Basketball Scoring Fundamentals: The Efficiency Equation
Basketball, at its core, is a game of points. The team that successfully puts the ball through the hoop more times than its opponent emerges victorious. Yet, not all scoring opportunities are equal. The geometry of the court and the rules of the game have created two distinct field goal types: the two-point field goal and the three-point field goal. While the latter awards an extra point, it comes with a lower probability of success. This fundamental trade-off between point value and accuracy lies at the heart of basketball strategy and has driven one of the most significant transformations in modern sports: the three-point revolution. This chapter establishes the foundational principles of basketball scoring efficiency, introducing the core metrics that allow us to quantify the value of each shot type. By understanding the mathematical underpinnings—specifically the concept of expected value and the break-even point for three-point accuracy—we can begin to decode why the three-point shot, despite its lower raw field goal percentage, can yield superior long-term efficiency. This efficiency equation is not merely a theoretical construct; it is the lens through which we must view the historical undervaluation of three-point attempts prior to the analytics era and the subsequent strategic shift that has reshaped the National Basketball Association (NBA).
The Two-Point and Three-Point Field Goals: Rules, Geometry, and Historical Context
The modern basketball court is a carefully calibrated rectangle of dimensions and markings that define the conditions for scoring. At each end, a hoop elevated 10 feet above the floor serves as the target. Surrounding this hoop is a key area, but the most critical demarcation for scoring purposes is the three-point line. In the NBA, this arc is not a uniform distance from the basket; it is a composite shape. From the center of the basket, the line measures 23.75 feet (7.24 meters) to the top of the arc, but it narrows to 22 feet (6.70 meters) in the corners near the baselines. This design reflects a balance between encouraging perimeter play and maintaining the historical value of inside scoring. Any field goal made with a player's feet on or outside this line counts for three points; any field goal made with both feet inside the line counts
for three points. Conversely, any field goal attempted with both feet on or inside this line counts for two points. This seemingly simple distinction creates a strategic dichotomy that has defined offensive basketball for decades. The historical evolution of the three-point line itself reveals much about the NBA's shifting philosophy. When the league introduced the three-point shot in the 1979-80 season, it was initially viewed as a gimmick—a novelty to boost scoring and fan interest. The original line was a uniform 23 feet 9 inches from the basket around the entire perimeter, significantly farther than today's corner distance. Over the years, the line has been adjusted multiple times: moved inward to 22 feet in 1994 (a move widely criticized for cheapening the shot), then restored to its current configuration in 2008. These adjustments were not arbitrary; they were attempts to balance the shot's difficulty against its strategic value, to calibrate the efficiency equation at the heart of this chapter.
Understanding the raw geometry of the court is essential, but equally important is understanding the human element—the physical and technical demands of each shot type. A two-point field goal attempt can take many forms: a close-range layup or dunk, a mid-range jump shot, or a post-up move. The average two-point field goal percentage in the NBA for the 2022-23 season was approximately 52.1%. In contrast, the average three-point field goal percentage was 36.2%. At first glance, the two-point shot appears vastly superior—players make over half of their two-point attempts but only about one-third of their three-point attempts. This raw percentage comparison, however, is precisely the flawed logic that the analytics revolution sought to correct. It compares apples to oranges by ignoring the fundamental difference in point value. To make an apples-to-apples comparison, we must normalize scoring output relative to shooting attempts, which brings us to the first critical efficiency metric: points per shot.
The Efficiency Equation: From Field Goal Percentage to Points Per Shot
The most basic efficiency metric is field goal percentage (FG%), calculated as:
$$\text{FG\%} = \frac{\text{Field Goals Made}}{\text{Field Goal Attempts}} \times 100\%$$
This metric, while simple and intuitive, is incomplete because it treats all field goals as equal. A player who shoots 50% on two-point attempts and a player who shoots 33% on three-point attempts would appear, under FG% alone, to be equally efficient scorers. Yet, we know intuitively that making one-third of three-point attempts yields the same number of points (3 × 0.333 ≈ 1 point per attempt) as making half of two-point attempts (2 × 0.5 = 1 point per attempt). The raw FG% masks this equivalence because it ignores point value.
To compare scoring efficiency across shot types, we must calculate points per shot (PPS), which directly answers the question: "On average, how many points does a player score each time they shoot the ball?" The formula is straightforward:
$$\text{PPS} = \text{FG\%} \times \text{Points per Attempt}$$
For two-point shots: $\text{PPS}_{2} = \text{FG\%}_{2} \times 2$ For three-point shots: $\text{PPS}_{3} = \text{FG\%}_{3} \times 3$
Using league averages from the 2022-23 season:
- Two-point PPS: $0.521 \times 2 = 1.042$ points per two-point attempt
- Three-point PPS: $0.362 \times 3 = 1.086$ points per three-point attempt
This calculation reveals a counterintuitive truth: despite a significantly lower field goal percentage, the average three-point attempt yielded approximately 4.2% more points than the average two-point attempt that season. This is the essence of the three-point revolution's mathematical foundation—the three-point shot, as a collective league average, was already more efficient than the two-point shot, even though most casual observers and many traditional coaches would have assumed the opposite based on raw percentages alone.
The disparity becomes even more pronounced when we consider effective field goal percentage (eFG%), a metric explicitly designed to account for the extra value of three-pointers. The formula is:
$$\text{eFG\%} = \frac{\text{Two-Point FG Made} + 1.5 \times \text{Three-Point FG Made}}{\text{Total FG Attempts}} \times 100\%$$
Or equivalently:
$$\text{eFG\%} = \text{FG\%} \times \text{Adjustment Factor}$$
where the adjustment factor is 1 for two-point attempts and 1.5 for three-point attempts (since 3/2 = 1.5). This metric answers: "If we treated three-pointers as 1.5 two-pointers, what would the field goal percentage be?" Using the same league averages, we can calculate the overall eFG%:
$$\text{eFG\%} = \frac{\text{Total FG Made} + 0.5 \times \text{3P Made}}{\text{Total FG Attempts}} \times 100\%$$
For the 2022-23 NBA season, the league-wide eFG% was approximately 54.5%, compared to a raw FG% of 48.5%. The 6-percentage-point gap directly reflects the impact of three-point shooting on overall efficiency. More importantly, eFG% allows for direct comparison between players and teams with different shot distributions. A player with a 45% FG% who takes many three-pointers may be a more efficient scorer than a player with a 48% FG% who only takes two-point shots, because the former's eFG% could be higher.
While eFG% improves upon raw FG%, it still does not account for another critical aspect of scoring efficiency: free throws. A complete picture of scoring efficiency must incorporate all points scored relative to all scoring opportunities. This brings us to true shooting percentage (TS%), the gold standard efficiency metric in basketball analytics.