The Latitude 38 Performance Trophy Explained
The Latitude 38 performance trophy is awarded to the yacht with the most convincing win relative to its own division.
One of the problems with multiple starts on different days is that different divisions often encounter significantly different weather conditions in the early part of the race. When this happens, some divisions have an inherent advantage over other divisions, and the overall winner is often determined by luck-of-the-draw with respect to offshore weather on their start date.
An element of randomness is not necessarily a bad thing in sailboat racing, but there has always been interest in comparing the winners of different divisions by looking at their performance against only the boats in their own division, thereby leveling out this source of error.
Another factor is the relative level of competition in each division. Some divisions are populated with relatively similar boats sailed by more competitive crews, and a large winning margin in these divisions is difficult. On the other hand, some divisions have more variability in both the type of boat and the degree of racing intensity, and winning margins in these divisions tend to be large.
The Latitude 38 Performance Award is an attempt to compare the winners of different divisions on more equitable terms by evaluating the degree of competitiveness in each division so that winning margins in each division can be realistically compared.
The central assumption here is that a more competitive division will have more closely clustered corrected time finishes than a less competitive division.
Standard deviation has been used in past races to evaluate the competitiveness or "tightness" of the division finishers. However standard deviation is strongly affected by outliers, and a straggler in an otherwise competitive division might skew the calculated standard deviation. For this reason the previous method arbitrarily rejected the bottom 20% of each division in calculating the results.
For 2012, the standard deviation method is replaced by the "median absolute deviation" (MAD) method. (With thanks to the crew of Moonshine, 2010 Performance Award runner-up, for pointing out the advantages of this method.)
From Wikipedia: "The median absolute deviation is a measure of statistical dispersion. It is a more robust estimator of scale than the sample variance or standard deviation... For instance, the MAD is a robust statistic, being more resilient to outliers in a data set than the standard deviation. In the standard deviation, the distances from the mean are squared, so on average, large deviations are weighted more heavily, and thus outliers can heavily influence it. In the MAD, the magnitude of the distances of a small number of outliers is irrelevant."
In other words, trailing outliers (stragglers) do not skew the results as they do with the standard deviation method.
Here's how the MAD-based Performance Index is calculated:
1) Find the median of the corrected times in the division. It is not necessary to filter out any of the trailing outliers.
2) Calculate the deviation of each corrected time from the median. Convert negative deviations to positive (i.e., take the absolute value of each deviation).
3) Find the median of all the deviations. This is the median absolute deviation.
In indicial notation: Median Absolute Deviation = mediani (|Xi - medianj(Xj)|)
4) Now look at how far the division winner was ahead of the median, and divide this by the MAD. The result is the performance index for that boat.
The boat with the highest performance index wins the latitude 38 Performance Trophy. (Which, incidentally, is still inscribed with the formula for standard deviation because it is a much more attractive formula when written out, and still includes a slide rule, even though a slide rule is essentially useless for calculating either standard deviation or median absolute deviation.)
Paul Kamen, April 23 2012