e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
198 lines
8.8 KiB
HTML
198 lines
8.8 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/numbers/geometric-mean.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:41:51 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Geometric Mean</title>
|
||
<meta name="description" content="Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.">
|
||
|
||
<script language="JavaScript" type="text/javascript">reSpell=[["millimeter","millimetre"],["millimeters","millimetres"]];</script>
|
||
|
||
<!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
||
<meta name="HandheldFriendly" content="true">
|
||
<meta name="referrer" content="always">
|
||
<link rel="preload" href="../images/style/font-champ-bold.ttf" as="font" type="font/ttf" crossorigin="">
|
||
<link rel="preload" href="../style4.css" as="style">
|
||
<link rel="preload" href="../main4.js" as="script">
|
||
<link rel="stylesheet" href="../style4.css">
|
||
<script src="../main4.js" defer="defer"></script>
|
||
<!-- Global site tag (gtag.js) - Google Analytics -->
|
||
<script async="" src="https://www.googletagmanager.com/gtag/js?id=UA-29771508-1"></script>
|
||
<script>
|
||
window.dataLayer = window.dataLayer || [];
|
||
function gtag(){dataLayer.push(arguments);}
|
||
gtag('js', new Date());
|
||
gtag('config', 'UA-29771508-1');
|
||
</script>
|
||
</head>
|
||
|
||
<body id="bodybg">
|
||
|
||
<div id="stt"></div>
|
||
<div id="adTop"></div>
|
||
<header>
|
||
<div id="hdr"></div>
|
||
<div id="tran"></div>
|
||
<div id="adHide"></div>
|
||
<div id="cookOK"></div>
|
||
</header>
|
||
|
||
<div class="mid">
|
||
|
||
<nav>
|
||
<div id="menuWide" class="menu"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
|
||
|
||
<div id="search" role="search"></div>
|
||
<div id="linkto"></div>
|
||
|
||
<div id="menuSlim" class="menu"></div>
|
||
<div id="menuTiny" class="menu"></div>
|
||
</nav>
|
||
|
||
<div id="extra"></div>
|
||
|
||
<article id="content" role="main">
|
||
|
||
<!-- #BeginEditable "Body" -->
|
||
|
||
|
||
<h1 class="center">Geometric Mean</h1>
|
||
|
||
<p>The <span class="center">Geometric Mean</span> is a special type of average where we multiply the numbers together and then take a square root (for two numbers), cube root (for three numbers) etc.</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is the <span class="center">Geometric Mean</span> of <b>2 and 18</b>?</h3>
|
||
<ul>
|
||
<li>First we multiply them: 2 × 18 = 36</li>
|
||
<li>Then (as there are two numbers) take the square root: √36 = <b>6</b></li>
|
||
</ul>
|
||
<p>In one line:</p>
|
||
<p class="center"><b>Geometric Mean of 2 and 18 = √(2 × 18) = 6</b></p>
|
||
<p>It is like the area is the same!</p>
|
||
<p class="center"><img src="images/geometric-mean-2.svg" alt="geometric mean 2x18 = 6x6" height="96" width="292"></p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is the <span class="center">Geometric Mean</span> of <b>10, 51.2 and 8</b>?</h3>
|
||
<ul>
|
||
<li>First we multiply them: 10 × 51.2 × 8 = 4096</li>
|
||
<li>Then (as there are three numbers) take the cube root: <sup>3</sup>√4096 = <b>16</b></li>
|
||
</ul>
|
||
<p>In one line:</p>
|
||
<p class="center"><b>Geometric Mean = <sup>3</sup>√(10 × 51.2 × 8) = 16</b></p>
|
||
<p>It is like the volume is the same:</p>
|
||
<p class="center"><img src="images/geometric-mean-3.svg" alt="geometric mean 10x51.2x8 = 16x16x16" height="101" width="312"></p> </div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is the <span class="center">Geometric Mean</span> of <b>1, 3, 9, 27 and 81</b>?</h3>
|
||
<ul>
|
||
<li>First we multiply them: 1 × 3 × 9 × 27 × 81 = 59049</li>
|
||
<li>Then (as there are 5 numbers) take the 5th root: <sup>5</sup>√59049 = <b>9</b></li>
|
||
</ul>
|
||
<p>In one line:</p>
|
||
<p class="center"><b>Geometric Mean = <sup>5</sup>√(1 × 3 × 9 × 27 × 81) = 9</b></p>
|
||
|
||
<p>I can't show you a nice picture of this, but it is still true that:</p>
|
||
<p class="center">1 × 3 × 9 × 27 × 81 = 9 × 9 × 9 × 9 × 9</p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is the <span class="center">Geometric Mean</span> of a <b>Molecule and a Mountain</b></h3>
|
||
<p class="center"><a href="../measure/lengths-small-large.html"><img src="../measure/images/length-continuum.svg" alt="length continuum" style="max-" height="136" width="630"></a></p>
|
||
<p>Using <a href="scientific-notation.html">scientific notation</a>:</p>
|
||
<ul>
|
||
<li>A molecule of water (for example) is 0.275 × 10<sup>-9</sup> m</li>
|
||
<li>Mount Everest (for example) is 8.8 × 10<sup>3</sup> m</li>
|
||
</ul>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left"><span class="center"><b>Geometric Mean</b></span></span><span class="right"><span class="center"><b>= √(0.275 × 10<sup>-9</sup> × 8.8 × 10<sup>3</sup>)</b></span></span></div>
|
||
<div class="row"><span class="left"> </span><span class="right"><span class="center"><b>= √(2.42 × 10<sup>-6</sup>)</b></span></span></div>
|
||
<div class="row"><span class="left"> </span><span class="right"><span class="center"><b>≈ 0.0016 m</b></span></span></div>
|
||
</div>
|
||
<p>Which is <b>1.6 millimeters</b>, or about the thickness of a coin.</p>
|
||
<p>We could say, in a rough kind of way,</p>
|
||
<p class="center larger">"a millimeter is half-way between a molecule and a mountain!"</p>
|
||
|
||
</div>
|
||
<p>Another cool one:</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is the <span class="center">Geometric Mean</span> of a <b>Cell and the Earth?</b></h3>
|
||
|
||
<ul>
|
||
<li>A skin cell is about 3 × 10<sup>-8</sup> m across</li>
|
||
<li>The Earth's diameter is 1.3 × 10<sup>7</sup> m</li>
|
||
</ul>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left"><span class="center"><b>Geometric Mean</b></span></span><span class="right"><span class="center"><b>= √(3 × 10<sup>-8</sup> × 1.3 × 10<sup>7</sup>)</b></span></span></div>
|
||
<div class="row"><span class="left"> </span><span class="right"><span class="center"><b>= √(3.9 × 10<sup>-1</sup>)</b></span></span></div>
|
||
<div class="row"><span class="left"> </span><span class="right"><span class="center"><b>= √0.39</b></span></span></div>
|
||
<div class="row"><span class="left"> </span><span class="right"><span class="center"><b>≈ 0.6 m</b></span></span></div>
|
||
</div>
|
||
<p>A child is about <b>0.6 m</b> tall! So we could say, in a rough kind of way,</p>
|
||
<p class="center larger">"A child is half-way between a cell and the Earth"</p>
|
||
</div>
|
||
<p>So the geometric mean gives us a way of finding a value in between widely different values.</p>
|
||
|
||
|
||
<h2>Definition</h2>
|
||
|
||
<p>For <b>n</b> numbers: multiply them all together and then take the <a href="nth-root.html">nth root</a> (written <span class="center large"><sup>n</sup>√</span> )</p>
|
||
<p>More formally, the geometric mean of <b>n</b> numbers <b>a<sub>1</sub> to a<sub>n</sub></b> is:</p>
|
||
<p class="center large"><sup>n</sup>√(a<sub>1</sub> × a<sub>2</sub> × ... × a<sub>n</sub>)</p>
|
||
|
||
|
||
<h2>Useful</h2>
|
||
|
||
<p>The Geometric Mean is useful when we want to compare things with very different properties.</p>
|
||
|
||
<div class="example">
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/top-view-camera.jpg" alt="top view camera" height="165" width="180"></p>
|
||
|
||
<h3>Example: you want to buy a new camera.</h3>
|
||
<ul>
|
||
<li>One camera has a zoom of 200 and gets an 8 in reviews,</li>
|
||
<li>The other has a zoom of 250 and gets a 6 in reviews.</li>
|
||
</ul>
|
||
<p>Comparing using the usual <a href="../mean.html">arithmetic mean</a> gives (200+8)/2 = <b>104</b> vs (250+6)/2 = <b>128</b>. The zoom is such a big number that the user rating gets lost.</p>
|
||
<p>But the geometric means of the two cameras are:</p>
|
||
<ul>
|
||
<li>√(200 × 8) = <b>40</b></li>
|
||
<li>√(250 × 6) = <b>38.7...</b></li>
|
||
</ul>
|
||
<p>So, even though the zoom is 50 bigger, the lower user rating of 6 is still important.</p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="questions">9379, 9380, 9381, 9382, 9383, 9384, 9385, 9386, 9387, 9388</div>
|
||
|
||
<div class="related">
|
||
<a href="../mean.html">Arithmetic Mean</a>
|
||
<a href="harmonic-mean.html">Harmonic Mean</a>
|
||
<a href="index.html">Number Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
<div id="adend" class="centerfull noprint"></div>
|
||
<footer id="footer" class="centerfull noprint"></footer>
|
||
<div id="copyrt">Copyright © 2022 Rod Pierce</div>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/numbers/geometric-mean.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:41:52 GMT -->
|
||
</html> |