In this post, I explore the relationship between the score achieved on the first ball and the resulting frame score. Generally speaking, it would be expected that a higher pin count on the 1st toss would lead to higher frame scores for obvious reasons. 1) You knock down more pins; 2) You increase the likelihood of achieving a mark; 3) If you get a mark, you have additional chances to enhance your frame score.
I begin with a histogram of first ball scores. It may be somewhat surprising given our relatively low skill level at bowling that the most common first ball outcome for our team is a strike. Strikes account for 27.2% of first balls thrown. Obviously, this does not mean that we strike most frames. If framed in terms strike vs. non-strike, we have not recorded strikes on approximately 73% of first balls. The distribution is highly left-skewed and thankfully first ball scores of 0-4 in total account for less than 4% of attempts. Scores of 5-10 are increasingly common in our database, again with strike being the most common, and therefore, the most likely of any possible outcome. Maybe we aren’t as bad at bowling as we thought.
How does the likelihood of picking up a mark change as a function of the first ball score? Although we don’t have a lot of data for first ball scores less than five, generally speaking, a small pin total after one throw means a low likelihood of picking up a spare. For us, it’s less than 20%. In our current database, with first ball pin totals of five or greater, the probability of picking up a spare increases steadily from approximately 25% for a 5 pin first ball to approximately 50% for a 9 pin first throw. It appears that we may be slightly better at picking up spares when three pins remain than when two are left. This slight discrepancy could be due to a small sample size, but I expect that this reversal is due to split effects, that the most common and difficult splits to pick up involve two pins. Of course, all ten pin first tosses are marks, so the 100% mark frequency for a 10 pin first toss is meaningless.
The score received on the first ball determines the minimum and maximum score possible for a frame. The minimum score possible for a frame is the first ball score (i.e., it is impossible to receive less than an eight for a frame if the first ball is an eight). The maximum score is constant for first balls of 0 through 9; it is 20 if a spare is picked up followed by a strike. For a first ball of 10, the maximum score is 30 if two strikes follow. In the scatter plot above, the area in yellow shows the total possible range of frame scores than can be achieved given a certain first ball score. The gray dots show our actual frame scores for 720 frames.
Given the considerations above, it should be no surprise that higher first ball scores lead to higher frame scores, although for our team, this effect does not appear until a 1st toss score of five. The graph above shows the average frame score for our team as a function of the first ball score. When less than five pins are recorded, we generally average less than 10 pins per frame with a mean outcome around eight. For all first throws above four, our average frame scores are above ten and increase steadily. For five pin first balls, we average a score of 10.3. For nine pins, our average frame score is 13.5. For strikes recorded on the first throw, there is a huge jump, and we average 21.4 pins per frame.
So what does all of this mean? Well, this story has no moral, except that I obviously have too much time on my hands. If I had to conclude something, I guess I was surprised that our most common first ball outcome is a strike. I would have guessed that we throw more nine balls. I suppose the most obvious conclusion is that success begets success. Do well on the first throw and good things tend to follow.
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11 months ago