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September 27, 1997
The winding down of the current baseball season seems an appropriate time to take a look at a curious inconsistency that sometimes appears in team standings.
Teams are generally ranked on the basis of both winning percentage and "games behind" the leader. Here’s a recent example:
EAST
Baltimore
New York
Detroit
Boston
Toronto
W
92
88
74
74
71
L
59
63
77
78
80
Pct.
.609
.583
.490
.487
.470
GB
--
4
18
18 1/2
21
There’s no inconsistency here between the ranking according to percentage and the ranking according to games behind, and that’s typical. Once in a while, however, the team with the higher winning percentage may be at least one-half a game behind in the standings.
That anomaly intrigued Peter A Rogerson, a geographer at the State University of New York at Buffalo, so he analyzed the underlying mathematics and identified the conditions necessary for such an occurrence. His results appear in the latest issue of the Journal of Recreational Mathematics.
In formal terms, Team 1 is said to be k games behind Team 2 when one-half the difference between the number of wins by each team plus one-half the difference in the number of losses by each team is equal to k: [(w2 - w1) + (l1 - l2)]/2.
In effect, the games behind statistic represents the number of games that Team 1 would have had to win, instead of lose, in order to be tied with Team 2. Rogerson provides a real-life example of the type of inconsistency in rankings that can arise (Triple A baseball standings in the American Association on May 14, 1991):
Team
Buffalo
Indianapolis
Nashville
Louisville
W
14
18
14
12
L
9
12
14
20
Pct.
.609
.600
.500
.375
GB
1/2
--
2 1/2
6 1/2
He concludes that such an inconsistency is most likely to occur when one team has played substantially fewer games than the other. "In the case of baseball, this can often happen early in the season, when one team has had many games postponed by adverse weather conditions," Rogerson says. Usually the cause is rain, but in Buffalo, that may sometimes be snow, he adds.
The inconsistency can also come up later in the season, especially when two teams have relatively high winning percentages. A table showing the winning percentage of Team 2, p, and how many more games Team 2 must have played than Team 1 for the inconsistency to arise, c, illustrates why.
Winning percentage, p
.510
.530
.550
.570
.590
.610
.630
.650
.670
Critical number of games, c
51
17
11
9
7
7
5
5
3
"Having two teams with extremely high winning percentages is also more common early in the season," Rogerson notes. He adds that if Team 2 has a winning percentage of exactly .500, the inconsistency can’t arise, because Team 1 could only be half a game behind Team 2 if Team 1’s winning percentage were less than .500. Similarly, the inconsistency cannot come up when Team 1 has a winning percentage of exactly .500.
Of course, baseball isn’t the only sport in which teams are ranked according to winning percentage and games behind. The same anomaly could occur in football, basketball, and hockey team standings. In which sport are they most likely to come up?
Copyright © 1997 by Ivars Peterson.
References
Rogerson, P.A. 1997. Inconsistencies in league standings. Journal of Recreational Mathematics 28(No. 2):81.
Comments are welcome. Please send messages to Ivars Peterson at ip@sciserv.org.
Ivars Peterson is the mathematics and physics writer and on-line editor at Science News (http://www.sciencenews.org/). He is the author of The Mathematical Tourist, Islands of Truth, Newton’s Clock, Fatal Defect, and *The Jungles of Randomness: A Mathematical Safari. His current works in progress are an updated, 10th anniversary edition of The Mathematical Tourist (to be published in 1998 by W.H. Freeman) and The House at Infinity: Imagination, Mathematics, and Art (to be published in 1999 by Wiley).
*NOW AVAILABLE: The Jungles of Randomness: A Mathematical Safari by Ivars Peterson. New York: Wiley, 1997. ISBN 0-471-16449-6. $24.95 US.
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