A thought experiment (1 Viewer)

[Chase Stuart]

You are told that someone pulled a sample of 25 games where the quarterback threw exactly X number of pass attempts. You are told that you will win $500 if you can correctly guess the number of interceptions, on average, the group of quarterbacks threw to within one-tenth of an interception.

Now, you are also informed that each quarterback in the sample of games threw exactly 12 incomplete passes.

You are told that, if you so desire, you can be informed of what X is above -- that is, how many pass attempts each quarterback had.

Do you want to know this last bit of information before you guess? Why or why not?

 

[Ignoratio Elenchi]

My initial reaction is that of course I'd want as much info as possible before guessing The worst I can imagine is that X is irrelevant, in which case there's still no real harm in knowing it. Due to the nature of the question, though, I assume you have some reason why I shouldn't want to know X? I'll have to give it some thought.

 

[Chase Stuart]

My initial reaction is that of course I'd want as much info as possible before guessing The worst I can imagine is that X is irrelevant, in which case there's still no real harm in knowing it. Due to the nature of the question, though, I assume you have some reason why I shouldn't want to know X? I'll have to give it some thought.

If you wanted that information, how would you use it?

 

[Kool-Aid Larry]

why wouldn't you ask for it?

you should change the problem so that if you ask for that info you only stand to win 250 bux instead of 500, so there's some incentive passing on it.

 

[GregR]

Intuition would expect a correlation between completion rate and int rate. Which if true would make it useful to know # of attempts.

I don't know if intuition is correct here though.

 

[Ignoratio Elenchi]

My initial reaction is that of course I'd want as much info as possible before guessing The worst I can imagine is that X is irrelevant, in which case there's still no real harm in knowing it. Due to the nature of the question, though, I assume you have some reason why I shouldn't want to know X? I'll have to give it some thought.

If you wanted that information, how would you use it?

Honestly, off the top of my head I'd have no idea what a reasonable guess would be for the average number of interceptions, whether I know X or not. I'm not saying I would definitely benefit from knowing X, but I still don't see how it would be bad to have that information. So I'd take it, and then decide if it was useful.

I think maybe you're trying to make the point that X is irrelevant, which is interesting if true, but that wouldn't mean I still wouldn't ask for the information anyway. The only reason I'd prefer not to have the information is if it's somehow "bad" to know it (i.e. knowing it would reduce my chances of guessing correctly) but I'm having a hard time imagining how that would be the case.

Probably a stupid clarification, but is an interception counted as an incomplete pass? In other words, is their total number of interceptions included in the 12 incompletions they threw, or is it really 12 + interceptions + completions = X?

 

[Chase Stuart]

why wouldn't you ask for it?you should change the problem so that if you ask for that info you only stand to win 250 bux instead of 500, so there's some incentive passing on it.

Because you don't like wasting your breath with irrelevant questions. If your answer to the question "how many interceptions did he throw" is the same if he threw 20, 30, 40, or 50 interceptions, then you have no need to ask the question.

 

[Chase Stuart]

My initial reaction is that of course I'd want as much info as possible before guessing The worst I can imagine is that X is irrelevant, in which case there's still no real harm in knowing it. Due to the nature of the question, though, I assume you have some reason why I shouldn't want to know X? I'll have to give it some thought.

If you wanted that information, how would you use it?

Honestly, off the top of my head I'd have no idea what a reasonable guess would be for the average number of interceptions, whether I know X or not. I'm not saying I would definitely benefit from knowing X, but I still don't see how it would be bad to have that information. So I'd take it, and then decide if it was useful.

I think maybe you're trying to make the point that X is irrelevant, which is interesting if true, but that wouldn't mean I still wouldn't ask for the information anyway. The only reason I'd prefer not to have the information is if it's somehow "bad" to know it (i.e. knowing it would reduce my chances of guessing correctly) but I'm having a hard time imagining how that would be the case.

Probably a stupid clarification, but is an interception counted as an incomplete pass? In other words, is their total number of interceptions included in the 12 incompletions they threw, or is it really 12 + interceptions + completions = X?

An interception is counted as an incompletion, yes.FWIW, the avg QB threw 2.9 interceptions per 100 pass attempts last year.

Think of it this way:

I told you he threw 20 passes. What is your answer?

I told you he threw 50 passes. What is your answer?

 

[Chase Stuart]

Intuition would expect a correlation between completion rate and int rate. Which if true would make it useful to know # of attempts.I don't know if intuition is correct here though.

He was 8/20. How many interceptions do you guess?He was 38/50. How many interceptions do you guess?

 

[Ignoratio Elenchi]

My initial reaction is that of course I'd want as much info as possible before guessing The worst I can imagine is that X is irrelevant, in which case there's still no real harm in knowing it. Due to the nature of the question, though, I assume you have some reason why I shouldn't want to know X? I'll have to give it some thought.

If you wanted that information, how would you use it?

Honestly, off the top of my head I'd have no idea what a reasonable guess would be for the average number of interceptions, whether I know X or not. I'm not saying I would definitely benefit from knowing X, but I still don't see how it would be bad to have that information. So I'd take it, and then decide if it was useful.

I think maybe you're trying to make the point that X is irrelevant, which is interesting if true, but that wouldn't mean I still wouldn't ask for the information anyway. The only reason I'd prefer not to have the information is if it's somehow "bad" to know it (i.e. knowing it would reduce my chances of guessing correctly) but I'm having a hard time imagining how that would be the case.

Probably a stupid clarification, but is an interception counted as an incomplete pass? In other words, is their total number of interceptions included in the 12 incompletions they threw, or is it really 12 + interceptions + completions = X?

An interception is counted as an incompletion, yes.FWIW, the avg QB threw 2.9 interceptions per 100 pass attempts last year.

Think of it this way:

I told you he threw 20 passes. What is your answer?

I told you he threw 50 passes. What is your answer?

Let's say my answer is 1.2 either way. Average QB threw 3 per 100 attempts last year. So this would lead you to believe that on 20 attempts, the average QB would throw 0.6 INTs. But a guy who goes 8/20 has presumably done worse than average, so you'd adjust that estimate upwards.

You'd also believe that on 50 attempts, the average QB would throw about 1.5 INTs. But a guy who goes 38/50 has presumably done better than average, so you'd adjust that estimate downwards.

The problem is, if you hadn't told me, I'd have no idea how many INTs/attempt the average QB threw last year. That's really the useful information that would help me make a better guess, not X.

So I'm still reaching the conclusion that your point is that the number of attempts is irrelevant, all you need to know is the number of incompletions. If true, I think that's a fun and interesting point. I just think you posed it in a weird way in the OP. Unless I seriously cared about "not wasting my breath with irrelevant questions" I still think it does no harm to ask for X. I might ultimately determine that X is irrelevant, but I'd still want to know it before guessing.

 

[Chase Stuart]

My initial reaction is that of course I'd want as much info as possible before guessing The worst I can imagine is that X is irrelevant, in which case there's still no real harm in knowing it. Due to the nature of the question, though, I assume you have some reason why I shouldn't want to know X? I'll have to give it some thought.

If you wanted that information, how would you use it?

Honestly, off the top of my head I'd have no idea what a reasonable guess would be for the average number of interceptions, whether I know X or not. I'm not saying I would definitely benefit from knowing X, but I still don't see how it would be bad to have that information. So I'd take it, and then decide if it was useful.

I think maybe you're trying to make the point that X is irrelevant, which is interesting if true, but that wouldn't mean I still wouldn't ask for the information anyway. The only reason I'd prefer not to have the information is if it's somehow "bad" to know it (i.e. knowing it would reduce my chances of guessing correctly) but I'm having a hard time imagining how that would be the case.

Probably a stupid clarification, but is an interception counted as an incomplete pass? In other words, is their total number of interceptions included in the 12 incompletions they threw, or is it really 12 + interceptions + completions = X?

An interception is counted as an incompletion, yes.FWIW, the avg QB threw 2.9 interceptions per 100 pass attempts last year.

Think of it this way:

I told you he threw 20 passes. What is your answer?

I told you he threw 50 passes. What is your answer?

Let's say my answer is 1.2 either way. Average QB threw 3 per 100 attempts last year. So this would lead you to believe that on 20 attempts, the average QB would throw 0.6 INTs. But a guy who goes 8/20 has presumably done worse than average, so you'd adjust that estimate upwards.

You'd also believe that on 50 attempts, the average QB would throw about 1.5 INTs. But a guy who goes 38/50 has presumably done better than average, so you'd adjust that estimate downwards.

The problem is, if you hadn't told me, I'd have no idea how many INTs/attempt the average QB threw last year. That's really the useful information that would help me make a better guess, not X.

So I'm still reaching the conclusion that your point is that the number of attempts is irrelevant, all you need to know is the number of incompletions. If true, I think that's a fun and interesting point. I just think you posed it in a weird way in the OP. Unless I seriously cared about "not wasting my breath with irrelevant questions" I still think it does no harm to ask for X. I might ultimately determine that X is irrelevant, but I'd still want to know it before guessing.

Fair enough. I'll also let you know that for these purposes, knowing the league average rate is also irrelevant. I have no point. I don't know the answer. It's a poll to see what people think.

 

[Kool-Aid Larry]

Intuition would expect a correlation between completion rate and int rate. Which if true would make it useful to know # of attempts.I don't know if intuition is correct here though.

He was 8/20. How many interceptions do you guess?He was 38/50. How many interceptions do you guess?

if I knew he was 8/20 I'd just look up blaine gabbert's int %

 

[LittlePhatty]

I'll also let you know that for these purposes, knowing the league average rate is also irrelevant.

Maybe knowing the exact number is irrelevant, but having a rough idea is necessary to make an educated guess.That is, IF these games were chosen at random.

And you didn't say "random" in your original question - you just said "someone pulled a sample of 25 games".

And since money is involved, I could assume that this person found every game with X number of passes and then specifically picked the 25 with the lowest or highest INTs in order to skew the results and make it harder to guess.

 

[Time Kibitzer]

Intuition would expect a correlation between completion rate and int rate. Which if true would make it useful to know # of attempts.I don't know if intuition is correct here though.

He was 8/20. How many interceptions do you guess?He was 38/50. How many interceptions do you guess?

Well given that going 38/50 suggests the QB was pretty accurate in hitting his targets, and the QB who went 8/20 wasn't, I think it's more than fair to assume the guy who went 8/20 is more likely to have thrown more INTs.

 

[flc735]

He was 8/20. How many interceptions do you guess?He was 38/50. How many interceptions do you guess?

He was 8/20. How many interceptions do you guess?

looks like a run dominated game. looks like a tim tebow game actually i'd guess .8 int's per game

He was 38/50. How many interceptions do you guess?

looks like a catch up game. more forced passes in desperation means more int's. even more forced passes to make up for the previous int's. i'd guess 1.3 int'sso ya, more information is better as long as you can apply it correctly. in this case, i think i can do that.

 

[FreeBaGeL]

Intuition would expect a correlation between completion rate and int rate. Which if true would make it useful to know # of attempts.I don't know if intuition is correct here though.

He was 8/20. How many interceptions do you guess?He was 38/50. How many interceptions do you guess?

if I knew he was 8/20 I'd just look up blaine gabbert's int %

 

[Ignoratio Elenchi]

He was 8/20. How many interceptions do you guess?He was 38/50. How many interceptions do you guess?

He was 8/20. How many interceptions do you guess?

looks like a run dominated game. looks like a tim tebow game actually i'd guess .8 int's per game

He was 38/50. How many interceptions do you guess?

looks like a catch up game. more forced passes in desperation means more int's. even more forced passes to make up for the previous int's. i'd guess 1.3 int'sso ya, more information is better as long as you can apply it correctly. in this case, i think i can do that.

So you think on average, guys who go 38/50 throw more INTs than guys who go 8/20? I have no idea what the right answer is so you could be right, but I'd be surprised.

 

[Hairy Snowman]

Intuition would expect a correlation between completion rate and int rate. Which if true would make it useful to know # of attempts.I don't know if intuition is correct here though.

He was 8/20. How many interceptions do you guess?He was 38/50. How many interceptions do you guess?

What if they only threw 13 passes and 12 were incomplete?Edit to Add: If he was 8/20 or 38/50 then he had ZERO interceptions because all the rest were complete. Just saying.

 

[zandbak]

You don't need X. 25 games is a big enough sample size to deduce the average amount of completions to incompletions (being 12)

Once you know amount of passes (see above) you can make an informed guess as to how many interceptions.

Unless the originator of the question cherry picks 25 games where there was 12 pass attempts (12 incompletions) in which case the thought experiment is pointless. The only method to give an informed answer to the experiment is to assume it was 25 random games of 12 pass attempts and above.

 

[SaintsInDome2006]

You are told that someone pulled a sample of 25 games where the quarterback threw exactly X number of pass attempts. You are told that you will win $500 if you can correctly guess the number of interceptions, on average, the group of quarterbacks threw to within one-tenth of an interception.

Now, you are also informed that each quarterback in the sample of games threw exactly 12 incomplete passes.

You are told that, if you so desire, you can be informed of what X is above -- that is, how many pass attempts each quarterback had.

Do you want to know this last bit of information before you guess? Why or why not?

Is it one QB we're looking at or multiple?How big is the sample?

And is the sample random, or specifically chosen and by whom? Because if the person offering the $500 is the person making the sample then he could skew the sample to throw the offeree off by showing the attempts number. The Tebow example is a good one. Another good example would be Peyton Manning vs the Falcons - who went 24/37 with 1 INT you ask?

If it's not a truly staistically random sample and the person making the offer is choosing the sample, then, no, I don't want the attempts number.

 

[Chase Stuart]

You are told that someone pulled a sample of 25 games where the quarterback threw exactly X number of pass attempts. You are told that you will win $500 if you can correctly guess the number of interceptions, on average, the group of quarterbacks threw to within one-tenth of an interception.

Now, you are also informed that each quarterback in the sample of games threw exactly 12 incomplete passes.

You are told that, if you so desire, you can be informed of what X is above -- that is, how many pass attempts each quarterback had.

Do you want to know this last bit of information before you guess? Why or why not?

Is it one QB we're looking at or multiple?How big is the sample?

And is the sample random, or specifically chosen and by whom? Because if the person offering the $500 is the person making the sample then he could skew the sample to throw the offeree off by showing the attempts number. The Tebow example is a good one. Another good example would be Peyton Manning vs the Falcons - who went 24/37 with 1 INT you ask?

If it's not a truly staistically random sample and the person making the offer is choosing the sample, then, no, I don't want the attempts number.

While I appreciate all of your questions, I'm still not sure any of them matter. What would your answer be if it's one QB? If it's multiple? If it's a random sample? If it's not a random sample?

 

[flc735]

He was 8/20. How many interceptions do you guess?He was 38/50. How many interceptions do you guess?

He was 8/20. How many interceptions do you guess?

looks like a run dominated game. looks like a tim tebow game actually i'd guess .8 int's per game

He was 38/50. How many interceptions do you guess?

looks like a catch up game. more forced passes in desperation means more int's. even more forced passes to make up for the previous int's. i'd guess 1.3 int'sso ya, more information is better as long as you can apply it correctly. in this case, i think i can do that.

So you think on average, guys who go 38/50 throw more INTs than guys who go 8/20? I have no idea what the right answer is so you could be right, but I'd be surprised.

yes, thats just my guess. if you look at all those 50 att games, im sure you will find a few tom brady games with 0 int's (thats why i said 1.3 instead of 1.8), but assuming these games were selected blindly based only on the stats, i would expect the majority of these games to be 'matt cassel trying to come back down 21 point' games

 

[SSOG]

My initial reaction is that of course I'd want as much info as possible before guessing The worst I can imagine is that X is irrelevant, in which case there's still no real harm in knowing it. Due to the nature of the question, though, I assume you have some reason why I shouldn't want to know X? I'll have to give it some thought.

If you wanted that information, how would you use it?

Disclaimer: I have not read past this post yet, because I don't want to bias my thought process by exposing it to outside viewpoints yet. I apologize in advance if my points have already been discussed. Anyway, my initial reaction is that a QB that threw 12 incompletions on 36 attempts is substantially more accurate and makes better decisions than a QB that threw 12 incompletions on 20 attempts. I would expect more INTs from the latter group, since it seems that they're playing over their head. The former group, on the other hand, seems like it knew what it was doing, so I'd expect a higher percentage of the incompletions to be throwaways or spikes (and, therefore, not INTs). My second reaction is that the number of attempts can give you important information about game situations, too. If the sample all threw 60 passes, I would from this infer that most of those QBs were down by a ton, and throwing to catch up. Combine that information with my knowledge that trailing QBs throw more INTs, and I would reason that the sample probably contained more INTs than average. Of course, this would only apply if X was sufficiently large- I think a standard 20-40 attempt game does not give us enough meaningful information about game situation to be taken into consideration.

 

[Das Boot]

I think you would absolutely want to know X, the number of attempts.

First, it gives you a clue to the game situation.

2nd, it could logically include/exclude some QBs/teams from consideration - i.e. teams that average a lot of passes or fewer could be excluded/included depending on X.

3rd, just on a logical basis, a really low or a really high X value would almost necessarily imply a lower or higher guess at avg. number of interceptions.

It wasn't stated, but knowing the year the stats were from would also be key info.

 

[ScottNorwood]

This is dumb

 

[ZWK]

My first thought is that it's relatively unimportant information - if you look at interceptions divided by incompletions my guess is that number is only weakly related to number of attempts (since there are factors going in both directions). So, as a first approximation, I'd just look up that number (INT/INC) and multiply it by 12.

But the information is probably not completely irrelevant. My guess is that QBs with more attempts tend to have slightly more interceptions, holding number of incompletions constant. Rare plays with an unusually high chance of an interception - tipped balls, hit-while-throwing, etc. - have more opportunities to happen when there are more pass attempts, and they impact a QB's interception rate a lot more than his completion percentage. And playing from behind tends to lead to more interceptions and more attempts, and can involve a high completion rate. There are also some styles of play (like the Niners') which involve a low number of attempts and few interceptions, and not a great completion percentage.

If you let me spend a half hour using the PFR database before I gave my guess, I could look up some numbers to try to make a quantitative estimate. But with only a hypothetical $500 at stake I think I'll pass.

 

[Stompin' Tom Connors]

My initial reaction is that of course I'd want as much info as possible before guessing The worst I can imagine is that X is irrelevant, in which case there's still no real harm in knowing it. Due to the nature of the question, though, I assume you have some reason why I shouldn't want to know X? I'll have to give it some thought.

If you wanted that information, how would you use it?

Honestly, off the top of my head I'd have no idea what a reasonable guess would be for the average number of interceptions, whether I know X or not. I'm not saying I would definitely benefit from knowing X, but I still don't see how it would be bad to have that information. So I'd take it, and then decide if it was useful.

I think maybe you're trying to make the point that X is irrelevant, which is interesting if true, but that wouldn't mean I still wouldn't ask for the information anyway. The only reason I'd prefer not to have the information is if it's somehow "bad" to know it (i.e. knowing it would reduce my chances of guessing correctly) but I'm having a hard time imagining how that would be the case.

Probably a stupid clarification, but is an interception counted as an incomplete pass? In other words, is their total number of interceptions included in the 12 incompletions they threw, or is it really 12 + interceptions + completions = X?

An interception is counted as an incompletion, yes.FWIW, the avg QB threw 2.9 interceptions per 100 pass attempts last year.

Think of it this way:

I told you he threw 20 passes. What is your answer?

I told you he threw 50 passes. What is your answer?

Let's say my answer is 1.2 either way. Average QB threw 3 per 100 attempts last year. So this would lead you to believe that on 20 attempts, the average QB would throw 0.6 INTs. But a guy who goes 8/20 has presumably done worse than average, so you'd adjust that estimate upwards.

You'd also believe that on 50 attempts, the average QB would throw about 1.5 INTs. But a guy who goes 38/50 has presumably done better than average, so you'd adjust that estimate downwards.

The problem is, if you hadn't told me, I'd have no idea how many INTs/attempt the average QB threw last year. That's really the useful information that would help me make a better guess, not X.

So I'm still reaching the conclusion that your point is that the number of attempts is irrelevant, all you need to know is the number of incompletions. If true, I think that's a fun and interesting point. I just think you posed it in a weird way in the OP. Unless I seriously cared about "not wasting my breath with irrelevant questions" I still think it does no harm to ask for X. I might ultimately determine that X is irrelevant, but I'd still want to know it before guessing.

Fair enough. I'll also let you know that for these purposes, knowing the league average rate is also irrelevant. I have no point. I don't know the answer. It's a poll to see what people think.

Interesting discussion, and I apologize in advance as know I am going to come off sounding like a real ### here, but reading through this up to this point made me think: 1) It's not a poll

2) It's seems not to be about seeing what people think but how they think -- i.e. who is able to point out relevant vs irrelevant statistical information in order to optimize probability of winning a theoretical bet.

3) Given 2, there does seem to be a point of this, but to what end? This doesn't impart any greater insight or knowledge football, fantasy or otherwise, or even engender discussion about theoretical aspects of the game, so as far as thought experiments, the Ship of Theseus or Schrodinger's Cat it's not.

Sorry, ignore my peeing in this part of the pool, carry on.

 

[DropKick]

You are told that someone pulled a sample of 25 games where the quarterback threw exactly X number of pass attempts. You are told that you will win $500 if you can correctly guess the number of interceptions, on average, the group of quarterbacks threw to within one-tenth of an interception.Now, you are also informed that each quarterback in the sample of games threw exactly 12 incomplete passes. You are told that, if you so desire, you can be informed of what X is above -- that is, how many pass attempts each quarterback had.Do you want to know this last bit of information before you guess? Why or why not?

In all the games, exactly "X" passes were attempted and exactly 12 were incomplete. This lack of any deviation (variance) can lead to the conclusion this group is statistically "average" (in line with NFL averages - the population they were chosen from)... So, in theory, you could compute the number of picks based on a ratio of INTs per incompletion. (Or, indirectly, attempts based on average completion percentage and then INTS per attempt)

 

[GregR]

Intuition would expect a correlation between completion rate and int rate. Which if true would make it useful to know # of attempts.I don't know if intuition is correct here though.

He was 8/20. How many interceptions do you guess?He was 38/50. How many interceptions do you guess?

There's a lot of factors that can have an affect. And I suspect some may be reflected in some ranges of attempts more strongly than at other places.I don't think I could say there would be a single dominant factor with a lot of passes, like that the team is trying to come from behind so is passing into coverages meant to stop the pass and so more INTs. That's one possibility, but I can think of other reasons teams would throw that much where INTs wouldn't be high necessarily.But I do think there might be a dominant factor at the bottom end. Teams will normally fall behind if they are completing 40% of their passes (8 of 20). And teams who are trailing would probably tend to pass more than 20 times in most situations. So I would think if a given team is passing that poorly and only attempting 20 passes, there is probably a good chance they are turning the ball over and prematurely ending their offensive series faster than a simple incompletion would have. But I'm not sure.So just to get a glimpse I did some quick queries on PFR. Query tools aren't enough to get a result for each number of attempts without checking it and dumping it into Excel one by one, so I only did a few to get a glimpse. I started going by 5s, and then did a few one by one to see how it varied around a given data point. I limited it back to 1960 to present to try to keep it down to 100 games or less for each category:

Att     Ints/game18	1.1019	1.3720	1.2921	1.2725	1.0929	1.0430	0.8831	1.1232	1.0035	1.1240	0.9445      (only 2 games qualified)

It does look like there might be a higher INT rate in lower attempt games like I was hypothesizing. Really though would need to do all numbers of attempts to get a better idea.

 

[ZWK]

Intuition would expect a correlation between completion rate and int rate. Which if true would make it useful to know # of attempts.I don't know if intuition is correct here though.

He was 8/20. How many interceptions do you guess?He was 38/50. How many interceptions do you guess?

There's a lot of factors that can have an affect. And I suspect some may be reflected in some ranges of attempts more strongly than at other places.I don't think I could say there would be a single dominant factor with a lot of passes, like that the team is trying to come from behind so is passing into coverages meant to stop the pass and so more INTs. That's one possibility, but I can think of other reasons teams would throw that much where INTs wouldn't be high necessarily.But I do think there might be a dominant factor at the bottom end. Teams will normally fall behind if they are completing 40% of their passes (8 of 20). And teams who are trailing would probably tend to pass more than 20 times in most situations. So I would think if a given team is passing that poorly and only attempting 20 passes, there is probably a good chance they are turning the ball over and prematurely ending their offensive series faster than a simple incompletion would have. But I'm not sure.So just to get a glimpse I did some quick queries on PFR. Query tools aren't enough to get a result for each number of attempts without checking it and dumping it into Excel one by one, so I only did a few to get a glimpse. I started going by 5s, and then did a few one by one to see how it varied around a given data point. I limited it back to 1960 to present to try to keep it down to 100 games or less for each category:

Att     Ints/game18	1.1019	1.3720	1.2921	1.2725	1.0929	1.0430	0.8831	1.1232	1.0035	1.1240	0.9445      (only 2 games qualified)

It does look like there might be a higher INT rate in lower attempt games like I was hypothesizing. Really though would need to do all numbers of attempts to get a better idea.

Year is a confound here - in the 1960s, they had fewer pass attempts and a higher INT rate (compared to the 2000s).

 

[GregR]

I just checked my "teams who are losing are likely to pass more than 20 times" thought.

Since 2000, teams had 20 attempts or less 340 times (with no restriction on completions/incompletions). However, 81% of those teams won their games. Only 64 teams out of the 340 lost and attempted 20 passes or fewer. Of those 64... 26 were leading or tied at the half, and 18 were leading or tied at the end of the 3rd quarter.

And for the QBs with exactly 20 attempts and only 8 completions since 1960... 47 losses and 20 wins for their NFL teams.

 

[CalBear]

So, I pulled some data: Games where a team completed less than 10 out of more than 20 passes, and games where a team completed at least 35 out of at least 45 passes. Some of the results are obvious; the first group had a 39.7% completion rate, the second a 69.4% completion rate. The first group had a 2.52% TD rate per attempt and a 4.98% INT rate per attempt, and the second group had a 4.17% TD rate and a 2.60% INT rate.

But the somewhat unexpected result is that the interception rate per incompletion was about the same: 8.27% in the first group, 8.48% in the second group. Similarly, the TD rate per completion was about the same; 6.33% in the first group, 6.02% in the second group. I didn't do the calculation but I expect those numbers are within the margin of error for the sample (n=162, n=61). So a completion is about as likely to be a TD, and an incompletion is about as likely to be an INT, whether you're throwing 8 for 20 or 38 for 50.

So while I'd want to know the number of attempts because more information can't be bad, it looks like the number of incompletions is a lot more predictive than the number of attempts.

 

[DropKick]

You are told that someone pulled a sample of 25 games where the quarterback threw exactly X number of pass attempts. You are told that you will win $500 if you can correctly guess the number of interceptions, on average, the group of quarterbacks threw to within one-tenth of an interception.Now, you are also informed that each quarterback in the sample of games threw exactly 12 incomplete passes. You are told that, if you so desire, you can be informed of what X is above -- that is, how many pass attempts each quarterback had.Do you want to know this last bit of information before you guess? Why or why not?

In all the games, exactly "X" passes were attempted and exactly 12 were incomplete. This lack of any deviation (variance) can lead to the conclusion this group is statistically "average" (in line with NFL averages - the population they were chosen from)... So, in theory, you could compute the number of picks based on a ratio of INTs per incompletion. (Or, indirectly, attempts based on average completion percentage and then INTS per attempt)

I'll expand on this. The average NFL INTs per game is .83. You could simply guess .83 per game. What other data is meaningful? No variance (as mentioned above) and the number of incompletions. The latter is important because you could intuit that there is a relationship between passes thrown and interceptions. More passing = more INTs. If you assume an average completion rate of 59.36%, we have roughly 29.5 passes per game. With a 2.9% interception rate, I would estimate 21.4 total interceptions or .856 per game (slightly above the NFL average).Of course, we don't have fractional interceptions, so put me down for 21 total interceptions or 0.84 per game.I get the $500 if the total number of INTs is between 19 (0.76) and 23 (0.92). I like my chances.ETA: If I actually guessed a little higher (.86), I would win the money if 24 picks were thrown too (.76 (19 INTs) <-> .86 <=> .96 (24 INTs))

 

[Ignoratio Elenchi]

You are told that someone pulled a sample of 25 games where the quarterback threw exactly X number of pass attempts. You are told that you will win $500 if you can correctly guess the number of interceptions, on average, the group of quarterbacks threw to within one-tenth of an interception.

Now, you are also informed that each quarterback in the sample of games threw exactly 12 incomplete passes.

You are told that, if you so desire, you can be informed of what X is above -- that is, how many pass attempts each quarterback had.

Do you want to know this last bit of information before you guess? Why or why not?

In all the games, exactly "X" passes were attempted and exactly 12 were incomplete. This lack of any deviation (variance) can lead to the conclusion this group is statistically "average" (in line with NFL averages - the population they were chosen from)... So, in theory, you could compute the number of picks based on a ratio of INTs per incompletion. (Or, indirectly, attempts based on average completion percentage and then INTS per attempt)

I'll expand on this. The average NFL INTs per game is .83. You could simply guess .83 per game. What other data is meaningful? No variance (as mentioned above) and the number of incompletions. The latter is important because you could intuit that there is a relationship between passes thrown and interceptions. More passing = more INTs. If you assume an average completion rate of 59.36%, we have roughly 29.5 passes per game. With a 2.9% interception rate, I would estimate 21.4 total interceptions or .856 per game (slightly above the NFL average).Of course, we don't have fractional interceptions, so put me down for 21 total interceptions or 0.84 per game.

I get the $500 if the total number of INTs is between 19 (0.76) and 23 (0.92). I like my chances.

I think the bolded is a faulty premise.

 

[DropKick]

You are told that someone pulled a sample of 25 games where the quarterback threw exactly X number of pass attempts. You are told that you will win $500 if you can correctly guess the number of interceptions, on average, the group of quarterbacks threw to within one-tenth of an interception.

Now, you are also informed that each quarterback in the sample of games threw exactly 12 incomplete passes.

You are told that, if you so desire, you can be informed of what X is above -- that is, how many pass attempts each quarterback had.

Do you want to know this last bit of information before you guess? Why or why not?

In all the games, exactly "X" passes were attempted and exactly 12 were incomplete. This lack of any deviation (variance) can lead to the conclusion this group is statistically "average" (in line with NFL averages - the population they were chosen from)... So, in theory, you could compute the number of picks based on a ratio of INTs per incompletion. (Or, indirectly, attempts based on average completion percentage and then INTS per attempt)

I'll expand on this. The average NFL INTs per game is .83. You could simply guess .83 per game. What other data is meaningful? No variance (as mentioned above) and the number of incompletions. The latter is important because you could intuit that there is a relationship between passes thrown and interceptions. More passing = more INTs. If you assume an average completion rate of 59.36%, we have roughly 29.5 passes per game. With a 2.9% interception rate, I would estimate 21.4 total interceptions or .856 per game (slightly above the NFL average).Of course, we don't have fractional interceptions, so put me down for 21 total interceptions or 0.84 per game.

I get the $500 if the total number of INTs is between 19 (0.76) and 23 (0.92). I like my chances.

I think the bolded is a faulty premise.

Why? In reality it is very unlikely to draw a random sample of 25 games with "X" passes thrown and to find that each and every game had exactly 12 incompletions. Obviously, we're dealing with a contrived problem. But, given this uniformity, I've concluded this group of QBs is consistent with the NFL average (not well above or below).

 

[Ignoratio Elenchi]

You are told that someone pulled a sample of 25 games where the quarterback threw exactly X number of pass attempts. You are told that you will win $500 if you can correctly guess the number of interceptions, on average, the group of quarterbacks threw to within one-tenth of an interception.

Now, you are also informed that each quarterback in the sample of games threw exactly 12 incomplete passes.

You are told that, if you so desire, you can be informed of what X is above -- that is, how many pass attempts each quarterback had.

Do you want to know this last bit of information before you guess? Why or why not?

In all the games, exactly "X" passes were attempted and exactly 12 were incomplete. This lack of any deviation (variance) can lead to the conclusion this group is statistically "average" (in line with NFL averages - the population they were chosen from)... So, in theory, you could compute the number of picks based on a ratio of INTs per incompletion. (Or, indirectly, attempts based on average completion percentage and then INTS per attempt)

I'll expand on this. The average NFL INTs per game is .83. You could simply guess .83 per game. What other data is meaningful? No variance (as mentioned above) and the number of incompletions. The latter is important because you could intuit that there is a relationship between passes thrown and interceptions. More passing = more INTs. If you assume an average completion rate of 59.36%, we have roughly 29.5 passes per game. With a 2.9% interception rate, I would estimate 21.4 total interceptions or .856 per game (slightly above the NFL average).Of course, we don't have fractional interceptions, so put me down for 21 total interceptions or 0.84 per game.

I get the $500 if the total number of INTs is between 19 (0.76) and 23 (0.92). I like my chances.

I think the bolded is a faulty premise.

Why? In reality it is very unlikely to draw a random sample of 25 games with "X" passes thrown and to find that each and every game had exactly 12 incompletions. Obviously, we're dealing with a contrived problem. But, given this uniformity, I've concluded this group of QBs is consistent with the NFL average (not well above or below).

OP didn't say it was a random sample.

 

[DropKick]

You are told that someone pulled a sample of 25 games where the quarterback threw exactly X number of pass attempts. You are told that you will win $500 if you can correctly guess the number of interceptions, on average, the group of quarterbacks threw to within one-tenth of an interception.

Now, you are also informed that each quarterback in the sample of games threw exactly 12 incomplete passes.

You are told that, if you so desire, you can be informed of what X is above -- that is, how many pass attempts each quarterback had.

Do you want to know this last bit of information before you guess? Why or why not?

In all the games, exactly "X" passes were attempted and exactly 12 were incomplete. This lack of any deviation (variance) can lead to the conclusion this group is statistically "average" (in line with NFL averages - the population they were chosen from)... So, in theory, you could compute the number of picks based on a ratio of INTs per incompletion. (Or, indirectly, attempts based on average completion percentage and then INTS per attempt)

I'll expand on this. The average NFL INTs per game is .83. You could simply guess .83 per game. What other data is meaningful? No variance (as mentioned above) and the number of incompletions. The latter is important because you could intuit that there is a relationship between passes thrown and interceptions. More passing = more INTs. If you assume an average completion rate of 59.36%, we have roughly 29.5 passes per game. With a 2.9% interception rate, I would estimate 21.4 total interceptions or .856 per game (slightly above the NFL average).Of course, we don't have fractional interceptions, so put me down for 21 total interceptions or 0.84 per game.

I get the $500 if the total number of INTs is between 19 (0.76) and 23 (0.92). I like my chances.

I think the bolded is a faulty premise.

Why? In reality it is very unlikely to draw a random sample of 25 games with "X" passes thrown and to find that each and every game had exactly 12 incompletions. Obviously, we're dealing with a contrived problem. But, given this uniformity, I've concluded this group of QBs is consistent with the NFL average (not well above or below).

OP didn't say it was a random sample.

OP didn't say a lot of things. That is why it is a thought experiment.OP could have said "a sample of 25 games was drawn with "x" passes thrown and no interceptions" - guess how many interceptions were thrown? Don't you have to make some assumptions?

 

[CalBear]

Following up on this, I found some interesting stats. The data set I was using was 2000-2012, a period in which passing offense rose dramatically, with passing yardage up 14% league-wide and passing TDs up 17% in that time frame. But passing INTs are basically flat. What's happened is that passing attempts are up a bit (6.6%), and passing completions are up a bit (from 58.2% to 60.1%), and so it turns out that the number of incomplete passes has been entirely flat for the past 12 years. And it looks like INTs are more strongly aligned with incompletions than with attempts; the incomplete/INT correlation is about double the strength of the attempt/INT correlation. (This is a small data set for yearly variations but I'd guess the result holds for larger sets as well).

So I'd probably wind up guessing that they threw something like .93 INTs per 12 incompletions (7.74%), which is the league average over the past 12 years, and a little higher than the average for INTs per game (presumably because 12 is higher than the league average for incompletions per game).

 

[DropKick]

Following up on this, I found some interesting stats. The data set I was using was 2000-2012, a period in which passing offense rose dramatically, with passing yardage up 14% league-wide and passing TDs up 17% in that time frame. But passing INTs are basically flat. What's happened is that passing attempts are up a bit (6.6%), and passing completions are up a bit (from 58.2% to 60.1%), and so it turns out that the number of incomplete passes has been entirely flat for the past 12 years. And it looks like INTs are more strongly aligned with incompletions than with attempts; the incomplete/INT correlation is about double the strength of the attempt/INT correlation. (This is a small data set for yearly variations but I'd guess the result holds for larger sets as well).

So I'd probably wind up guessing that they threw something like .93 INTs per 12 incompletions (7.74%), which is the league average over the past 12 years, and a little higher than the average for INTs per game (presumably because 12 is higher than the league average for incompletions per game).

The way the numbers fall, you'd be better off with a guess of .94 since you win with total INTs in the range of 21-26. I'm slightly lower... but we've answered the fundamental question: You don't need to know "X" to make an intelligent guess.

 

[GregR]

You are told that someone pulled a sample of 25 games where the quarterback threw exactly X number of pass attempts. You are told that you will win $500 if you can correctly guess the number of interceptions, on average, the group of quarterbacks threw to within one-tenth of an interception.

Now, you are also informed that each quarterback in the sample of games threw exactly 12 incomplete passes.

You are told that, if you so desire, you can be informed of what X is above -- that is, how many pass attempts each quarterback had.

Do you want to know this last bit of information before you guess? Why or why not?

In all the games, exactly "X" passes were attempted and exactly 12 were incomplete. This lack of any deviation (variance) can lead to the conclusion this group is statistically "average" (in line with NFL averages - the population they were chosen from)... So, in theory, you could compute the number of picks based on a ratio of INTs per incompletion. (Or, indirectly, attempts based on average completion percentage and then INTS per attempt)

I'll expand on this. The average NFL INTs per game is .83. You could simply guess .83 per game. What other data is meaningful? No variance (as mentioned above) and the number of incompletions. The latter is important because you could intuit that there is a relationship between passes thrown and interceptions. More passing = more INTs. If you assume an average completion rate of 59.36%, we have roughly 29.5 passes per game. With a 2.9% interception rate, I would estimate 21.4 total interceptions or .856 per game (slightly above the NFL average).Of course, we don't have fractional interceptions, so put me down for 21 total interceptions or 0.84 per game.

I get the $500 if the total number of INTs is between 19 (0.76) and 23 (0.92). I like my chances.

I think the bolded is a faulty premise.

Why? In reality it is very unlikely to draw a random sample of 25 games with "X" passes thrown and to find that each and every game had exactly 12 incompletions. Obviously, we're dealing with a contrived problem. But, given this uniformity, I've concluded this group of QBs is consistent with the NFL average (not well above or below).

OP didn't say it was a random sample.

OP didn't say a lot of things. That is why it is a thought experiment.OP could have said "a sample of 25 games was drawn with "x" passes thrown and no interceptions" - guess how many interceptions were thrown? Don't you have to make some assumptions?

I've already posted data that shows interceptions per incompletion varies by number of attempts for 12 incompletion games. Which would disprove the assumption.

 

[Ignoratio Elenchi]

OP didn't say it was a random sample.

OP didn't say a lot of things. That is why it is a thought experiment.OP could have said "a sample of 25 games was drawn with "x" passes thrown and no interceptions" - guess how many interceptions were thrown? Don't you have to make some assumptions?

I see no reason to assume the sample mentioned in the OP is random, though. Given the question it just seems like a strange assumption to make. Besides, it seems you only used this assumption to make an estimate for X anyway. OP is offering to tell you X, so there's no need to used a possibly flawed assumption to take a roundabout way to estimating X.

 

[DropKick]

You are told that someone pulled a sample of 25 games where the quarterback threw exactly X number of pass attempts. You are told that you will win $500 if you can correctly guess the number of interceptions, on average, the group of quarterbacks threw to within one-tenth of an interception.

Now, you are also informed that each quarterback in the sample of games threw exactly 12 incomplete passes.

You are told that, if you so desire, you can be informed of what X is above -- that is, how many pass attempts each quarterback had.

Do you want to know this last bit of information before you guess? Why or why not?

In all the games, exactly "X" passes were attempted and exactly 12 were incomplete. This lack of any deviation (variance) can lead to the conclusion this group is statistically "average" (in line with NFL averages - the population they were chosen from)... So, in theory, you could compute the number of picks based on a ratio of INTs per incompletion. (Or, indirectly, attempts based on average completion percentage and then INTS per attempt)

I'll expand on this. The average NFL INTs per game is .83. You could simply guess .83 per game. What other data is meaningful? No variance (as mentioned above) and the number of incompletions. The latter is important because you could intuit that there is a relationship between passes thrown and interceptions. More passing = more INTs. If you assume an average completion rate of 59.36%, we have roughly 29.5 passes per game. With a 2.9% interception rate, I would estimate 21.4 total interceptions or .856 per game (slightly above the NFL average).Of course, we don't have fractional interceptions, so put me down for 21 total interceptions or 0.84 per game.

I get the $500 if the total number of INTs is between 19 (0.76) and 23 (0.92). I like my chances.

I think the bolded is a faulty premise.

Why? In reality it is very unlikely to draw a random sample of 25 games with "X" passes thrown and to find that each and every game had exactly 12 incompletions. Obviously, we're dealing with a contrived problem. But, given this uniformity, I've concluded this group of QBs is consistent with the NFL average (not well above or below).

OP didn't say it was a random sample.

OP didn't say a lot of things. That is why it is a thought experiment.OP could have said "a sample of 25 games was drawn with "x" passes thrown and no interceptions" - guess how many interceptions were thrown? Don't you have to make some assumptions?

I've already posted data that shows interceptions per incompletion varies by number of attempts for 12 incompletion games. Which would disprove the assumption.

My assumption is that the QB performance in the sample is representative of average NFL performance. Your "study" doesn't disprove that. You are dwelling on a minor point. The bottom line is that you can make an educated guess at the interception rate without the additional information of "x".

 

[GregR]

...

My assumption is that the QB performance in the sample is representative of average NFL performance. Your "study" doesn't disprove that. You are dwelling on a minor point. The bottom line is that you can make an educated guess at the interception rate without the additional information of "x".

The numbers I posted showed that some values of X the performance may be representative of average NFL performance. They also showed that for many values of X, they likely are not representative of average performance. If you know that values of X around 20 mean higher INT rates than values of X around 30 or 40, then we definitely would want to know X.

 

[DropKick]

OP didn't say it was a random sample.

OP didn't say a lot of things. That is why it is a thought experiment.OP could have said "a sample of 25 games was drawn with "x" passes thrown and no interceptions" - guess how many interceptions were thrown? Don't you have to make some assumptions?

I see no reason to assume the sample mentioned in the OP is random, though. Given the question it just seems like a strange assumption to make. Besides, it seems you only used this assumption to make an estimate for X anyway. OP is offering to tell you X, so there's no need to used a possibly flawed assumption to take a roundabout way to estimating X.

Seems stranger to assume it isn't random... Stats from 25 games were examined were exactly "x" passes were thrown by hall of fame QBs in their best statistical seasons...If it isn't random, the OP withheld a significant piece of information. I don't see a need for that type of curve ball in this type of exercise. I agree that you could simply ask for "x" since its free and why should we ignore additional information? However, it wouldn't be much of a topic, would it? What if the original question were posed differently and the pay-off were cut to $100 if you wanted "X"?

 

[Ignoratio Elenchi]

Seems stranger to assume it isn't random... Stats from 25 games were examined were exactly "x" passes were thrown by hall of fame QBs in their best statistical seasons...

If it isn't random, the OP withheld a significant piece of information. I don't see a need for that type of curve ball in this type of exercise.

Not at all. Who said anything about "hall of fame QBs in their best statistical seasons?" Do you really think the scenario OP intended to pose is, "I took a random sample of 25 QB performances from NFL history. Incredibly, they all attempted the same exact number of passes (X) and all threw the same exact number of incompletions (12)! How many interceptions do you think they averaged?" That would be absurdly unlikely and is obviously not the point of the OP imo. There's really no reason to interpret the question the way you have.

I agree that you could simply ask for "x" since its free and why should we ignore additional information? However, it wouldn't be much of a topic, would it?

It would be a great topic. I said earlier I think OP framed it in a weird way, but I think the crux of the question is whether interceptions are correlated much more strongly with incompletions than with attempts. This idea that it's a random sample is a total red herring imo. In fact I think OP already insinuated upthread that it doesn't matter whether the sample was arrived at randomly or not. That's not the point of the question. You are simply given a sample of 25 QB performances in which they all threw the same number of attempts and incompletions. It's an interesting question whether you need to know the number of attempts to accurately determine the number of interceptions they threw.

 

[CalBear]

Unfortnately PFR doesn't let you search on incompletions, but here are some stats reverse-engineered from completion percentages:

Since 1995:

9/21: 8 games, 9 INTs (1.12/game)

12/24: 23 games, 11 INTs (.48/game)

15/27: 43 games, 21 INTs (.49/game)

18/30: 60 games, 45 INTs (.75/game)

23/35: 44 games, 51 INTs (1.16/game)

28/40: 12 games, 10 INTs (.83/game)

The results for 12 and 15 completions seem way out of whack, so I looked at some more:

13/25: 31 games, 32 INTs (1.03/game)

14/26: 40 games, 33 INTs (.83/game)

16/28: 53 games, 61 INTs (1.15/game)

We're clearly seeing the effects of small sample sizes here; overall it doesn't look like there's a trend in either direction but the points are pretty widely scattered. The entire sample is 314 games with 283 INTs, .90 per game, which puts the league average for INTs/incompletion within the margin of error.

If you happened to have this data set, knowing the number of attempts could be useful to come to a more accurate guess; for example, if you knew that several of the games included exactly 24 or 27 attempts you could guess a lower number of INTs for the sample. But I don't think you'd want to adjust anything based on the total number of attempts for the sample.

 

[Warrior]

I'd want to know the # of completions. More completions = more attempts. More attempts makes the defense's job of guessing whether or not the next play will be a pass easier. Easier guess = more interceptions. Not only that, but more pass attempts implies that the team is behind. When they are behind, they will force more throws. Again, more attempts (and thus completions) = more interceptions.

Lets take it to the extreme. 12 incompletions for both. I'd guess there are more interceptions for the guy who was 40 for 52 than the guy who was 2 for 14.

Then again, the team who threw more likely had a higher time of posession. The 2 for 14 QB probably didn't get many opportunities to throw the ball because the other team had posession for the vast majority of the game. That's likely because he kept turing the ball over. The 40 for 52 QB probably kept moving the ball down the field with short completions, allowing his team to control the t.o.p. Therefore, more completions = less interceptions.

Conclusion:

 

[dgreen]

OP didn't say it was a random sample.

OP didn't say a lot of things. That is why it is a thought experiment.OP could have said "a sample of 25 games was drawn with "x" passes thrown and no interceptions" - guess how many interceptions were thrown? Don't you have to make some assumptions?

I see no reason to assume the sample mentioned in the OP is random, though. Given the question it just seems like a strange assumption to make.

Yes, it's odd to assume that a random sample of all games resulted in 25 games that all have the same number of attempts and incompletions. However, I think it's ok to assume that it's a random sample of all the games where QBs were 18/30.