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Abstract

Given its popularity and large market value, professional basketball has attracted increased academic attention over the past few decades. We employ the Extreme Value Theory (EVT) to compare two almost unreachable records in NBA: all-time and 10-time season scoring champions. The surpassing probabilities for each record were measured using EVT. We find that the surpassing probability of the NBA all-time scoring champion is more than eight times higher than the surpassing probability of the NBA 10-time season scoring champion, indicating that the latter is the more unlikely record.

Keywords

Basketball extreme value theory greatest of all time player (GOAT)

Introduction

Sports have a great economic, political and cultural influence on daily life. The global sports market had a value of $459 billion in 2019 and is expected to reach $826 billion by 2030.¹ In more than 3,000 sports around the world,² basketball is among the most popular ones. According to Adam Silver, the current commissioner of the National Basketball Association (NBA), NBA’s revenue was over $10 billion in the 2021/22 season. Given its popularity and large market value, professional basketball has attracted increased academic attention over the past few decades. There are various topics in the field of basketball data analytics: modeling score processes,^3–5 identify momentum,^6–8 player and team performance evaluation,^9–14 position performance assessment,^15–17 home-court advantage analysis,^18,19 and the impact of team physical characteristics,²⁰ just to name a few. For a comprehensive overview of these domains, we refer to systematic reviews such as Courel-Ibáñez et al.²¹ and Chen et al.²²

To the best of our knowledge, there are few articles focusing on single achievements, which are nevertheless striking: Wilt Chamberlain scored 100 points in a single game on March 2, 1962; Earvin Johnson won the NBA Finals Most Valuable Player Award during his rookie season (1979/80); Stephen Curry broke 300 three-pointers barrier and made 402 three-pointers in the 2015/16 season; Michael Jordan led the NBA in scoring 10 times. Even after many years, people still discuss and honor these achievements in the NBA history.

On February 7, 2023, LeBron James broke Kareem Abdul-Jabbar’s NBA all-time scoring record of 38,387 points which remained unbroken for 38 years (James’ record is 38,424 up to the end of Feb 2023), and the game was unprecedentedly stopped for a brief ceremony. LeBron James was excited about this achievement and said “I would never, ever in a million years dreamed it’s better than what it is tonight.”²³ When he was asked if this achievement makes him the greatest player of all time (GOAT), he answered: “…I’m going to take myself against anybody that’s ever played this game.” Before we proceed, it is important to note that we do not conduct an all-around comparison between Michael Jordan and LeBron James in this paper.

Both players have earned numerous honors throughout their careers. Their NBA championships and honors are visually summarized in Figure 1. These achievements naturally lead to frequent comparisons between them. However, such a comparison would involve too many factors and some of which are difficult to measure, e.g. dominance on the court or global influence off the court. Here, we compare the NBA all-time scoring record made by LeBron James with the 10-time season scoring record made by Michael Jordan. Even though the goal of this paper is not on the GOAT, the comparison between the two achievements obviously contributes to the GOAT debate in the scoring category in an objective manner.

Figure 1.

Number of NBA championships and honors earned by LeBron James (22 seasons, career ongoing) and Michael Jordan (15 seasons).

These two almost unreachable achievements are extremes. Thus, we use the Extreme Value Theory (EVT) to measure how unlikely these records are. Specifically, we calculate the probability of surpassing these records; the lower the probability, the more improbable the record is. For a comprehensive review of EVT, we refer to Gomes and Guillou.²⁴

The article is organized as follows. Section “Statistical model based on EVT” describes the methodology and Section “Data description and results” presents the data and results. Section “Discussion” provides the discussion and conclusions.

Statistical model based on EVT

Our method employs EVT to investigate the basketball players’ achievements, similar to the approach used in the early work of Einmahl and Magnus,²⁵ which applied EVT to predict sport world records. We present a brief overview of EVT here and refer to Embrechts et al.²⁶ for a detailed discussion.

Let $X_{1}, X_{2}, \dots, X_{n}$ be independent random variables with the same cumulative distribution function $F$ , and the associated order statistics are $X_{1, n} \leq X_{2, n} \leq \dots \leq X_{n, n}$ , with $X_{n, n}$ being the maximum. Based on EVT, the probability of surpassing a record can be approximated by

P {X_{n, n} > x} \approx \frac{k}{n} {(1 + γ \frac{x - b_{n, k}}{a_{n, k}})}^{- \frac{1}{γ}},

(1)

where

n

is the sample size and

k

is the number of upper observations used for estimation. We assume

k \to \infty

and

k / n \to 0

n \to \infty

γ

is the tail index of

F

;

a_{n, k} > 0

and

b_{n, k}

\in

R

are two sequences of normalized constants. We denote this surpassing probability as SP.

For a fixed $k$ , we let

M_{n}^{(j)} : = \frac{1}{k} \sum_{i = 0}^{k - 1} {(\log X_{n - i, n} - \log X_{n - k, n})}^{j}, j = 1, 2.

The estimator for $γ$ is

\hat{γ} = M_{n}^{(1)} + 1 - \frac{1}{2} {(1 - \frac{{(M_{n}^{(1)})}^{2}}{M_{n}^{(2)}})}^{- 1},

(2)

and the estimators for

a_{n, k}

, and

b_{n, k}

are

{\hat{a}}_{n, k} = {\begin{matrix} M_{n}^{(1)} X_{n - k, n} (1 - \hat{γ}), & \hat{γ} < 0 \\ M_{n}^{(1)} X_{n - k, n}, & \hat{γ} \geq 0 \end{matrix},

and

{\hat{b}}_{n, k} = X_{n - k, n} .

The choice of $k$ involves balancing bias and variance of $\hat{γ}$ , more discussions can be found in Dekkers et al.²⁷ and Scarrott et al.²⁸ This paper uses the standard numerical method offered by EVT to select $k$ , that is, plotting $\hat{γ}$ against $k$ and choosing $k$ from the range where $\hat{γ}$ first hits a stable region. This approach is motivated by the need to balance bias and variance in the estimation of the tail index $\hat{γ}$ . A small $k$ introduces high variance, while a large $k$ leads to bias. The first stable region identifies a robust interval where estimates balance this trade-off, ensuring that inferences regarding tail behavior remain reliable even under slight variations in $k$ .

The SP of the NBA all-time scoring champion can be estimated by Equation (1), directly with plug-in estimators provided above. The lower the SP of the record is, the more improbable is to exceed the record. The same applies to the SP of the NBA season scoring champion. However, Michael Jordan earned the title of NBA season scoring champion 10 times, which is a joint event. Although becoming the scoring leader depends more directly on all players’ performance within a specific season rather than past seasons, individual performances may still be correlated across years. To address this, we assume that the SP of one season scoring record is yearly independent conditioning on the second-highest point totals in the respective seasons. The second-highest score reflects the overall level of scoring of a season. Conditioning on this variable allows us to account for differences in scoring environments across seasons and to isolate record events from temporal dependencies.

Data description and results

In this section, we first describe the data construction and then the estimations of the SP of those two records using the aforementioned approach.

Data description

Numerous factors—including rule modifications, pace of play, and number of games–influence scoring performance and could be relevant when evaluating records. However, integrating these variables requires subjective decisions regarding selection and weighting. This study focuses on measuring the surpassing probability of records based on observed outcomes. We therefore concentrate specifically on the scoring data, following an outcome-oriented rationale: player’ scorers represent a final, integrated measure that captures the net effect of all influencing factors.

The data of NBA all-time points and regular season points are collected from NBA’s website at www.nba.com using data scraping techniques. As of February 2023, there are 4,787 players on the NBA all-time scoring list and we analyze the top half of the all-time scoring list (PTS $_{alltime}$ ), as the lower half does not impact the upper-tail estimation relevant to EVT. EVT focuses on the asymptotic behavior in the tails of a distribution. Restricting the dataset to the top half is therefore appropriate, as it adequately preserves tail characteristics while avoids noise from non-tail observations. Moreover, lower scores are less relevant to the competitive environment in which scoring records are set. Figure 2 shows the histogram of the top half of PTS $_{alltime}$ with a logarithmic y-axis. The histogram reveals that PTS $_{alltime}$ of most players are less than 10,000. More specifically, only $7$ (approximately $0.14 %$ of the) players have PTS $_{alltime}$ above 30,000. This extreme rarity confirms that the NBA all-time scoring record is almost unreachable.

Figure 2.

Histogram of the top half of PTS $_{alltime}$ , plotted on a logarithmic y-axis scale.

Unlike LeBron James, who is still an active NBA player, Michael Jordan has retired. Figure 3 shows their career spans. LeBron James has played continuously since 2003, whereas Michael Jordan’s career was divided into distinct periods, including 1984/85-1992/93 seasons, 1994/95-1997/98 seasons, and 2001/02-2002/03 seasons. During the 1986/87-1992/93 and 1995/96-1997/98 seasons, Michael Jordan won the season scoring leader. The NBA scoring title is given to the player who has the highest points per game average during the regular season. To measure the SP of the NBA 10-time season scoring champion, the datasets from these ten seasons are collected. Again, we use the points per game average in a regular season, denoted by PPG $_{season}$ , from the top half of the respective seasons scoring lists.

Figure 3.

The career spans of LeBron James and Michael Jordan.

Figure 4 shows histograms of the PPG $_{season}$ for four NBA seasons as examples. Although the shapes of the these histograms vary across seasons, reflecting distinct competitive environments in each season, the histograms consistently show that PPG $_{season}$ for the majority of players is less than $20$ . In addition, only two players (Michael Jordan and Wilt Chamberlain) have won more than six scoring titles in NBA history.

Figure 4.

Distribution of PPG $_{season}$ values for four selected NBA seasons: (a) $1988 / 89$ , (b) $1990 / 91$ , (c) $1992 / 93$ , (d) $1995 / 96$ .

Results

We now apply EVT to our samples to measure the SP of the NBA all-time scoring champion and NBA 10-time season scoring champion.

We first estimate $γ$ for PTS $_{alltime}$ and PPG $_{season}$ in each season, while $k$ is selected from the first stable region in the diagnostic plot as mentioned in Section “Statistical model based on EVT”. Figure 5 provides the diagnostic plot of $\hat{γ}$ for PTS $_{alltime}$ with a limit range of $k$ as an example, indicating the first stable region from $k$ =100 to $k$ =120 by the dashed lines. Generally, any value of $k$ in this region is reasonable. Although it might be argued that the choice of $k$ is subjective, this subjective nature allows us to select $k$ more consistently across years. The values of selected $k$ and the corresponding $\hat{γ}$ are presented in the Table 1. All $\hat{γ}$ values are negative, indicating that the data have finite upper bounds of PTS $_{alltime}$ and PPG $_{season}$ that align with the reality of basketball games.

Figure 5.

The plot of $\hat{γ}$ versus $k$ for PTS $_{alltime}$ . The selected first stable region is between the two dashed vertical lines.

Table 1.

Results for NBA scoring points.

Sample	$k$	$\hat{γ}$	SP
All-time scoring	111	$-$ 0.0693	$7.4 \times 10^{- 5}$
1986/87 season scoring	25	$-$ 0.0345	0.0005
1987/88 season scoring	16	$-$ 0.0224	0.0017
1988/89 season scoring	26	$-$ 0.0581	0.0016
1989/90 season scoring	12	$-$ 0.0280	0.0013
1990/91 season scoring	29	$-$ 0.0968	0.0021
1991/92 season scoring	30	$-$ 0.0182	0.0017
1992/93 season scoring	27	$-$ 0.0526	0.0012
1995/96 season scoring	29	$-$ 0.0197	0.0011
1996/97 season scoring	30	$-$ 0.0575	0.0013
1997/98 season scoring	34	$-$ 0.0264	0.0014

Given the value of $\hat{γ}$ , the SP of the NBA all-time scoring champion (as of February 2023) and the NBA season scoring champion in the respective seasons are calculated and shown in the last column of Table 1. The SP of the NBA 10-time season scoring champion is around $6.3 \times 10^{- 6}$ , which is the product of the conditional SP of NBA season scoring champion based on the second highest point in each of the ten seasons. This value indicates that the probability of a player winning ten season scoring titles is extremely low. Achieving this requires a player to be the league’s best scorer not just for one season, but to maintain that high level for ten years. The SP value effectively captures this extreme difficulty.

Table 1 also shows that the SP of the NBA all-time scoring champion is about $7.4 \times 10^{- 5}$ . This small value reflects the extreme difficulty of achieving this record. Historically, Kareem Abdul-Jabbar, who broke the all-time scoring record in 1984 and continued to extend it until his retirement in 1989, held the record for nearly 40 years before it was broken by LeBron James. Unlike winning a seasonal scoring title, this record does not require leading the league in any single season, but rather demands stable, high-level performance over a long career.

The lower the SP of the record is, the more improbable is to exceed the record. Thus, the NBA all-time scoring record made by LeBron James is more likely than the 10-time season scoring record made by Michael Jordan. Furthermore, as LeBron James is still an active NBA player, he has the opportunity to raise the SP of the NBA all-time scoring record above the SP of NBA 10-time season scoring champion. Therefore, we are also interested in how many more points LeBron James needs to reach the SP of the NBA 10-time season scoring record. The result is shown in Table 2. By making the SP of the NBA all-time scoring champion equal to $6.3 \times 10^{- 6}$ , the corresponding PTS $_{alltime}$ is about 44,600. Therefore, when LeBron James broke the NBA all-time scoring record, he needed at least 6,176 more points to reach the SP of NBA 10-time regular season scoring record. As of November 9, 2025, with his total points at 42,184, this requirement has decreased to 2,416.

Table 2.

The result for comparison of player achievements.

SP	PTS $_{alltime}$
$7.4 \times 10^{- 5}$	38,424
$6.3 \times 10^{- 6}$	44,600

Sensitivity analysis

To further contextualize the result, we examine the SP of the NBA’s all-time scoring record at the end of each regular season from $1971 / 72$ to $2021 / 22$ . As shown in Figure 6, the SP declined during the initial decade. This trend is attributable to the rising cumulative record and the expanding player pool. Subsequently, as the record continued to increase, the performance level of the league’s elite players reached an equilibrium. This led to a stabilization phase in the SP, which is consistent with the long-standing dominance of Kareem Abdul-Jabbar’s scoring achievement.

Figure 6.

The plot displays the SP for the NBA all-time scoring at the end of each regular season from 1971/72 to 2021/22, with a dashed line indicating the value of $7.4 \times 10^{- 5}$ .

Although only Michael Jordan has won the NBA 10-time season scoring champion, we performed a simulation to estimate the SP of a player achieving this record. By randomly selecting 10 seasons from the 1971/72 to 2020/21 seasons over 500 iterations, we generated the results and presented in Figure 7. The period begins with the $1971 / 72$ season, which was chosen because the all-time list had by then accumulated sufficient data for analysis. It ends with the $2021 / 22$ season, immediately preceding the season in which LeBron James broke the NBA all-time scoring record. Figure 7 shows that the simulated SP for a player winning the NBA season scoring title ten times is an extreme event. This is consistent with the SP of Michael Jordan’s NBA 10-time season scoring champion, as indicated by the dashed line. This result is also supported by the history of the scoring title: Wilt Chamberlain holds the second-most titles with seven, followed by George Gervin and Kevin Durant with four each. The fact that no player has exceeded four titles in over twenty years since Jordan retired further shows the challenge of reaching Michael Jordan’s record. Furthermore, these simulated SPs are lower than the SP of the NBA all-time scoring champion, as indicated by the long dashed line in Figure 7.

Figure 7.

Distribution of Simulated SP for NBA 10-time scoring champion. A dashed line and a long dashed line mark the values of $6.3 \times 10^{- 6}$ and $7.4 \times 10^{- 5}$ , respectively.

The above results are derived from the top half of PTS $_{alltime}$ and PPG $_{season}$ . For the purpose of exposition, the comparison of the SP for the all-time scoring record derived from the top half of the data against the results from the top 40%, 60%, 80%, and the full dataset (100%) is presented to demonstrate the robustness of this selection. Table 3 shows the comparison results and indicates that our conclusions do not depend on the specific top half selection. For the top 40%, 60%, 80%, and 100% selections, the SP values show low variation across k in the first stable region. The variance remains consistently below $1 \times 10^{- 9}$ for all k values considered. In addition, a sensitivity analysis was conducted using the top 40%, 60%, 80%, and the full dataset of season scoring lists. The results showed negligible differences in the surpassing probabilities, only appearing at the sixth decimal place.

Table 3.

The result for comparison of data selection.

Data selection	k	SP
100%	246	$6.6 \times 10^{- 5}$
80%	226	$6.1 \times 10^{- 5}$
60%	221	$7.1 \times 10^{- 5}$
40%	106	$9.6 \times 10^{- 5}$

An important consideration is the sensitivity of the SP to the choice of $k$ . When $k$ is selected from a reliable first stable region, the SP is not sensitive. Figures 8 and 9 display the variation of $\hat{γ}$ and SP over a range of $k$ values. The vertical lines in these figures denote the chosen $k$ for each season, while the horizontal lines indicate the corresponding values of $\hat{γ}$ and SP values, respectively. We selected four seasons as examples, including those with relatively stable performance and others with higher volatility.

Figure 8.

The plot of $\hat{γ}$ versus $k$ for PPG $_{season}$ : (a) $1988 / 89$ , (b) $1990 / 91$ , (c) $1992 / 93$ , (d) $1995 / 96$ .

Figure 9.

The plot of SP versus $k$ for PPG $_{season}$ as example: (a) $1988 / 89$ , (b) $1990 / 91$ , (c) $1992 / 93$ , (d) $1995 / 96$ .

Since each season’s top-half data contains about 250 points, the first stable region usually lies between k = 10 and 40. However, the exact position of the first stable region varies by season due to differences in seasonal scoring environments. We observe that around the vertical line in each of the plots, both $\hat{γ}$ and SP do not fluctuate much as a function of $k$ . Moreover, by using our selected $k$ , the SP values are relatively higher than those produced by most other $k$ values within the stable region. Thus, although the choice of $k$ affects the number of points LeBron James needs to match the SP of the 10-time scoring record, it does not change our main conclusion.

Discussion

The sport records are associated with extremes. In this paper, we apply EVT to measure how unlikely the records are by estimating the SP of the records. Given that Michael Jordan is the only player who earned the NBA 10-time season scoring champion and LeBron James holds the all-time scoring record as of 2025, we focus on these two specific, almost unreachable achievements: LeBron James’s NBA all-time scoring champion and Michael Jordan’s NBA 10-time season scoring champion. We calculate the probability of surpassing these records, which are both extremely low, corresponding to the exceptional difficulty of achieving them. In the more than twenty years since Michael Jordan’s retirement, no player has won more than four season scoring titles. Before LeBron James became the NBA’s all-time scoring champion, Kareem Abdul-Jabbar held the record for nearly 40 years. In addition, the lower the probability is, the more improbable the record is. Therefore, the SP of Lebron James’s NBA all-time scoring champion is more than eight times higher than the SP of Michael Jordan’s NBA 10-time season scoring champion. Meanwhile, we find that the SP for becoming the NBA all-time scoring champion is higher than that for becoming the NBA 10-time random-season scoring champion.

Unlike Michael Jordan who is retired, LeBron James is still an active NBA player and continues to extend his own records. Consequently, the SP for the NBA all-time scoring champion title is expected to decrease. Therefore, we estimated the points LeBron James needs to reach the SP of the NBA 10-time season scoring champion. We found that he would need to score at least 2,416 more points to reach the SP of the NBA 10-time season scoring champion.

While some studies have applied EVT to record sports,^25,29,30 this paper is the first to use EVT to compare the achievements of basketball players. The SP measures the difficulty of achieving a record, providing an objective basis for decision-making. For players, SP provides a clear assessment of the challenges, translating abstract goals into probabilities. This quantitative perspective assists players in designing tailored development plans and making more rational pathway choices. For coaches, SP serves as an objective metric to evaluate the risks associated with supporting individual honors. From an organizational perspective, the difficulty reflected by SP correlates with the commercial and brand value of achievements, informing decisions in player recruitment, brand positioning, and long-term business strategy.

A further application of this method can be the cross-seasonal comparison of achievements. For instance, it could be used to analyze the relative difficulty of earning a scoring title in separate seasons. Such a comparison would reveal differences in the competitive environment, helping to quantify the impact of contextual variables such as rule changes.

When extending this method to cross-seasonal comparisons, dependencies across seasons become an important factor. This paper assumes that the SP for the season scoring record is conditionally independent across seasons, given the second-highest score. Recognizing the potential limitations of this simplification, future work could strengthen the framework by directly incorporating these longitudinal dependencies.

Footnotes

ORCID iDs

Zhuojing Zhang

Tao Chen

Ethical approval

Not applicable.

Consent to participate

Not applicable

Authors contributions

The authors have contributed equally.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Availability of supporting data

The data that support the findings of this study are publicly available on NBA’s website, at www.nba.com.

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NBA all-time scoring champion vs. 10-time season scoring champion: Which record is more unlikely?

Abstract

Keywords

Introduction

Statistical model based on EVT

Data description and results

Data description

Results

Sensitivity analysis

Discussion

Footnotes

ORCID iDs

Ethical approval

Consent to participate

Authors contributions

Funding

Declaration of conflicting interests

Availability of supporting data

References