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<p>You can calculate it in one pass through the data. Just sum up <b>x</b>, <b>y</b>, <b>x<sup>2</sup></b>, <b>y<sup>2</sup></b> and <b>xy</b> (no need for <b>a</b> or <b>b</b> calculations above) then use the formula:</p>
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<p class="center"><img src="images/correlation-formula-onepass.gif" style="max-width:100%" alt="correlation formula onepass"></p>
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</div>
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<h3>Other Methods</h3>
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<p>There are other ways to calculate a correlation coefficient, such as "Spearman's rank correlation coefficient".</p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(3072, 3776, 8757, 8758, 8759, 8760, 3080, 3081, 3810, 3811);</script>
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</div>
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<div class="related">
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<a href="correlation-calculator.html">Correlation Calculator</a>
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<a href="index.html">Data Index</a>
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