A Graphic Demonstration of the TOP-Formula Accuracy
(One of several "White Papers" on baseball analysis to be found on this site--see the Site Directory below.)|
and it's a nuisance to re-make the graph; but almost half a century of beautiful matches ought to suffice.) Summary of ResultsThe graph below shows the results of applying the full High Boskage House formula for runs scored (derived from team Plate Appearances, Hits, Total Bases, Walks, Steals, Caught Stealing, and Sacrifice Bunts) to almost a half-century, 48 full years, of actual major-league data: all teams in all years from 1954 through 2001 inclusive (there were scoring-rule changes in earlier years that make certain raw data suspect). Shown below are actual team runs scored, predicted team runs scored (rounded to the nearest whole number), and the difference between prediction and actuality as a percentage. As expected, the average size of runs projection errors is well under 3% (about 2½%). For those who know statistical theory, the Standard Deviation is 22.31 runs (without any regression "curve-fitting" cooking of the results). The overall results are these:
The "Error Sizes" disregard whether the error is high or low--they measure its size. The "Cumulative Error" allows plus and minus errors to cancel; as we should expect, it is virtually zero, far less than a run a team a season. (The statistical formulae used to calculate those data are shown at the bottom of this page.) Full Results
Notice especially that accuracy remains excellent at the extremes: down at 329 annual runs (the lowest) and also up at 993 annual runs (the second highest), there are points bang on the line. That indicates a truly general predictive method of accuracy, not just one that is OK where the data bunch up. For those who might want to see those data in tabular form, we have provided it on a separate page (separate owing to its length--it may take some time to fully download). If you want to check it, though here's the tabulated data. The CalculationsAs to the statistical measures: The expected error is calculated using the standard statistical probability formulae. The expected average error is 79.79% of one Standard Deviation. The Standard Deviation for any one team-season is, in turn, the square root of npq, where n is the number of data samples, p is the probability of a success, and q is the probability of a failure (by definition, then, q=1-p). For this tabulation, a "success" is a run scored and a "data sample" is a batter at the plate. Thus, the probability of a success is the calculated expected runs divided by the team's total of batter plate appearances (we use the calculated, not the actual, runs because we want to know the S.D. for the projection, although in practice it would matter little which was used). So, the expected average error per team-season is just: We then average the individual expected errors for an overall average expected error figure (the individual expected-error figures per team-season will be rather similar because neither PAs nor TOPs vary all that much from one to another). |
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