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<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>
<p> </p>
<h2p><span class="mw-headline">Ellipsoidal Objects</span></h2><dl><dd><span classfont size="boilerplate seealso5"><em>See also: Earth radiusEllipsoidal Objects</emfont></span></dd></dlp>
<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 =
<p><em>(where a, b are the semi-major, semi-minor axes, respectively.</em></p>
<p> </p>
<h2p><span class="mw-headline"><font size="5">Derivation</font></span></h2p>
<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="\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>
<h2p><span class="mw-headline"><font size="5">Sphericity of common objects</font></span></h2p>
<table style="MARGIN: 0pt auto; BORDER-COLLAPSE: collapse; TEXT-ALIGN: center" cellpadding="7" border="1">
<tbody>
</table>
<p> </p>
<h2p><span class="mw-headline"><font size="5">Sphericity in Statistics</font></span></h2p>
<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>
<h2p><span class="mw-headline"><font size="5">References</font></span></h2p>
<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>
