Well, if you want a measure of "Gaussianness" of a dataset, in this case the set of (actual-predicted) margins, I'm fond of constructing a Normal Quantile Plot and then using Simple Linear Regression to fit the resulting graph. In more detail:
1) Construct N-element (actual-predicted) dataset, sorted numerically 2) Assign probabilities from 1/N to (N-1)/N in sorted order 3) Convert probabilities to sigma using the Inverse Cumulative Normal Function 4) Fit y=mx+b to the (data,sigma) graph using SLR
The resulting r^2 (and corresponding confidence statistics) from the SLR will give you a measure of Normalcy.
You mean (actual-predicted) final margins? It's not entirely clear how you're trying to relate scores to the probability of winning the game.
The standard approach (in Economics, at least) that assumes Normality would be to fit a Probit model to the data; the probability of winning as a function of the predicted margin. Note that this doesn't require the actual margin to have a Normal distribution, only the unobserved probability of victory (of which the actual margin is a special case).
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1) Construct N-element (actual-predicted) dataset, sorted numerically
2) Assign probabilities from 1/N to (N-1)/N in sorted order
3) Convert probabilities to sigma using the Inverse Cumulative Normal Function
4) Fit y=mx+b to the (data,sigma) graph using SLR
The resulting r^2 (and corresponding confidence statistics) from the SLR will give you a measure of Normalcy.
Reply
The standard approach (in Economics, at least) that assumes Normality would be to fit a Probit model to the data; the probability of winning as a function of the predicted margin. Note that this doesn't require the actual margin to have a Normal distribution, only the unobserved probability of victory (of which the actual margin is a special case).
I'll think about this some more.
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Now I just have to figure out how to use a spreadsheet...
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