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## Correlated Gaussian Method

### Definition

A method that relates winning percentage to points scored, points allowed, the standard deviations of points scored and allowed, and the correlation between points scored and allowed. Points scored can be replaced with offensive rating and points allowed can be replaced with defensive rating.

### Applicability

This method applies in a straightforward manner to teams and probably applies in a similar manner to individuals.

### Calculating Expected Win% using the Correlated Gaussian Method

```            __                                 __
|           (Rtg-Opp.Rtg)           |
Win% = NORM |-----------------------------------|
|       SD(Rating Difference)       |
--                                 --
SD(Rating Difference) = SD(Rtg - Opp.Rtg)
= SQRT[Var(Rtg)+Var(Opp.Rtg)
-2*Cov(Rtg,Opp.Rtg)]

NORM means to take the percentile of
a mean-zero variance-one normal distribution
corresponding to a value given by that in
the brackets I faked.

Rtg:		Points scored per 100
possessions (offensive rating)
Opp.Rtg:	Points allowed per 100
possessions (defensive rating)
SD():		Statistical standard deviation
of quantity in parentheses ()
Var():		Statistical variance of
quantity in parentheses ()
Cov():		Statistical covariance of
quantities in parentheses ()
```
This method can be found in Microsoft Excel as NORMSDIST(z). Given a value of z, this function evaluates what percentage of a Normal (or Gaussian) Distribution is smaller than z.

### Discussion

See Basketball's Bell Curve for a discussion of this method. It has also been used in other articles, such as The Effect of Controlling Tempo and What Strategies Are Risky?

This method accomplishes the same thing as the Pythagorean Method, providing more insights, but also requiring more information.

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