Basketball Arithmetic: This Equation Is Always True

[JoePgh]

With only a few days until the new season starts, we seem to be scrounging desperately for thread topics. It has reached the point where we are even talking about umlauts.

The topic of this post falls in the category of "Rather obvious if you think about it", but I have reason to believe that not everybody is aware of it. So, as a public service, and to dissipate the BY boredom before the first game, I offer the following:

Did you know that every single basketball game that has ever been played or will ever be played (under current scoring rules) will satisfy the following identity equation?

Margin of victory = 2 * [Difference in field goals made] +
1 * [Difference in 3-point makes] +
1 * [Difference in free throw makes]

This applies to EVERY game: men or women; pro, college, high school, or midget league; played in Europe, Asia, or America.

To make it concrete with an example, let's use the Arizona game, which (as a poster has helpfully pointed out) occurred seven months ago. In that game, the statistics in the above identity were:

FG's made: UConn 20-of-56, Arizona 20-of-50
3-pointers made: UConn 5-of-12, Arizona 7-of-19
FT's made: UConn 14-of-20, Arizona 22-of-31

So, plugging this data into the above equation, and showing differences in favor of Arizona as positive and differences in favor of UConn as negative, we have:

Margin of victory = 2 * [20 - 20] +
1 * [7 - 5] +
1 * [22 - 14]
= 0 + 2 + 8 = 10

... which, miraculously, is AZ's margin of victory.

In another thread, a poster said that my arithmetic must be erroneous because I only multiplied the difference in 3-point makes by 1 instead of 3. The reason for this is that 3-point makes are included in the total of FG's made, i.e., the 20 field goals for UConn include 5 3-pointers, and the 20 FG's made by Arizona include 7 3-pointers. So 2 of the 3 points scored on a 3-point basket are accounted for in the first term of the equation. Hence only the "extra" third point from a 3-point basket needs to be specifically calculated for 3-point field goals. Thus it is correct to multiply the difference in 3-point makes by 1 rather than 3.

Note that shooting percentage has no bearing on this equation. Misses are disregarded entirely; only made shots count. Of course this is perfectly sensible, since missed shots have zero impact on the final score.

Using this identity helps to identify what aspect of the game most strongly influenced the final score/result. In this case, the difference in free throw opportunities (NOT free throw percentage) accounts for 80% of AZ's margin of victory, which you might not suspect after just watching the game.

As I said, this identity equation is fairly obvious when you think about it, but sometimes we react to a game without thinking about it, and we miss something in that process.

 

[CocoHusky]

 

[Littlemin]

Again proving that statistics mean nothing without context, the difference in free throws attempted was primarily the product of Uconn fouling down the stretch to stop the clock. That also served to add points to McDonalds final stats (although obviously she had to make the pressure shots) to give some the impression she had a dominant game.

 

[MD#12Fan]

Your formula is all about scoring. Therefore your formula is akin to the idea that whoever scores the most points wins. That is truly a revelation! I need some there there. I need to know what happened and why it happened. If you're so wedded to stats, what about turnovers, rebounds, especially offensive rebounds, shooting percentages, and fouls?
So, why did Arizona get more FT's than UConn. UConn historically commits among the least fouls in WCBB.
Of those 7 made 3-pointers, how many were by their AA?
Ugh, I think I've been set-up.

 

[JoePgh]

Your formula is all about scoring. Therefore your formula is akin to the idea that whoever scores the most points wins. That is truly a revelation! I need some there there. I need to know what happened and why it happened. If you're so wedded to stats, what about turnovers, rebounds, especially offensive rebounds, shooting percentages, and fouls?
So, why did Arizona get more FT's than UConn. UConn historically commits among the least fouls in WCBB.
Of those 7 made 3-pointers, how many were by their AA?
Ugh, I think I've been set-up.

I think that if you read my post (as the initial poster) in the "Lessons and Non-Lessons From Arizona Game" thread, you will see that I answered the above questions.

 

[oldude]

My Jr HS algebra teacher would drill us constantly to simplify our equations. Mrs Elliot would point out that you do not require the multiplication sign outside of the parentheses for 2-pt shot differential or the extraneous multipliers for 3 pt & FT shot differential. So your AZ/UConn example would simply be:

2(20-20) + (7-5) + (22-14) = 10

 

[Happydad745]

This sounds like someone is trying to make something that is simple into something that complicated.

 

[eebmg]

Equations are great but I like inequalities.

UConn > (x). x variable.

 

[Bama fan]

With only a few days until the new season starts, we seem to be scrounging desperately for thread topics. It has reached the point where we are even talking about umlauts.

The topic of this post falls in the category of "Rather obvious if you think about it", but I have reason to believe that not everybody is aware of it. So, as a public service, and to dissipate the BY boredom before the first game, I offer the following:

Did you know that every single basketball game that has ever been played or will ever be played (under current scoring rules) will satisfy the following identity equation?

Margin of victory = 2 * [Difference in field goals made] +
1 * [Difference in 3-point makes] +
1 * [Difference in free throw makes]

This applies to EVERY game: men or women; pro, college, high school, or midget league; played in Europe, Asia, or America.

To make it concrete with an example, let's use the Arizona game, which (as a poster has helpfully pointed out) occurred seven months ago. In that game, the statistics in the above identity were:

FG's made: UConn 20-of-56, Arizona 20-of-50
3-pointers made: UConn 5-of-12, Arizona 7-of-19
FT's made: UConn 14-of-20, Arizona 22-of-31

So, plugging this data into the above equation, and showing differences in favor of Arizona as positive and differences in favor of UConn as negative, we have:

Margin of victory = 2 * [20 - 20] +
1 * [7 - 5] +
1 * [22 - 14]
= 0 + 2 + 8 = 10

... which, miraculously, is AZ's margin of victory.

In another thread, a poster said that my arithmetic must be erroneous because I only multiplied the difference in 3-point makes by 1 instead of 3. The reason for this is that 3-point makes are included in the total of FG's made, i.e., the 20 field goals for UConn include 5 3-pointers, and the 20 FG's made by Arizona include 7 3-pointers. So 2 of the 3 points scored on a 3-point basket are accounted for in the first term of the equation. Hence only the "extra" third point from a 3-point basket needs to be specifically calculated for 3-point field goals. Thus it is correct to multiply the difference in 3-point makes by 1 rather than 3.

Note that shooting percentage has no bearing on this equation. Misses are disregarded entirely; only made shots count. Of course this is perfectly sensible, since missed shots have zero impact on the final score.

Using this identity helps to identify what aspect of the game most strongly influenced the final score/result. In this case, the difference in free throw opportunities (NOT free throw percentage) accounts for 80% of AZ's margin of victory, which you might not suspect after just watching the game.

As I said, this identity equation is fairly obvious when you think about it, but sometimes we react to a game without thinking about it, and we miss something in that process.

Nice work and helpful explanations. However, a team has to "make" their free throw opportunities to gain the goody from the opportunities. And in this case , the Wildcats certainly did. Because their percentage was sufficient they gained the advantage of Uconn's desperation.

 

[CL82]

With only a few days until the new season starts, we seem to be scrounging desperately for thread topics. It has reached the point where we are even talking about umlauts.

The topic of this post falls in the category of "Rather obvious if you think about it", but I have reason to believe that not everybody is aware of it. So, as a public service, and to dissipate the BY boredom before the first game, I offer the following:

Did you know that every single basketball game that has ever been played or will ever be played (under current scoring rules) will satisfy the following identity equation?

Margin of victory = 2 * [Difference in field goals made] +
1 * [Difference in 3-point makes] +
1 * [Difference in free throw makes]

This applies to EVERY game: men or women; pro, college, high school, or midget league; played in Europe, Asia, or America.

To make it concrete with an example, let's use the Arizona game, which (as a poster has helpfully pointed out) occurred seven months ago. In that game, the statistics in the above identity were:

FG's made: UConn 20-of-56, Arizona 20-of-50
3-pointers made: UConn 5-of-12, Arizona 7-of-19
FT's made: UConn 14-of-20, Arizona 22-of-31

So, plugging this data into the above equation, and showing differences in favor of Arizona as positive and differences in favor of UConn as negative, we have:

Margin of victory = 2 * [20 - 20] +
1 * [7 - 5] +
1 * [22 - 14]
= 0 + 2 + 8 = 10

... which, miraculously, is AZ's margin of victory.

In another thread, a poster said that my arithmetic must be erroneous because I only multiplied the difference in 3-point makes by 1 instead of 3. The reason for this is that 3-point makes are included in the total of FG's made, i.e., the 20 field goals for UConn include 5 3-pointers, and the 20 FG's made by Arizona include 7 3-pointers. So 2 of the 3 points scored on a 3-point basket are accounted for in the first term of the equation. Hence only the "extra" third point from a 3-point basket needs to be specifically calculated for 3-point field goals. Thus it is correct to multiply the difference in 3-point makes by 1 rather than 3.

Note that shooting percentage has no bearing on this equation. Misses are disregarded entirely; only made shots count. Of course this is perfectly sensible, since missed shots have zero impact on the final score.

Using this identity helps to identify what aspect of the game most strongly influenced the final score/result. In this case, the difference in free throw opportunities (NOT free throw percentage) accounts for 80% of AZ's margin of victory, which you might not suspect after just watching the game.

As I said, this identity equation is fairly obvious when you think about it, but sometimes we react to a game without thinking about it, and we miss something in that process.

“I can see it all now. It’s so beautiful!”

 

[JoePgh]

My Jr HS algebra teacher would drill us constantly to simplify our equations. Mrs Elliot would point out that you do not require the multiplication sign outside of the parentheses for 2-pt shot differential or the extraneous multipliers for 3 pt & FT shot differential. So your AZ/UConn example would simply be:

2(20-20) + (7-5) + (22-14) = 10

My high school math teachers, Brother Henry and Brother Carl, would have agreed with you. I presume they are now resting in peace, despite the mayhem that Brother Henry in particular inflicted on us as high school freshmen.

However, while math is constant, mathematical communication has changed a bit since those halcyon days. The biggest change has been the near universal use of spreadsheet software, which requires an explicit multiplication operator ("*", an asterisk) in cell formulas. So, for a spreadsheet maven (or for that matter a Fortran programmer) the use of the asterisk would be natural.

Regarding the explicit use of "1" as the multiplying factor for the 3-point and free throw expressions, I did that for the 3-point difference to emphasize that the correct factor there is indeed 1 and not 3, as one might initially expect. Having included it for the 3-point term, it seemed merely consistent to include it also for the free throw term.

Maybe someone of more tender years will find a way to write this equation which is even more natural and communicative for the millenial generation, but I cannot aspire to that role.

 

[MD#12Fan]

My high school math teachers, Brother Henry and Brother Carl, would have agreed with you. I presume they are now resting in peace, despite the mayhem that Brother Henry in particular inflicted on us as high school freshmen.

However, while math is constant, mathematical communication has changed a bit since those halcyon days. The biggest change has been the near universal use of spreadsheet software, which requires an explicit multiplication operator ("*", an asterisk) in cell formulas. So, for a spreadsheet maven (or for that matter a Fortran programmer) the use of the asterisk would be natural.

Regarding the explicit use of "1" as the multiplying factor for the 3-point and free throw expressions, I did that for the 3-point difference to emphasize that the correct factor there is indeed 1 and not 3, as one might initially expect. Having included it for the 3-point term, it seemed merely consistent to include it also for the free throw term.

Maybe someone of more tender years will find a way to write this equation which is even more natural and communicative for the millenial generation, but I cannot aspire to that role.

Can you explain mathematically why it is sound strategy to shoot 3's rather than 2's. I think the first big-time coach to employ shooting 3's in abundance was Coach K. Anyone watch an NBA game lately? There's the "point" with the ball and basically everyone else spread out outside the 3 point line. And the NBA 3 is a long shot! (at least by our standards in the late 70's) Is the 2 point shot merely a vehicle to open up the 3 point shot? Like a line plunge in football might open up a sweep.)
Can you help me with that from a mathematical slant?

 

[oldude]

Can you explain mathematically why it is sound strategy to shoot 3's rather than 2's. I think the first big-time coach to employ shooting 3's in abundance was Coach K. Anyone watch an NBA game lately? There's the "point" with the ball and basically everyone else spread out outside the 3 point line. And the NBA 3 is a long shot! (at least by our standards in the late 70's) Is the 2 point shot merely a vehicle to open up the 3 point shot? Like a line plunge in football might open up a sweep.)
Can you help me with that from a mathematical slant?

The basic math of the 3 pt shot is pretty straightforward. Every 3 pt basket is worth 1 1/2 times more than every 2 pt basket. So if a team shoots 40% from 3, that’s the equivalent of shooting 60% from 2.

There are a couple other factors that also come into play. For a team that can shoot the ball, but generally lacks the size to match up with an opponent, they can stay in the game and perhaps win by outshooting their bigger opponent from the 3-pt line.

One other factor that is often mentioned is that 3-pt shots that are missed result in longer rebounds, that have a greater potential to be controlled by the offensive team, essentially creating more offensive possessions, hence more chances to score.

 

[MD#12Fan]

The basic math of the 3 pt shot is pretty straightforward. Every 3 pt basket is worth 1 1/2 times more than every 2 pt basket. So if a team shoots 40% from 3, that’s the equivalent of shooting 60% from 2.

There are a couple other factors that also come into play. For a team that can shoot the ball, but generally lacks the size to match up with an opponent, they can stay in the game and perhaps win by outshooting their bigger opponent from the 3-pt line.

One other factor that is often mentioned is that 3-pt shots that are missed result in longer rebounds, that have a greater potential to be controlled by the offensive team, essentially creating more offensive possessions, hence more chances to score.

Thanks for the points! (pun not intended)
I always use the example of 10 shots from 2 point range and 10 shots from 3 point range and comparing the percentages to have the same basic score.
How about the evolution of the game itself. Like how the giraffe developed a long neck in order to take advantage of food resources higher up vertically. Think of what has happened to the center. Do great coaches still teach basic post moves? And so much specialization especially regarding 3-point shooting. Come to think of it, one appeal of the women's game is that it's behind the men's game in these ways. Still seeing the development of real centers, like Aja.
I just don't think the game is as aesthetically pleasing as it once was. (arguably Golden State was but when the whole league tries to play that style, not so much)

 

[Bama fan]

My high school math teachers, Brother Henry and Brother Carl, would have agreed with you. I presume they are now resting in peace, despite the mayhem that Brother Henry in particular inflicted on us as high school freshmen.

However, while math is constant, mathematical communication has changed a bit since those halcyon days. The biggest change has been the near universal use of spreadsheet software, which requires an explicit multiplication operator ("*", an asterisk) in cell formulas. So, for a spreadsheet maven (or for that matter a Fortran programmer) the use of the asterisk would be natural.

Regarding the explicit use of "1" as the multiplying factor for the 3-point and free throw expressions, I did that for the 3-point difference to emphasize that the correct factor there is indeed 1 and not 3, as one might initially expect. Having included it for the 3-point term, it seemed merely consistent to include it also for the free throw term.

Maybe someone of more tender years will find a way to write this equation which is even more natural and communicative for the millenial generation, but I cannot aspire to that role.

Last I heard Brother Henry was living in retirement near Baltimore, Md.