Monday, December 28, 2009

The Rules of Bowling: Averages

When I first started writing this blog, I thought I would try (and often fail) to be humorous. It seems lately, though, that I have been in a more serious mood. So instead of making fun of some aspect of the USBC rulebook, today I'd actually like to make a suggestion for a rule change. It is not a major revision. I am not suggesting that we add an 11th pin, though a prime number of targets would be dandy. Instead, I'd like to make a case for why a change to the rule regarding the calculation of averages, and therefore handicaps, is in order. My intent is to make the game more fair for all, and it would affect everyone who bowls in a handicapped league. So in that sense, it is actually a very serious change.

In that light, I hope that somebody who has the power to change rules notices. Admittedly, I am somewhat selfishly inspired since I am going to use my own average as case and point (as I have done before), but if this has happened to me, then no doubt it has also happened to thousands of other bowlers. So, USBC Executive Team, you have some motivation to read on. For those of you who are ego-googlers, I'm talking to you Kevin Dornberger, Susan Merrill, Jim Oberholtzer, Neil Stremmel, and Pete Tredwell. Let's begin...

In scratch bowling, there is no need to keep an average. It is irrelevant to the outcome of any match. In handicapped bowling, however, an average is necessary because it establishes the skill level of a bowler. From the average, the handicap is derived. The intent of the handicap is make it possible for bad bowlers (like me) to compete with good bowlers (like you). Straight from the new rulebook, here is how averages are calculated:


It's like any old average. Sum your game scores and divide by the number of games. On the surface, this seems perfectly reasonable. The resultant value should be roughly in the middle of all of your game scores with about half of your scores falling below the average and half above. Therefore, somebody with a 170 average is better than somebody with a 150 average, right?

Wrong. In most cases, this would be true, but there are some obvious ways it could be incorrect. For example, what if the person who has averaged 150 has intentionally bowled poorly to establish a low average? This is why there is a rule against sandbagging. What if only one game has been bowled? Can we really judge the relative skill of these bowlers based only on one game?

What if 1000 games have been bowled and no sandbagging has occurred? Can you be confident that the 170 bowler is more skilled? No. You can be confident that over the last 1000 games, the 170 bowler has bowled much better, but the question is who is a better bowler now. What if the 150 bowler started in her first hundred games with a 100 average, and for each subsequent 100 games bowled, she increased her average by ten pins. That means that for the last 100 games, she has averaged 190 pins. In fact, Ms. 150 might now be the better bowler, but her average does not well reflect her recent games and increased skill.

Sandbagging aside, these examples are intended to highlight two problems with the way that averages are currently calculated: 1) For small number of games, an average can be highly inaccurate; 2) For large numbers of games, an average is very insensitive to recent performance. Both of these factors can make competition very unfair.

Let's start with the first. When a league begins, an average must be established. According to Rule 118b, "Each league shall adopt a rule to determine the number of games required to establish an average in that league." In our league, for veterans, an average is established after nine games. Until nine games have been bowled, your average from the prior season is used. For new bowlers, only three games are needed to establish. If by chance, a bowler begins the league with unusually good or bad bowling, their average will be highly inaccurate. The bowler is then put at an advantage if they bowled poorly or a disadvantage if they bowled well. This system provides strong incentive to sandbag at the start of a league.

Regarding the second problem, bowling leagues can be lengthy endeavors. We bowl three games a week for 32 weeks, or a total of 96 games. For many bowlers, skill changes. Some improve; others get worse. But because the average is calculated for the entire duration of the league, it becomes increasingly insensitive to these changes. Here's a simple example. Assume you enter a league with a prior 150 average. What happens if you bowl ten pins over average in your first game? Now, you have a 160 average. You average increased 10 pins, right (avg = 160/1 =160)? What if you have a 150 average after 99 games. If you bowl a 160, how much will your average change? It will increase by only 0.1 pins (avg = 15010/100 = 150.1).

The following graph shows the sensitivity of the average as a function of the number of games bowled. In a nutshell, it shows how many pins will be added to an average if a bowler bowls 10 pins over their average in a single game. When few games have been bowled, the average is extremely sensitive, but after a large number of games, it becomes essentially fixed. This insensitivity gave me and my team a major advantage at the end of last season for reasons I'll explain in a moment.


First, let me suggest the rule change. The rule should simply read "A bowling average is determined by dividing the total number of pins credited to a bowler for the previous 30 games of one USBC league by the number 30, inclusive of games bowled during the prior season. If a bowler has fewer than 30 games in the league, a bowling average is determined by dividing the total number of pins credited to a bowler in one USBC league by the number of games bowled in that league in a season."

Essentially, I suggest that the average is changed from a simple average to a moving average and should be based only the last 30 games. Once a bowler has 30 games in a league, both problems are solved. In statistics a sample size of 30 is considered sufficiently large to provide an accurate average. Furthermore, the sensitivity of the average does not change.

Here is a real world example of how this would affect a bowler. The bowler is myself. I started last season bowling terribly, but by the end of the 1st half, something changed, and I began to bowl much better. I started keeping records on the 1st of December, and I came into that night with a 136 average after 39 games. Over the next 18 weeks, my average steadily grew to a 147. Clearly, I was bowling better. To increase my average that much over such a short period of time, I was bowling considerably better than I had. The graph below shows how my average has changed. For comparison, I show how my moving average over the prior 30 games, the way I would prefer to see averages calculated.


Notice that for the end of last season, the moving average is anywhere from 8 to 10 pins higher than my official USBC average. Why? The USBC average was dragged down by my poor bowling at the start of the season. I was bowling 10 pins better than my average and handicap showed. This gave my team a significant advantage. It could be argued that I deserved it because I had improved... it is a reward for becoming a better bowler. Well, if the reverse had happened, if I had gone into a major slump, my USBC average would be too high. Simply put, the current method for average calculation puts too much weight on what happens early in the season, and as the season goes on, it becomes increasingly insensitive to what has happened recently.

At the start of this season, my average once again failed me. In this instance, I had my best night of bowling ever on Week 1 (a 596 series). By the time our averages were established in Week 4, my USBC average instantly jumped 25 pins to 172! I am not a 172 bowler. I was just really unlucky to have a seriously good day of bowling during the period when averages were being established. This put me at a major disadvantage, and my average has been decreasing ever since. Notice in contrast that the moving average shows smooth and continuous improvement. It is a much better reflection of my skill, and therefore, would be a much better basis for establishing a handicap.

I need to end this post because if it gets any longer, I'll have to bind it and sell it at Amazon. In short, I propose that Rule 118 is changed so that averages are calculated as a 30 game moving average. This would provide fairer competition because:

1) It would eliminate the problem of inaccurate averages at the start of leagues due to small sample sizes (but only for returning bowlers).

2) Once 30 games have been bowled, it is insensitive to the number of games bowled. It is therefore a better reflection of current bowling skill and provides a sounder basis for handicapping.

3) It would decrease the effectiveness of sandbagging, and reduce the incentive to do so.

Regarding the last point, I will explain in comments if anybody is interested. I figured you probably did not want to read any more because I hadn't made any jokes in a while.

14 comments:

  1. Great suggestion although it will confuse the mathematically challenged.

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  2. Thanks Doc. As for the mathematically challenged, in theory, they shouldn't have to deal with it. Our averages are magically calculated by someone or some computer already. I assume this is standard practice

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  3. Which is me (the mathematically challenged; my people have begotten themselves exponentially, but don't know how)...

    Hmmm. When it comes to bowling lab rats, no one squeaks louder than me (I, actually, but this a bowling blog, for Pete's sake). This travelling (floating, whatever) average intrigues me, but I'm not sure it would help me. My 175 avg. over the last 30 days, say, might make me seem a bit more competitive (actually, a 175 over 30 days would be kind of nice now that I think about it)...

    ...inasmuch as a 175 average over 30 days would make a true 190 bowler (you know, the guy who can make spares) seem less competitive and, accordingly, jack up his handicap. Assuming our luck holds out on day 31, I have more than a fighting chance; honestly, though-- I would probably get killed by scratch count THEN by handicap.

    I like to delude myself into thinking that I have a least a modicum of ability on the lanes, that for now, my paltry average may be camouflaging a better skill level...the consolation prize (and this is a sad statement) is my high handicap.

    To better put this into perspective (for me, the mathematically challenged), I have tried to think of what would happen in baseball if this same system were applied to batting averages... Maybe you're still only as good as you are at the moment you're at bat?

    It IS an interesting concept, Todd. Tell you what. If you want, I'll cross the moat, scale the wall, and hand deliver your suggestion to the king... (I amy as well visit the gift shop while I'm there...think they sell WRW Jr. bobbleheads?)

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  4. EB, I would be shocked if you have a 175 running average. I would guess you would likely come in somewhere in the 150's.

    Think of the batting average this way. What if career batting average was calculated for every at bat starting in tee ball, through little league, high school, and college. Depending on the player, the batting average would likely be highly skewed upward or downward. The running average is sort of like using a monthly batting average rather than an annual or career average. It tracks what's going on lately (over the last two months for our league). It's sort of like when a guy is on a hot streak, and they report that he is batting .450 over his last 50 at bats.

    As for delivering the message to the king, please no. You do not need to be associated with me any more than you already are. Plus, I consider the message already delivered.

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  5. I'm telling you, brother, I defy statistical curves... Here's a sample of my past you could lay over your theory to see what shakes out. These are actual scores of mine over a month (June 16 - July 7): 171 (series avg.), 165 (cumulative avg.); 187, 169; 172, 170; 185, 172. (And these are 4-game series averages.)

    [I'd like to propose to you that THAT guy is lurking somewhere in my current 151 average.]

    If I had a 30-day average based on that particular period, I would have been killed if I didn't keep it up (because of handicap, mostly)... So, again, this floating average is a curiosity, but I doubt it would work in my favor.

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  6. Well, at the start of the league this season, did your 165 average carry over? If so, you were at a disadvantage anyway. If not, it should have, and the moving average would quickly correct, after a few weeks of bowling, to reflect you new found suckitude. I hate to bring it up because serious bowling types do not like the golf comparison, but the USGA figured out this problem long ago. In golf, handicaps are based on recent rounds only because they are most relevant to establishing comparative abilities.

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  7. OK, this is bordering on a sickness, but using ancient calculation methods (not a computer, not a calculator, but a no. 2 lead pencil and a post it note), E.B.'s 30 game moving average is 151 (vs. 152 season to date average) and my 30 game moving average is 146 (vs. 146 YTD average). I think that may mean that we have plateau-ed. Not very comforting.

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  8. Doc- you got it. It means you guys are going nowhere fast I will note that toward the end of the season is where you will see the greatest divergence. -Todd

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  9. Mick, what have we told you about using the p-word in front of other bowlers... Where's your bedside manner?

    And, speaking of moving averages, mine is like a thief in the night.

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  10. I hope it's harmonic... the divergence, I mean... Get it? Harmonic divergence?

    Nevermind.

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  11. And to address a comment about three back, even if I had a rolling 175 avg., I'd STILL be at a disatvantage. My middle name is Handicap(ped).

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  12. You guys must be bored today. Oh the holidays... I'm off to the Lanes of Laramie!

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  13. Man, my tee-ball batting average was the only thing keeping me above .250 through high-school.

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  14. I agree. I started the league for the first time this year. I set my average at around 122. and after a few weeks I was bowling in the 130 range. I then purchase a new finger tip ball and learn how to hook about a 6 weeks ago. I'm bowling up in the 150-200 range now i don't think i have bowled anything less than a 150 in the pass month. Well my average is only 129 now and every week when i bowl in the 150-200 range it might go up a pin.. So yeah this is going to help my team out esp if i can bowl in the 190 with a 65 handicap.

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