Normal

 

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You might be normal, but are you hyperbolically normal?

A hyperbolic normal distribution is proposed to be one in which 1/3  of the observations, outcomes, fall within the median ± σ and 100% of the outcomes fall within the median ± 3σ.  As in any normal distribution, the median will be equal to the mean, but the mean can be computed without ordering the observations.  

This also indicates that the minimum number of outcomes must be 3 or a hyperbolic distribution will be abnormal.  For example for two outcomes, e.g. a two player game, a choice/transition/phase change, will have a variance, σ² , but it will be hyperbolically abnormal because while it passes the 100% test, it fails the 1/3 test.  By contrast three‑or‑more‑players will always pass  both the 100% and the 1/3 test.  The variance, σ²,  of a two outcome game is 0.277777 which makes the range, σ, 0.166667.  If choice one has a value of 1 and choice two has a value of 2, then the median and the mean are both 1.5.  However while both outcomes pass the 100% test in that they are within 1.5  ± 3*(0.166667), they fail the 1/3 test in that neither outcome is within 1.5 ±  0.166667.  By contrast, a three player outcome: 1, 2, or 3; has a variance, σ², of 0.111111, or 1/9, which means that σ=1/3.  This distribution has a mean of 2 and a median of 2.  One hundred percent of the outcomes are within 2 ± 1, and 1/3 of the outcomes are within 2 ± (1/3).

The hyperbolic skew is proposed to be the ratio of the mean and the median.  When the ratio is greater than 1, the distribution  favors higher outcomes.  When the ratio is less than 1, it favors lower outcomes.  A hyperbolic normal distribution is one where this ratio is between 0.75 and 1.5 . In these cases, the observations will pass both the 100% and 1/3 test but the lowest observation will also not be less than 0.  When the median is equal to the mean of course the hyperbolic skew is 1.

It is not surprising that the minimum number of outcomes in a hyperbolic normal distribution is 3. In game theory there is a different strategy for playing two-player games and three‑or‑more‑player games.  If there are only two players in a game, then there have to be three outcomes: e.g. win, loss, and tie.  Having only two outcomes is abnormal.  If there are only two outcomes, you can not tell if the outcome is due to chance or the winner is better than the loser.