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<p><strong>Sphericity</strong> is a measure of how spherical (round) an object is. As such, it is a specific example of a compactness measure of a shape. Defined by Wadell in 1935,<sup class="reference" id="cite_ref-0">[1]</sup> the sphericity, <span class="texhtml">Ψ</span>, of a particle is the ratio of the surface area of a sphere (with the same volume as the given particle) to the surface area of the particle:</p>
<dl><dd><img class="tex" alt="\Psi = \frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p}" src="http://upload.wikimedia.org/math/e/a/2/ea230750eba82fd8fe1b178ea651d242.png" /></dd></dl>
<p>where <span class="texhtml"><em>V</em><sub><em>p</em></sub></span> is volume of the particle and <span class="texhtml"><em>A</em><sub><em>p</em></sub></span> is the surface area of the particle</p>
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<p> </p>
<h2><span class="mw-headline">Ellipsoidal Objects</span></h2>
<dl><dd><span class="boilerplate seealso"><em>See also: Earth radius</em></span></dd></dl>
<p>The sphericity, <span class="texhtml">Ψ</span>, of an oblate spheroid (similar to the shape of the planet Earth) is defined as such:</p>
<dl><dd><img class="tex" alt="\Psi =
\frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p} =
\frac{2\sqrt[3]{ab^2}}{a+\frac{b^2}{\sqrt{a^2-b^2}}\ln{(\frac{a+\sqrt{a^2-b^2}}b)}}" src="http://upload.wikimedia.org/math/f/2/6/f268c95b6388dc2002a77bc7224c190d.png" /></dd></dl>
<p><em>(where a, b are the semi-major, semi-minor axes, respectively.</em></p>
<p> </p>
<h2><span class="mw-headline">Derivation</span></h2>
<p>Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the particle divided by the actual surface area of the particle.</p>
<p>First we need to write surface area of the sphere, <span class="texhtml"><em>A</em><sub><em>s</em></sub></span> in terms of the volume of the particle, <span class="texhtml"><em>V</em><sub><em>p</em></sub></span></p>
<dl><dd><img class="tex" alt="A_{s}^3 = \left(4 \pi r^2\right)^3 = 4^3 \pi^3 r^6 = 4 \pi \left(4^2 \pi^2 r^6\right) = 4 \pi \cdot 3^2 \left(\frac{4^2 \pi^2}{3^2} r^6\right) = 36 \pi \left(\frac{4 \pi}{3} r^3\right)^2 = 36\,\pi V_{p}^2" src="http://upload.wikimedia.org/math/8/2/7/827c80a821a4e9042325749cb7c461f7.png" /></dd></dl>
<p>therefore</p>
<dl><dd><img class="tex" alt="A_{s} = \left(36\,\pi V_{p}^2\right)^{\frac{1}{3}} = 36^{\frac{1}{3}} \pi^{\frac{1}{3}} V_{p}^{\frac{2}{3}} = 6^{\frac{2}{3}} \pi^{\frac{1}{3}} V_{p}^{\frac{2}{3}} = \pi^{\frac{1}{3}} \left(6V_{p}\right)^{\frac{2}{3}}" src="http://upload.wikimedia.org/math/8/6/c/86c49950a2c92504eabbd66a1336219b.png" /></dd></dl>
<p>hence we define <span class="texhtml">Ψ</span> as:</p>
<dl><dd><img class="tex" alt="\Psi = \frac{A_s}{A_p} = \frac{ \pi^{\frac{1}{3}} \left(6V_{p}\right)^{\frac{2}{3}} }{A_{p}}" src="http://upload.wikimedia.org/math/0/6/9/06986e86f9d53b2e4f3560f330909416.png" /></dd></dl>
<p> </p>
<h2><span class="mw-headline">Sphericity of common objects</span></h2>
<table style="MARGIN: 0pt auto; BORDER-COLLAPSE: collapse; TEXT-ALIGN: center" cellpadding="7" border="1">
<tbody>
<tr>
<th>Name</th>
<th>Picture</th>
<th>Volume</th>
<th>Area</th>
<th>Sphericity</th>
</tr>
<tr>
<td align="left" colspan="5"><strong>Platonic Solids</strong></td>
</tr>
<tr>
<td>tetrahedron</td>
<td><img height="47" alt="Tetrahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tetrahedron.jpg/50px-Tetrahedron.jpg" width="50" border="0" /></td>
<td><img class="tex" alt="\frac{\sqrt{2}}{12}\,s^3" src="http://upload.wikimedia.org/math/f/0/3/f03ea7115c243660a2ea99e73ad310db.png" /></td>
<td><img class="tex" alt="\sqrt{3}\,s^2" src="http://upload.wikimedia.org/math/0/a/6/0a6e48b85549c8b6512fdd3906ac8aa6.png" /></td>
<td><img class="tex" alt="\left(\frac{\pi}{6\sqrt{3}}\right)^{\frac{1}{3}} \approx 0.671" src="http://upload.wikimedia.org/math/c/e/9/ce99e8d11ec5357af8c21c1415f40cab.png" /></td>
</tr>
<tr>
<td>cube (hexahedron)</td>
<td><img height="56" alt="Hexahedron (cube)" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/78/Hexahedron.jpg/50px-Hexahedron.jpg" width="50" border="0" /></td>
<td><img class="tex" alt="\,s^3" src="http://upload.wikimedia.org/math/7/9/9/7998c630c83b898c1fdb72d667936996.png" /></td>
<td><img class="tex" alt="6\,s^2" src="http://upload.wikimedia.org/math/b/4/7/b472474af030c5d9cf3620a43c5e417b.png" /></td>
<td>
<p><img class="tex" alt="\left(
\frac{\pi}{6}
\right)^{\frac{1}{3}} \approx 0.806" src="http://upload.wikimedia.org/math/8/b/c/8bc7f285938e1b6ff146ed239698afce.png" /></p>
</td>
</tr>
<tr>
<td>octahedron</td>
<td><img height="50" alt="Octahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/0/07/Octahedron.svg/50px-Octahedron.svg.png" width="50" border="0" /></td>
<td><img class="tex" alt=" \frac{1}{3} \sqrt{2}\, s^3" src="http://upload.wikimedia.org/math/d/5/c/d5c9af02c97d15e54291f3ddce4d2211.png" /></td>
<td><img class="tex" alt=" 2 \sqrt{3}\, s^2" src="http://upload.wikimedia.org/math/2/9/3/2935e3febfb41a617e0fea6efc2bab02.png" /></td>
<td>
<p><img class="tex" alt="\left(
\frac{\pi}{3\sqrt{3}}
\right)^{\frac{1}{3}} \approx 0.846 " src="http://upload.wikimedia.org/math/b/4/c/b4cb491f3e93c2bcd8f345582be18ffa.png" /></p>
</td>
</tr>
<tr>
<td>dodecahedron</td>
<td><img height="48" alt="Dodecahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/6/66/POV-Ray-Dodecahedron.svg/50px-POV-Ray-Dodecahedron.svg.png" width="50" border="0" /></td>
<td><img class="tex" alt=" \frac{1}{4} \left(15 + 7\sqrt{5}\right)\, s^3" src="http://upload.wikimedia.org/math/c/2/7/c2750faa2f4f0b5a934390dd3d135dd2.png" /></td>
<td><img class="tex" alt=" 3 \sqrt{25 + 10\sqrt{5}}\, s^2" src="http://upload.wikimedia.org/math/1/9/b/19bcc8e5c2da1d6a9c9290e3bbb55011.png" /></td>
<td>
<p><img class="tex" alt="\left(
\frac{\left(15 + 7\sqrt{5}\right)^2 \pi}{12\left(25+10\sqrt{5}\right)^{\frac{3}{2}}}
\right)^{\frac{1}{3}} \approx 0.910" src="http://upload.wikimedia.org/math/b/2/1/b21c0ccd12191d494564898dffc6daf2.png" /></p>
</td>
</tr>
<tr>
<td>icosahedron</td>
<td><img height="48" alt="Icosahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Icosahedron.jpg/50px-Icosahedron.jpg" width="50" border="0" /></td>
<td><img class="tex" alt="\frac{5}{12}\left(3+\sqrt{5}\right)\, s^3" src="http://upload.wikimedia.org/math/7/7/e/77ee1cd7a4858ddfa2a994c29d7d2db5.png" /></td>
<td><img class="tex" alt="5\sqrt{3}\,s^2" src="http://upload.wikimedia.org/math/0/f/5/0f589b4ae2f26ce5c4b29705a02d3498.png" /></td>
<td><img class="tex" alt="\left(
\frac{ \left(3 + \sqrt{5} \right)^2 \pi}{60\sqrt{3}}
\right)^{\frac{1}{3}} \approx 0.939" src="http://upload.wikimedia.org/math/6/f/7/6f78ec8588faabadd52bc5946ebbbe58.png" /></td>
</tr>
<tr>
<td align="left" colspan="5"><strong>Round Shapes</strong></td>
</tr>
<tr>
<td>ideal cone<br />
<img class="tex" alt="(h=2\sqrt{2}r)" src="http://upload.wikimedia.org/math/f/a/0/fa0b47385960369df693112cc666e210.png" /></td>
<td> </td>
<td><img class="tex" alt="\frac{1}{3} \pi\, r^2 h " src="http://upload.wikimedia.org/math/c/b/b/cbb69401f7fae091004a0b025b5136fe.png" /><br />
<p><img class="tex" alt="= \frac{2\sqrt{2}}{3} \pi\, r^3" src="http://upload.wikimedia.org/math/3/0/0/300e3da8ae4b0341c28ef5e767eed9d3.png" /></p>
</td>
<td><img class="tex" alt="\pi\, r (r + \sqrt{r^2 + h^2}) " src="http://upload.wikimedia.org/math/d/0/3/d03ef3d12512676dbe6b51191c06be4b.png" /><br />
<p><img class="tex" alt="= 4 \pi\, r^2 " src="http://upload.wikimedia.org/math/9/d/b/9db0629297845830f39774970ed03073.png" /></p>
</td>
<td><img class="tex" alt="\left(
\frac{1}{2}
\right)^{\frac{1}{3}} \approx 0.794" src="http://upload.wikimedia.org/math/4/9/7/497312086e2798aba0f7232e6f3d3278.png" /></td>
</tr>
<tr>
<td>hemisphere<br />
(half sphere)</td>
<td> </td>
<td><img class="tex" alt="\frac{2}{3} \pi\, r^3" src="http://upload.wikimedia.org/math/d/e/5/de599428e7c43ba81b82e045513f8662.png" /></td>
<td><img class="tex" alt="3 \pi\, r^2" src="http://upload.wikimedia.org/math/7/1/d/71dce90d7880a8b7dfb4aeebf1df1b9a.png" /></td>
<td>
<p><img class="tex" alt="\left(
\frac{16}{27}
\right)^{\frac{1}{3}} \approx 0.840" src="http://upload.wikimedia.org/math/6/a/e/6aea0187b6dd454098ce2ee883db2c55.png" /></p>
</td>
</tr>
<tr>
<td>ideal cylinder<br />
<img class="tex" alt="(h=2\,r)" src="http://upload.wikimedia.org/math/2/6/d/26d4795ee1f02913f0dd9c0f4a0a68f5.png" /></td>
<td> </td>
<td><img class="tex" alt="\pi r^2 h = 2 \pi\,r^3" src="http://upload.wikimedia.org/math/f/9/f/f9f38314067649347ddc6c18a1c93232.png" /></td>
<td><img class="tex" alt="2 \pi r ( r + h ) = 6 \pi\,r^2" src="http://upload.wikimedia.org/math/8/8/0/88001809b1d9aaeb1327584f33baa711.png" /></td>
<td>
<p><img class="tex" alt="\left(
\frac{2}{3}
\right)^{\frac{1}{3}} \approx 0.874" src="http://upload.wikimedia.org/math/5/c/2/5c2b5f728c566f1f1913ed4f8279d46e.png" /></p>
</td>
</tr>
<tr>
<td>ideal torus<br />
<span class="texhtml">(<em>R</em> = <em>r</em>)</span></td>
<td> </td>
<td><img class="tex" alt="2 \pi^2 R r^2 = 2 \pi^2 \,r^3" src="http://upload.wikimedia.org/math/5/c/0/5c063b1a9e01901df957ece8ace81fd0.png" /></td>
<td><img class="tex" alt="4 \pi^2 R r = 4 \pi^2\,r^2" src="http://upload.wikimedia.org/math/1/4/0/14051c34379da213ca3b815a26e82e45.png" /></td>
<td>
<p><img class="tex" alt="\left(
\frac{9}{4 \pi}
\right)^{\frac{1}{3}} \approx 0.894" src="http://upload.wikimedia.org/math/c/b/4/cb4dfdaded145ac88dbe0d646a39f875.png" /></p>
</td>
</tr>
<tr>
<td>sphere</td>
<td> </td>
<td><img class="tex" alt="\frac{4}{3} \pi r^3" src="http://upload.wikimedia.org/math/9/3/a/93a32e52c9580c5f627ace8dc3ad6397.png" /></td>
<td><img class="tex" alt="4 \pi\,r^2" src="http://upload.wikimedia.org/math/1/0/9/109a7c5ee8613b982dc98a9d2f0a590d.png" /></td>
<td>
<p><img class="tex" alt="1\," src="http://upload.wikimedia.org/math/d/0/6/d06c48671eacd7f1e2afde7289e483d5.png" /></p>
</td>
</tr>
</tbody>
</table>
<p> </p>
<h2><span class="mw-headline">Sphericity in Statistics</span></h2>
<p>In statistical analyses, sphericity relates to the equality of the variances of the differences between levels of the repeated measures factor. Sphericity requires that the variances for each set of difference scores are equal. This is an assumption of an ANOVA with a repeated measures factor, where violations of this assumption can invalidate the analysis conclusions. Mauchly's sphericity test is the statistical test used to evaluate sphericity.</p>
<p> </p>
<h2><span class="mw-headline">References</span></h2>
<ol class="references">
<li id="cite_note-0"><strong>^</strong> <cite style="FONT-STYLE: normal">Wadell, Hakon (1935). "Volume, Shape and Roundness of Quartz Particles". <em>Journal of Geology</em> <strong>43</strong>: 250–280.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Volume%2C+Shape+and+Roundness+of+Quartz+Particles&rft.jtitle=Journal+of+Geology&rft.date=1935&rft.volume=43&rft.aulast=Wadell&rft.aufirst=Hakon&rft.pages=250%E2%80%93280"> </span></li>
</ol>
<p> </p>
<h2><span class="mw-headline">See also</span></h2>
<ul>
<li>Rounding (sediment)</li>
</ul>
<p><a id="External_links" name="External_links"></a></p>
<h2><span class="mw-headline">External links</span></h2>
<ul>
<li><a class="external text" title="http://www.howround.com/" href="http://www.howround.com/" rel="nofollow">How round is your circle?</a></li>
<li><a class="external text" title="http://people.uncw.edu/dockal/gly312/grains/grains.htm" href="http://people.uncw.edu/dockal/gly312/grains/grains.htm" rel="nofollow">Grain Morphology: Roundness, Surface Features, and Sphericity of Grains</a></li>
</ul>
<dl><dd><img class="tex" alt="\Psi = \frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p}" src="http://upload.wikimedia.org/math/e/a/2/ea230750eba82fd8fe1b178ea651d242.png" /></dd></dl>
<p>where <span class="texhtml"><em>V</em><sub><em>p</em></sub></span> is volume of the particle and <span class="texhtml"><em>A</em><sub><em>p</em></sub></span> is the surface area of the particle</p>
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<p> </p>
<h2><span class="mw-headline">Ellipsoidal Objects</span></h2>
<dl><dd><span class="boilerplate seealso"><em>See also: Earth radius</em></span></dd></dl>
<p>The sphericity, <span class="texhtml">Ψ</span>, of an oblate spheroid (similar to the shape of the planet Earth) is defined as such:</p>
<dl><dd><img class="tex" alt="\Psi =
\frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p} =
\frac{2\sqrt[3]{ab^2}}{a+\frac{b^2}{\sqrt{a^2-b^2}}\ln{(\frac{a+\sqrt{a^2-b^2}}b)}}" src="http://upload.wikimedia.org/math/f/2/6/f268c95b6388dc2002a77bc7224c190d.png" /></dd></dl>
<p><em>(where a, b are the semi-major, semi-minor axes, respectively.</em></p>
<p> </p>
<h2><span class="mw-headline">Derivation</span></h2>
<p>Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the particle divided by the actual surface area of the particle.</p>
<p>First we need to write surface area of the sphere, <span class="texhtml"><em>A</em><sub><em>s</em></sub></span> in terms of the volume of the particle, <span class="texhtml"><em>V</em><sub><em>p</em></sub></span></p>
<dl><dd><img class="tex" alt="A_{s}^3 = \left(4 \pi r^2\right)^3 = 4^3 \pi^3 r^6 = 4 \pi \left(4^2 \pi^2 r^6\right) = 4 \pi \cdot 3^2 \left(\frac{4^2 \pi^2}{3^2} r^6\right) = 36 \pi \left(\frac{4 \pi}{3} r^3\right)^2 = 36\,\pi V_{p}^2" src="http://upload.wikimedia.org/math/8/2/7/827c80a821a4e9042325749cb7c461f7.png" /></dd></dl>
<p>therefore</p>
<dl><dd><img class="tex" alt="A_{s} = \left(36\,\pi V_{p}^2\right)^{\frac{1}{3}} = 36^{\frac{1}{3}} \pi^{\frac{1}{3}} V_{p}^{\frac{2}{3}} = 6^{\frac{2}{3}} \pi^{\frac{1}{3}} V_{p}^{\frac{2}{3}} = \pi^{\frac{1}{3}} \left(6V_{p}\right)^{\frac{2}{3}}" src="http://upload.wikimedia.org/math/8/6/c/86c49950a2c92504eabbd66a1336219b.png" /></dd></dl>
<p>hence we define <span class="texhtml">Ψ</span> as:</p>
<dl><dd><img class="tex" alt="\Psi = \frac{A_s}{A_p} = \frac{ \pi^{\frac{1}{3}} \left(6V_{p}\right)^{\frac{2}{3}} }{A_{p}}" src="http://upload.wikimedia.org/math/0/6/9/06986e86f9d53b2e4f3560f330909416.png" /></dd></dl>
<p> </p>
<h2><span class="mw-headline">Sphericity of common objects</span></h2>
<table style="MARGIN: 0pt auto; BORDER-COLLAPSE: collapse; TEXT-ALIGN: center" cellpadding="7" border="1">
<tbody>
<tr>
<th>Name</th>
<th>Picture</th>
<th>Volume</th>
<th>Area</th>
<th>Sphericity</th>
</tr>
<tr>
<td align="left" colspan="5"><strong>Platonic Solids</strong></td>
</tr>
<tr>
<td>tetrahedron</td>
<td><img height="47" alt="Tetrahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tetrahedron.jpg/50px-Tetrahedron.jpg" width="50" border="0" /></td>
<td><img class="tex" alt="\frac{\sqrt{2}}{12}\,s^3" src="http://upload.wikimedia.org/math/f/0/3/f03ea7115c243660a2ea99e73ad310db.png" /></td>
<td><img class="tex" alt="\sqrt{3}\,s^2" src="http://upload.wikimedia.org/math/0/a/6/0a6e48b85549c8b6512fdd3906ac8aa6.png" /></td>
<td><img class="tex" alt="\left(\frac{\pi}{6\sqrt{3}}\right)^{\frac{1}{3}} \approx 0.671" src="http://upload.wikimedia.org/math/c/e/9/ce99e8d11ec5357af8c21c1415f40cab.png" /></td>
</tr>
<tr>
<td>cube (hexahedron)</td>
<td><img height="56" alt="Hexahedron (cube)" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/78/Hexahedron.jpg/50px-Hexahedron.jpg" width="50" border="0" /></td>
<td><img class="tex" alt="\,s^3" src="http://upload.wikimedia.org/math/7/9/9/7998c630c83b898c1fdb72d667936996.png" /></td>
<td><img class="tex" alt="6\,s^2" src="http://upload.wikimedia.org/math/b/4/7/b472474af030c5d9cf3620a43c5e417b.png" /></td>
<td>
<p><img class="tex" alt="\left(
\frac{\pi}{6}
\right)^{\frac{1}{3}} \approx 0.806" src="http://upload.wikimedia.org/math/8/b/c/8bc7f285938e1b6ff146ed239698afce.png" /></p>
</td>
</tr>
<tr>
<td>octahedron</td>
<td><img height="50" alt="Octahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/0/07/Octahedron.svg/50px-Octahedron.svg.png" width="50" border="0" /></td>
<td><img class="tex" alt=" \frac{1}{3} \sqrt{2}\, s^3" src="http://upload.wikimedia.org/math/d/5/c/d5c9af02c97d15e54291f3ddce4d2211.png" /></td>
<td><img class="tex" alt=" 2 \sqrt{3}\, s^2" src="http://upload.wikimedia.org/math/2/9/3/2935e3febfb41a617e0fea6efc2bab02.png" /></td>
<td>
<p><img class="tex" alt="\left(
\frac{\pi}{3\sqrt{3}}
\right)^{\frac{1}{3}} \approx 0.846 " src="http://upload.wikimedia.org/math/b/4/c/b4cb491f3e93c2bcd8f345582be18ffa.png" /></p>
</td>
</tr>
<tr>
<td>dodecahedron</td>
<td><img height="48" alt="Dodecahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/6/66/POV-Ray-Dodecahedron.svg/50px-POV-Ray-Dodecahedron.svg.png" width="50" border="0" /></td>
<td><img class="tex" alt=" \frac{1}{4} \left(15 + 7\sqrt{5}\right)\, s^3" src="http://upload.wikimedia.org/math/c/2/7/c2750faa2f4f0b5a934390dd3d135dd2.png" /></td>
<td><img class="tex" alt=" 3 \sqrt{25 + 10\sqrt{5}}\, s^2" src="http://upload.wikimedia.org/math/1/9/b/19bcc8e5c2da1d6a9c9290e3bbb55011.png" /></td>
<td>
<p><img class="tex" alt="\left(
\frac{\left(15 + 7\sqrt{5}\right)^2 \pi}{12\left(25+10\sqrt{5}\right)^{\frac{3}{2}}}
\right)^{\frac{1}{3}} \approx 0.910" src="http://upload.wikimedia.org/math/b/2/1/b21c0ccd12191d494564898dffc6daf2.png" /></p>
</td>
</tr>
<tr>
<td>icosahedron</td>
<td><img height="48" alt="Icosahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Icosahedron.jpg/50px-Icosahedron.jpg" width="50" border="0" /></td>
<td><img class="tex" alt="\frac{5}{12}\left(3+\sqrt{5}\right)\, s^3" src="http://upload.wikimedia.org/math/7/7/e/77ee1cd7a4858ddfa2a994c29d7d2db5.png" /></td>
<td><img class="tex" alt="5\sqrt{3}\,s^2" src="http://upload.wikimedia.org/math/0/f/5/0f589b4ae2f26ce5c4b29705a02d3498.png" /></td>
<td><img class="tex" alt="\left(
\frac{ \left(3 + \sqrt{5} \right)^2 \pi}{60\sqrt{3}}
\right)^{\frac{1}{3}} \approx 0.939" src="http://upload.wikimedia.org/math/6/f/7/6f78ec8588faabadd52bc5946ebbbe58.png" /></td>
</tr>
<tr>
<td align="left" colspan="5"><strong>Round Shapes</strong></td>
</tr>
<tr>
<td>ideal cone<br />
<img class="tex" alt="(h=2\sqrt{2}r)" src="http://upload.wikimedia.org/math/f/a/0/fa0b47385960369df693112cc666e210.png" /></td>
<td> </td>
<td><img class="tex" alt="\frac{1}{3} \pi\, r^2 h " src="http://upload.wikimedia.org/math/c/b/b/cbb69401f7fae091004a0b025b5136fe.png" /><br />
<p><img class="tex" alt="= \frac{2\sqrt{2}}{3} \pi\, r^3" src="http://upload.wikimedia.org/math/3/0/0/300e3da8ae4b0341c28ef5e767eed9d3.png" /></p>
</td>
<td><img class="tex" alt="\pi\, r (r + \sqrt{r^2 + h^2}) " src="http://upload.wikimedia.org/math/d/0/3/d03ef3d12512676dbe6b51191c06be4b.png" /><br />
<p><img class="tex" alt="= 4 \pi\, r^2 " src="http://upload.wikimedia.org/math/9/d/b/9db0629297845830f39774970ed03073.png" /></p>
</td>
<td><img class="tex" alt="\left(
\frac{1}{2}
\right)^{\frac{1}{3}} \approx 0.794" src="http://upload.wikimedia.org/math/4/9/7/497312086e2798aba0f7232e6f3d3278.png" /></td>
</tr>
<tr>
<td>hemisphere<br />
(half sphere)</td>
<td> </td>
<td><img class="tex" alt="\frac{2}{3} \pi\, r^3" src="http://upload.wikimedia.org/math/d/e/5/de599428e7c43ba81b82e045513f8662.png" /></td>
<td><img class="tex" alt="3 \pi\, r^2" src="http://upload.wikimedia.org/math/7/1/d/71dce90d7880a8b7dfb4aeebf1df1b9a.png" /></td>
<td>
<p><img class="tex" alt="\left(
\frac{16}{27}
\right)^{\frac{1}{3}} \approx 0.840" src="http://upload.wikimedia.org/math/6/a/e/6aea0187b6dd454098ce2ee883db2c55.png" /></p>
</td>
</tr>
<tr>
<td>ideal cylinder<br />
<img class="tex" alt="(h=2\,r)" src="http://upload.wikimedia.org/math/2/6/d/26d4795ee1f02913f0dd9c0f4a0a68f5.png" /></td>
<td> </td>
<td><img class="tex" alt="\pi r^2 h = 2 \pi\,r^3" src="http://upload.wikimedia.org/math/f/9/f/f9f38314067649347ddc6c18a1c93232.png" /></td>
<td><img class="tex" alt="2 \pi r ( r + h ) = 6 \pi\,r^2" src="http://upload.wikimedia.org/math/8/8/0/88001809b1d9aaeb1327584f33baa711.png" /></td>
<td>
<p><img class="tex" alt="\left(
\frac{2}{3}
\right)^{\frac{1}{3}} \approx 0.874" src="http://upload.wikimedia.org/math/5/c/2/5c2b5f728c566f1f1913ed4f8279d46e.png" /></p>
</td>
</tr>
<tr>
<td>ideal torus<br />
<span class="texhtml">(<em>R</em> = <em>r</em>)</span></td>
<td> </td>
<td><img class="tex" alt="2 \pi^2 R r^2 = 2 \pi^2 \,r^3" src="http://upload.wikimedia.org/math/5/c/0/5c063b1a9e01901df957ece8ace81fd0.png" /></td>
<td><img class="tex" alt="4 \pi^2 R r = 4 \pi^2\,r^2" src="http://upload.wikimedia.org/math/1/4/0/14051c34379da213ca3b815a26e82e45.png" /></td>
<td>
<p><img class="tex" alt="\left(
\frac{9}{4 \pi}
\right)^{\frac{1}{3}} \approx 0.894" src="http://upload.wikimedia.org/math/c/b/4/cb4dfdaded145ac88dbe0d646a39f875.png" /></p>
</td>
</tr>
<tr>
<td>sphere</td>
<td> </td>
<td><img class="tex" alt="\frac{4}{3} \pi r^3" src="http://upload.wikimedia.org/math/9/3/a/93a32e52c9580c5f627ace8dc3ad6397.png" /></td>
<td><img class="tex" alt="4 \pi\,r^2" src="http://upload.wikimedia.org/math/1/0/9/109a7c5ee8613b982dc98a9d2f0a590d.png" /></td>
<td>
<p><img class="tex" alt="1\," src="http://upload.wikimedia.org/math/d/0/6/d06c48671eacd7f1e2afde7289e483d5.png" /></p>
</td>
</tr>
</tbody>
</table>
<p> </p>
<h2><span class="mw-headline">Sphericity in Statistics</span></h2>
<p>In statistical analyses, sphericity relates to the equality of the variances of the differences between levels of the repeated measures factor. Sphericity requires that the variances for each set of difference scores are equal. This is an assumption of an ANOVA with a repeated measures factor, where violations of this assumption can invalidate the analysis conclusions. Mauchly's sphericity test is the statistical test used to evaluate sphericity.</p>
<p> </p>
<h2><span class="mw-headline">References</span></h2>
<ol class="references">
<li id="cite_note-0"><strong>^</strong> <cite style="FONT-STYLE: normal">Wadell, Hakon (1935). "Volume, Shape and Roundness of Quartz Particles". <em>Journal of Geology</em> <strong>43</strong>: 250–280.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Volume%2C+Shape+and+Roundness+of+Quartz+Particles&rft.jtitle=Journal+of+Geology&rft.date=1935&rft.volume=43&rft.aulast=Wadell&rft.aufirst=Hakon&rft.pages=250%E2%80%93280"> </span></li>
</ol>
<p> </p>
<h2><span class="mw-headline">See also</span></h2>
<ul>
<li>Rounding (sediment)</li>
</ul>
<p><a id="External_links" name="External_links"></a></p>
<h2><span class="mw-headline">External links</span></h2>
<ul>
<li><a class="external text" title="http://www.howround.com/" href="http://www.howround.com/" rel="nofollow">How round is your circle?</a></li>
<li><a class="external text" title="http://people.uncw.edu/dockal/gly312/grains/grains.htm" href="http://people.uncw.edu/dockal/gly312/grains/grains.htm" rel="nofollow">Grain Morphology: Roundness, Surface Features, and Sphericity of Grains</a></li>
</ul>
