<h3>How the OU tier is constructed</h3>
<p>The OU tier contains, as already mentioned, the Pokemon that are used commonly in the standard metagame. The reader might be interested to know how the Pokemon in the OU tier are selected to make part of the tier.</p>
<p>For RB, GS and RS, the OU tier is formed from the experience of our community of which Pokemon are commonly used in Smogon's standard metagame tournaments. In tournaments, people use Pokemon that can compete at the highest level to allow them to win, so naturally they are an excellent means of determining the OU tier.</p>
<p>The OU tier for DP is constructed from the league statistics extracted from the current DP Pokemon battling medium. These statistics provide the number of times each Pokemon was used in the standard metagame league during each month, and those of the six months prior to the OU tier creation are utilised in particular to predict which Pokemon are going to be used in the near future. The Pokemon commonly used by expert players, who are highly-ranked in the league, receive heavier weighting than those used by less expert ones. Since, compared to the previous three generations, the DP standard metagame is still in its infancy, the OU tier for DP is continually updated on a three-month basis.</p>
<h3>The mathematical details of how the OU list is generated in DP</h3>
<p>Shoddybattle's weighted Pokemon usages lists of the six months prior to the OU update are first extracted from their website. The weighted usage of a Pokemon is the summation of the rankings of every player that used that Pokemon. For example, say both Blissey and Celebi are used three times during a day, and say that the players using Blissey had ranking 1650, 1360 and 1470, while those using Celebi had ranking 1520, 1360 and 1190. Their weighted usage for that day would be 4480 for Blissey and 4070 for Celebi. This would rank Blissey above Celebi in the weighted usage list, even though they were both used the same number of times during that day.</p>
<p>Each weighted usage of each month is then divided by the total of the weighted usages during that particular month, so that the probability of that particular Pokemon being used during that month is obtained. Suppose the probability for a particular Pokemon to be used in month m is <var>P_m</var>. These months are ordered in the following manner: if the list is being updated in July, then January, February, March, April, May and June correspond to months 1 to 6 in that order.</p>
<p>For each Pokemon, the predicted probability <var>P</var> is calculated as per the equation below, found using linear extrapolation techniques:</p>
<pre><var>P</var> = (50×<var>P_6</var> - 5×<var>P_5</var> - 4×<var>P_4</var> - 3×<var>P_3</var> - 2×<var>P_2</var> - <var>P_1</var>) ÷ 35</pre>
<p>These predicted probabilities are then sorted in descending order and made into a cumulative frequency, so that the 30th number, say, would be the probability that one of the top 30 used Pokemon is used in battle. Finally, the OU list is made to consist of all the Pokemon whose cumulative probability of being used does not exceed 0.75. This would mean that, whenever a Pokemon shows up in a battle for the first time, it has a 75% chance of being a member of the OU tier.</p>
<p>It can be shown that the probabilities of the usage of each Pokemon follow an exponential distribution. This means that the <var>n</var>th most used Pokemon will roughly have a <var>b</var> × exp(-<var>b</var>×<var>n</var>) probability of being used, where <var>b</var> is a constant that determines the shape of the exponential curve. The best value of <var>b</var> can be found using regression analysis on a spreadsheet program like Microsoft Excel. For Pokemon usages, it usually results in a value between 0.026 and 0.028.</p>
<p>In addition, the cumulative exponential distribution can be found using integration to be 1-exp(-<var>b</var>×<var>n</var>). This means that the probability that a Pokemon used in battle is among the first <var>n</var> most used Pokemon is 1-exp(-<var>b</var>×<var>n</var>). Thus, we can effectively solve the equation 1-exp(-<var>b</var>×<var>n</var>) = 0.75 to find the number of Pokemon that the OU tier will roughly contain in terms of <var>b</var>:</p>
<pre>
1 - exp(-<var>b</var> × <var>n</var>) = 0.75
exp(-<var>b</var> × <var>n</var>) = 1 - 0.75
exp(-<var>b</var> × <var>n</var>) = 0.25
-<var>b</var> × <var>n</var> = ln(0.25)
-<var>b</var> × <var>n</var> = -1.386
<var>b</var> × <var>n</var> = 1.386
<strong><var>n</var> = 1.386 ÷ <var>b</var></strong>
</pre>
<p>Assuming that <var>b</var> is between 0.026 and 0.028, the number of Pokemon in the OU tier would roughly contain between 1.386 ÷ 0.026 and 1.386 ÷ 0.028 Pokemon, or approximately 50 Pokemon.</p>
