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122 lines
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<h1>Harmonic Mean</h1>
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<p>The harmonic mean is:</p>
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<p class="center larger">the reciprocal of the average of the reciprocals</p>
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<p>Yes, that is a lot of reciprocals!</p>
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<p><a href="../reciprocal.html">Reciprocal</a> just means <span class="intbl"><em>1</em><strong>value</strong></span>.</p>
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<p>The formula is:</p>
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<p class="center"><img src="images/harmonic-mean.svg" alt="Harmonic Mean"></p>
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<p>Where <b>a,b,c,...</b> are the values, and <b>n</b> is how many values.</p>
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<p>Steps: </p>
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<ul>
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<li>Calculate the reciprocal (1/value) for every value.</li>
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<li>Find the average of those reciprocals (just add them and divide by how many there are)</li>
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<li>Then do the reciprocal of that average (=1/average)</li>
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</ul>
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<div class="example">
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<h3>Example: What is the harmonic mean of 1, 2 and 4?</h3>
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<p>The reciprocals of 1, 2 and 4 are:</p>
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<p class="center"><span class="intbl"><em>1</em><strong>1</strong></span> = 1,
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<span class="intbl"><em>1</em><strong>2</strong></span> = 0.5,
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<span class="intbl"><em>1</em><strong>4</strong></span> = 0.25
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</p>
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<p>Now add them up:</p>
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<p class="center">1 + 0.5 + 0.25 = 1.75</p>
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<p>Divide by how many:</p>
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<p class="center">Average = <span class="intbl"><em>1.75</em><strong>3</strong></span></p>
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<p>The reciprocal of that average is our answer:</p>
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<p class="center">Harmonic Mean = <span class="intbl"><em>3</em><strong>1.75</strong></span> = <b>1.714</b> (to 3 places) </p>
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</div>
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<h2>Why</h2>
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<p>In <i>some</i> rate type questions the harmonic mean gives the true answer!</p>
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<div class="example">
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<h3>Example: we travel 10 km at 60 km/h, than another 10 km at 20 km/h, what is our average speed?</h3>
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<p class="center">Harmonic mean = 2/(<span class="intbl"><em>1</em><strong>60</strong></span> + <span class="intbl"><em>1</em><strong>20</strong></span>) = <b>30 km/h</b></p>
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<p>Check: the 10 km at 60 km/h takes 10 minutes, the 10 km at 20 km/h takes 30 minutes, so the total 20 km takes 40 minutes, which is 30 km per hour</p>
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</div>
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<p>The harmonic mean is also good at handling large <a href="../data/outliers.html">outliers</a>.
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</p>
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<div class="example">
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<h3>Example: 2, 4, 6 and 100</h3>
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<p>The arithmetic mean is <span class="intbl"><em>2+4+6+100</em><strong>4</strong></span> = 28</p>
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<p>The harmonic mean is 4/(<span class="intbl"><em>1</em><strong>2</strong></span> + <span class="intbl"><em>1</em><strong>4</strong></span> + <span class="intbl"><em>1</em><strong>6</strong></span> + <span class="intbl"><em>1</em><strong>100</strong></span>) = 4.32 (to 2 places)</p>
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</div>
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<p>But small outliers will make things worse!</p>
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<h2>Another way to think of it</h2>
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<p>We can rearrange the formula above to look like this:</p>
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<p class="center"><img src="images/harmonic-mean-flip.svg" alt="Harmonic Mean"><br>
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</p>
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<p>It is <b>not</b> easy to use this way, but it does look more "balanced" (<b>n</b> on one side matched with n <b>1</b>s on the other, and the mean on one side matched with the values on the other side too).</p>
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<p> </p>
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<div class="related">
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<a href="../mean.html">Arithmetic Mean</a> <a href="geometric-mean.html">Geometric Mean</a> <a href="index.html">Numbers Index</a></div>
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