Basketball Math

BASKETBALL COMBINED
FIELD GOAL PERCENTAGE

During my thirty-three year math teaching career, I spent thirty of those years coaching basketball. One of the things that I found interesting was that when the three point shot became part of the game, it changed the concept of what was a good field goal percentage (meaning what percent of the field goals attempted were successful). Most high school coaches were happy if their team made at least 45% of their field goal attempts when all shots made were worth two points. The advent of the three point shot changed that because the team scored three points for a made shot from behind the arc as opposed to two points for a made shot from inside the arc. So making a three point shot was naturally more valuable to a team than making a two point shot.

The made three point shot is worth 1.5 times a much as a made two point shot so the three point percentage should be multiplied by 1.5 to fairly compare it to the percentage from two point range. So if a team or player shot 40% from three point range that would be equal to shooting 60% from two point range. As an example: making four out of ten (40%) from three point range would score 12 points (in ten shots) and making six out of ten (60%) from two point range would also score 12 points (in ten shots). Therefore if a coach would be happy with 45% shooting from two point range they would have to be happy with 30% shooting from three point range.

In the reality of basketball there might be some slight differences in those two situations. Three point shooters are less likely to be fouled than two point shooters. Also missed three point shots are not as often rebounded for baskets as are the missed two point shots. (One thing that some people consider to be negative about the three point rule is that the “in between” shot [between the lane and the three point arc] has almost completely disappeared from the game at the high school and collegiate level.) However for the balance of this discussion I am not going to consider these realities.

One day while I was watching a game I started to wonder if there was a way to take all of the field goal attempts and field goals made and come up with one percentage which reflected the different value of the shots. In other words it would not be realistic to simply add the two point shots made and the three point shots made and divide by the total number of field goals attempted. Here is why: let's say a team made ten of twenty from two point range (50% and scoring 20 points) and made five of twenty from three point range (25% and scoring 15 points). Just adding them normally that would be 15 made shots in 40 attempts which is 37.5%. However the team actually scored 35 points with those 40 shots and if they had made those same 15 out of 40 shots under the old rules (no shot worth three points) they would have only scored 30 points.

After some experimentation, thinking, and calculating, I finally came up with a way to actually figure out a combined field goal percentage method that accurately reflected the value of a three point shot as well as the two point shot. What it actually does is to give you a field goal percentage that would accurately reflect how many points you scored with your field goal attempts if all the attempts were from two point range. The original formula I had wasn't “user friendly” so I made an adjustment that was mathematically equal but much easier to use. Here is the “friendly” formula:

[Total points scored from all field goals] divided by [Twice the total number of all field goal attempts] Convert the resulting decimal to a percent.

In the above example the team scored 35 points with 40 shots so it would be 35 divided by (2x40) or 35 / 80 which is .4375 or 43.75%. That is higher than the 37.5 % which would be the normal way and that reflects those five made three pointers scored 15 points not ten points.

The interested reader might want to comment on why this formula works. I think I know but I would be anxious to hear what you have to say about it.

Here is one additional challenge. Can someone come up with a formula that also takes into account the free throws made (worth one point of course) and the free throws attempted along with all the field goals made and attempted? In other words, one percentage that would reflect one point shots, two point shots and three point shots. It is possible that no one in the world has ever done that, maybe because they had many other better things to do! As I am typing this, I do have an idea on what the formula would be but I would like to hear what other people think. If you figure it out or find that someone else has figured it out, please let everyone know by posting a comment on this blog.

One other way that basketball coaches and/or fans could look at this information is to simply compute what I would call the “PPS” for their team. “PPS” stands for “Points Per Shot.” Take the total number of points scored by a team and divide by the total number of shots (free throws, two point shots, and three point shots) that the team took during the game. Here is an example: A team makes 12 of 18 free throws scoring 12 points, 20 of 45 two point shots scoring 40 points, and 5 of 17 three point shots scoring 15 points. Therefore the team scored a total of 67 points and took a total of 80 shots. The PPS would be 67 / 80 which is .8375. So the team averaged .8375 points for each shot. That decimal should not be converted to a percent.

Let me know what you think, if you are still awake and make it this far!