Sunday, January 4, 2009

On the Curious Nature of Bowling Scores

Bowling scores have interesting quantitative properties. The scoring system was apparently formalized in 1895 by the American Bowling Congress in New York City. It is a remarkably elegant thing, establishing a maximum possible score of 300 over 10 frames. In this post, I want to explore the relationship between the actual number of pins knocked down versus the score received. Although generally speaking, recording more pins will result in a higher score, in actuality the relationship between these two variables is surprisingly weak. Consider the minimum and maximum scores that can be achieved when a bowler knocks down every possible pin. A total of 120 pins can be recorded in any game assuming three strikes in the 10th. The maximum score is easy; a perfect game with 12 strikes equates to a score of 300. The minimum score is achieved by nine frames comprised of gutter balls on the 1st toss and a spare on the second. The final frame must be a turkey. This game would result in a score of 120 (9x10+ 3x10=120). So, recording every pin in a game of bowling can theoretically result in scores with a range of 180. This is certainly true in theory, but is it also true in practice?

We have 60 games in our current database. Above is a scatter plot showing the relationship between actual pins recorded and score received. While it is generally true that more pins equate to a higher score, the relationship is quite weak. In fact pins recorded only explains approximately 55% of the variance in game score. I have circled two extreme games to illustrate why this is the case. Circled in blue is a game in which the bowler (who shall remain anonymous) recorded 96 pins and received a score of 105. In red is a game in which a bowler recorded fewer pins, only 94, and yet ended with a score of 189. The reason for this discrepancy should be obvious. The first game included only three marks, one strike and two spares, and the latter game seven marks, five strikes and two spares. Another obvious point follows. Your score is not determined so much by the number of pins knocked but instead by the manner in which they are knocked down.

Using the ratio of game score to pins recorded, I have created a statistic I call "pin efficiency". It refers to the number of points received per pin recorded. It's minimum and maximum values are 1.0 and 2.5 respectively. In a game with zero marks, the game score will be identical to the number of pins recorded, so the ratio is equal to 1. In a perfect game, 300 points are derived from 120 pins, so the ratio is 2.5 (300/120=2.5). The histogram below shows the distribution of pin efficiency for our 60 games on record.

The take home message of this graph is that we are bad bowlers, no big surprise there. We average approximately 97 pins per game and approximately 1.4 points per pin recorded, thus our pathetic team average. We have gone as low as 1.08 points per pin in a single game with two marks and as high as 2.01 in a game with 7 marks. Interestingly, we have two games with 9 marks on record, and these equate to pin efficiencies of 1.68 and 1.69 respectively. So what is the key to increasing pin efficiency? Here is the game with the highest recorded pin efficiency to date.

Please disregard the fact that I totally choked at the end of this game. What is important are the pin totals by frame in the last row. These are not the total number of pins but the score received for that frame. For frames 2-7, a score of at least 20 was received for achieving a mark, or knocking down ten pins (pin efficiency≥2.0). This can be done two ways, following a spare with a strike or following a strike with any mark. Either way, you need strikes. Putting up a game of all spares, even 9-spares, won't get you to a pin efficiency above 2.0, but it will get you close. If you want to reach a score of 200, you will likely need a pin efficiency of 2.0 or above.

One more graph, and then I'll end this pointless blather. It should be no surprise that the best single predictor of pin efficiency is the number of strikes in a game:
But there are other factors that are important, too, like timing. Stringing together strikes will make pin efficiency skyrocket. This is not easy to do when you are a bad bowler, but if we ever want to consistently bowl 200 games and get our pin efficiencies in its upper ranges (>1.75), we need to throw more strikes and take advantage of spares. That was a long a convoluted way to get to a really simple and obvious conclusion, so please forgive me. I like numbers.


  1. Joe and I vote that we just drink, roll, chew tobacco and have fun. Screw the over-analyzing b.s. We haven't been rolling long enough to be that good. Time will improve our scores and abilities. Rome wasn't built in a day.

  2. When you manage to extract your panties from your crack, let me know if I should still keep your scores.

  3. That should be plural (i.e. cracks).


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