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(This article was first published on <strong><a href="https://xianblog.wordpress.com/2016/12/07/the-incredible-accuracy-of-stirlings-approximation/"> R – Xi'an's Og</a></strong>, and kindly contributed to <a rel="nofollow" href="https://www.r-bloggers.com/">R-bloggers)</a> </div>
<p><strong>T</strong>he last riddle from <a target="_blank" rel="nofollow" href="http://fivethirtyeight.com/features/a-puzzle-will-you-yes-you-decide-the-election/">the Riddler</a> [last before <a target="_blank" rel="nofollow" href="https://xianblog.wordpress.com/2016/11/09/a-tyring-day/">The Election</a>] summed up to find the probability of a Binomial B(2N,½) draw ending up at the very middle, N. Which is</p>
<p><img title="\wp={2N \choose N}2^{-2N}" alt="\wp={2N \choose N}2^{-2N}" class="latex" src="https://s0.wp.com/latex.php?latex=%5Cwp%3D%7B2N+%5Cchoose+N%7D2%5E%7B-2N%7D&bg=000000&%23038;fg=B0B0B0&%23038;s=0"></p>
<p>If one uses the <a target="_blank" rel="nofollow" href="https://en.wikipedia.org/wiki/Stirling%27s_approximation">standard Stirling approximation</a> to the factorial function,</p>
<p>log(N!)≈Nlog(N) – N + ½log(2πN)</p>
<p>the approximation to ℘ is 1/√πN, which is not perfect for the small values of N. Introducing the <a target="_blank" rel="nofollow" href="https://en.wikipedia.org/wiki/Stirling%27s_approximation">second order Stirling approximation</a>,</p>
<p>log(N!)≈Nlog(N) – N + ½log(2πN) + 1/12N</p>
<p>the approximation become</p>
<p>℘≈exp(-1/8N)/√πN</p>
<p>which fits almost exactly from the start. This accuracy was already pointed out by <a target="_blank" rel="nofollow" href="https://xianblog.wordpress.com/2014/05/15/an-a-statistical-proof-of-a-binomial-identity/">William Feller</a>, Section II.9.</p>
<p>Filed under: <a target="_blank" rel="nofollow" href="https://xianblog.wordpress.com/category/kids/">Kids</a>, <a target="_blank" rel="nofollow" href="https://xianblog.wordpress.com/category/statistics/r-statistics/">R</a>, <a target="_blank" rel="nofollow" href="https://xianblog.wordpress.com/category/statistics/">Statistics</a> Tagged: <a target="_blank" rel="nofollow" href="https://xianblog.wordpress.com/tag/american-elections-2016/">American elections 2016</a>, <a target="_blank" rel="nofollow" href="https://xianblog.wordpress.com/tag/stirling-approximation/">Stirling approximation</a>, <a target="_blank" rel="nofollow" href="https://xianblog.wordpress.com/tag/the-riddler/">The Riddler</a> <a target="_blank" rel="nofollow" href="http://feeds.wordpress.com/1.0/gocomments/xianblog.wordpress.com/31935/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/xianblog.wordpress.com/31935/" border="0"></a> <img alt="" src="https://i1.wp.com/pixel.wp.com/b.gif?resize=1%2C1&ssl=1" border="0" data-recalc-dims="1"></p>
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