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We got an outright top on a board and yet on our personal score card it is shown as 98.98%. Why not 100%?
When all boards are not played the same number of times, the match points for the boards played fewer times are factored/increased to allow for this. The Neuberg formula is used to do the factoring. However, the effect of the formula is also to 'downgrade' the match points on these boards a bit to allow for the fact that the 'competition' wasn't as good as on the boards played more times.
The calculation is performed as follows:
On each traveller match points are allocated to each pair first. This is done by comparing this pair's score with all the other scores on the board. (Adjustments/averages and missing scores are excluded from this comparison.) 2 match points are allocated for each score that is worse than this pair's score and 1 match point for each score equal to it.
Therefore the maximum number of match points anybody can get on a board is:
Top = (2 * Number of Scores on the Board) - 2
Number of Scores on the Board excludes adjustments/missing scores.
So if there are 12 scores on a board and one of them was an adjustment/average, then the top score on this board will be 20, while with 12 'normal' scores, the top score would be 22.
The Neuberg formula is then applied to the resulting match points.
The Neuberg formula is:
FMpt = the factored match points to be given on this board to a specific pair
MPt = the original match points given to the pair as calculated above
MaxS = the maximum number of scores on any board in the event
ActS = the actual number of scores on this board. This excludes adjustments/averages and missing scores.
So, for example, if a pair gets 12 match points on a board that was played 11 times and the maximum number of times any board was played is 12, then this pair will get 13.18 match points after the Neuberg formula was applied or 59.9% (13.18/22) and not 60% (12/20). Equally if a pair gets a top on a board played 11 times and the maximum number of times any board was played is 12, then this pairs score will be 21.9 match points after the Neuberg formula was applied and not 22, which would be the top score for the board played 12 times. In percentage terms this would be 99.54% (21.9/22), and not 100% (20/20).
Wish you hadn't asked?
How random are our computer dealt hands?
I extracted statistics from the hand definition files - the PBN files - (kindly supplied by John Hunter) and fed them into an Excel spreadsheet. Data from 2007, 2008 and 2009 have been used, 16500 deals in all.
The results follow quite closely the theoretically expected values - see Summary.
I also produced a series of graphs which demonstrate random behaviour from set to set.
High Card Points Graphs
Suit Break Graphs for 7 and 8 card fits
Suit Break Graphs for 9, 10 and 11 card fits
Hand Patterns Graphs - 1
Hand Patterns Graphs - 2
Hand Patterns Graphs - 3