US Football: best formulas to evaluate a quarterback in US professional football

 Here is an exploration of the topic of mathematical formulas for evaluating and rating the ability of professional football quarterbacks. Below, I will provide a comprehensive summary of empirical data, studies, and established methodologies related to this subject. I will focus on the most widely recognized and data-driven approaches used in the NFL and academic research.

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Empirical Data and Studies on Quarterback Evaluation Formulas

The evaluation of quarterback performance in professional football has been a subject of significant interest for statisticians, sports analysts, and team management. Over the years, several mathematical formulas and metrics have been developed to quantify a quarterback's ability, often relying on game statistics such as passing yards, touchdowns, interceptions, completion percentage, and more. Below is a detailed summary of the key formulas, their empirical foundations, and relevant studies or applications.

1. NFL Passer Rating (Official NFL Formula)

• Overview: The NFL Passer Rating, also known as the Quarterback Rating, is the official metric used by the National Football League to evaluate quarterback performance since 1973. It was developed by Don Smith of the Pro Football Hall of Fame and is a composite score based on four statistical categories: completion percentage, yards per attempt, touchdown percentage, and interception percentage.
• Formula: The Passer Rating is calculated using the following steps, with each category scaled to a maximum of 2.375 (and a minimum of 0):
  1. Completion Percentage: $\text{a} = \left( \frac{\text{completions}}{\text{attempts}} \times 100 - 30 \right) \times 0.05$
  2. Yards per Attempt: $\text{b} = \left( \frac{\text{yards}}{\text{attempts}} - 3 \right) \times 0.25$
  3. Touchdown Percentage: $\text{c} = \frac{\text{touchdowns}}{\text{attempts}} \times 20$
  4. Interception Percentage: $\text{d} = 2.375 - \left( \frac{\text{interceptions}}{\text{attempts}} \times 25 \right)$
  • Final Passer Rating: $\text{Rating} = \frac{\text{a} + \text{b} + \text{c} + \text{d}}{6} \times 100$
  • The maximum possible rating is 158.3, and the minimum is 0.
• Empirical Basis: The formula was designed to balance different aspects of quarterback performance and has been used for decades to rank players. It is empirically grounded in historical NFL game data, with weights assigned based on perceived importance of each statistic at the time of its creation.
• Limitations and Criticism: Studies and analyses, such as those by sports statisticians like Brian Burke (formerly of Advanced NFL Stats), have criticized the Passer Rating for its arbitrary weighting and lack of context (e.g., it does not account for game situation, opponent strength, or rushing contributions). A 2011 article by Burke in the Washington Post highlighted that the formula overemphasizes completion percentage and does not adjust for era-specific differences in passing efficiency.
• Usage: Despite its limitations, the NFL Passer Rating remains a widely cited metric, often used in player contracts, Hall of Fame considerations, and media discussions. Historical data shows that elite quarterbacks like Tom Brady, Peyton Manning, and Aaron Rodgers consistently achieve ratings above 90, with exceptional seasons exceeding 100 (e.g., Aaron Rodgers’ 2011 rating of 122.5).

2. ESPN’s Total Quarterback Rating (QBR)

• Overview: Introduced by ESPN in 2011, the Total Quarterback Rating (QBR) is a proprietary metric designed to improve upon the traditional NFL Passer Rating by incorporating situational context, play-by-play contributions, and advanced statistical modeling. It rates quarterbacks on a 0-100 scale.
• Formula: The exact formula for QBR is not publicly disclosed, but ESPN has stated that it accounts for:
  • Expected Points Added (EPA) per play, which measures a quarterback’s contribution to scoring based on down, distance, field position, and game situation.
  • Clutch performance weighting, giving more credit for plays in critical moments.
  • Penalties and fumbles, which are debited from the score.
  • Rushing contributions, unlike the NFL Passer Rating.
• Empirical Basis: QBR is derived from detailed play-by-play data and advanced analytics developed by ESPN’s Stats & Information Group. It uses machine learning models to assign credit or blame for each play, factoring in teammate and opponent contributions. According to ESPN, QBR correlates more strongly with team wins than the traditional Passer Rating.
• Studies and Validation: A 2015 study published by ESPN analysts Dean Oliver and Alok Pattani demonstrated that QBR has a higher predictive power for future game outcomes compared to Passer Rating, with a correlation coefficient of approximately 0.6 for team success versus 0.4 for Passer Rating. Additionally, QBR has been shown to better differentiate quarterback performance in high-pressure situations.
• Limitations: Critics argue that QBR’s proprietary nature makes it less transparent and harder to replicate or scrutinize. Some analysts, as noted in discussions on platforms like FiveThirtyEight, suggest that QBR may overadjust for situational factors, leading to counterintuitive rankings at times.

3. Pro Football Focus (PFF) Grading System

• Overview: Pro Football Focus (PFF) provides a grading system for quarterbacks (and other positions) based on detailed film analysis and play-by-play evaluations. Unlike purely statistical formulas, PFF grades incorporate subjective assessments by analysts who review game footage.
• Methodology: Each play is graded on a scale from -2 to +2, with increments of 0.5, based on execution, decision-making, and impact. These grades are aggregated into an overall score for the season, normalized to a 0-100 scale.
• Empirical Basis: PFF’s system is grounded in extensive data collection, with over 10,000 plays graded per NFL season. The methodology also incorporates advanced metrics like “Adjusted Completion Percentage” (accounting for drops and throwaways) and “Big-Time Throws” (high-difficulty passes with significant impact).
• Studies and Validation: While PFF does not publish peer-reviewed studies, its grades are widely used by NFL teams and media. A 2020 analysis by PFF showed that their quarterback grades correlate strongly with team offensive success (R² ≈ 0.55). Moreover, PFF grades have been cited in academic discussions of player evaluation, such as in papers presented at the MIT Sloan Sports Analytics Conference.
• Limitations: The subjective nature of PFF grading introduces potential bias, and the lack of a fully transparent formula can be a drawback. However, its focus on context (e.g., evaluating a quarterback’s decision-making under pressure) provides insights beyond raw statistics.

4. DYAR (Defense-Adjusted Yards Above Replacement) and DVOA (Defense-Adjusted Value Over Average)

• Overview: Developed by Football Outsiders, DYAR and DVOA are advanced metrics that evaluate quarterback performance relative to a replacement-level player and adjust for opponent strength and game situation.
• Formula:
  • DYAR measures total value in terms of yards above a replacement-level quarterback, factoring in passing and rushing contributions.
  • DVOA measures efficiency per play, expressed as a percentage above or below league average, adjusted for opponent defense.
  • Specific formulas are proprietary, but they rely on play-by-play data, expected outcomes, and defensive adjustments.
• Empirical Basis: Football Outsiders uses historical NFL data to establish baselines for replacement-level performance and league averages. Their metrics are derived from regression models and situational analysis.
• Studies and Validation: Football Outsiders has published numerous articles demonstrating the predictive power of DYAR and DVOA, with correlations to team wins often exceeding those of traditional stats. A 2018 study by Aaron Schatz (founder of Football Outsiders) showed that DVOA is a better predictor of playoff success than raw passing yards or Passer Rating.
• Limitations: Like QBR, these metrics are proprietary and complex, making independent replication difficult. They also require extensive data, which may not be accessible to casual analysts.

5. Academic Studies and Custom Models

• Berri and Simmons (2009): In their paper, “Catching a Draft: On the Process of Selecting Quarterbacks in the National Football League Draft,” published in the Journal of Productivity Analysis, David Berri and Rob Simmons analyzed quarterback performance using a custom metric called “Net Points.” This metric evaluates a quarterback’s contribution to scoring while accounting for turnovers and situational efficiency. Their study found that college statistics have limited predictive power for NFL success, but certain metrics like completion percentage and yards per attempt are more reliable indicators.
• MIT Sloan Sports Analytics Conference Papers: Several papers presented at the MIT Sloan Conference have explored quarterback evaluation. For instance, a 2016 paper by Michael Lopez and Gregory Matthews introduced a Bayesian model to predict quarterback performance based on historical data, incorporating both passing and non-passing contributions. Their model outperformed traditional metrics in forecasting future success, with a reported accuracy improvement of 10-15% over Passer Rating.
• Expected Points Added (EPA) Models: Widely discussed in sports analytics literature, EPA quantifies a quarterback’s contribution to scoring on a per-play basis. Studies by researchers like Burke (Advanced NFL Stats) and Michael Schuckers (2014, Statistical Sports Models in R) have validated EPA as a robust measure of quarterback impact, showing strong correlations with team wins (R² ≈ 0.6-0.7).

Comparative Analysis of Metrics

• Predictive Power: Studies consistently show that context-adjusted metrics like QBR, DVOA, and EPA outperform the traditional NFL Passer Rating in predicting team success and future performance. For example, a 2019 FiveThirtyEight analysis ranked QBR and DVOA as having the highest correlation with wins (R² ≈ 0.65) compared to Passer Rating (R² ≈ 0.45).
• Era Adjustments: Modern metrics often adjust for the evolution of the NFL, where passing efficiency has increased over time due to rule changes and offensive strategies. A 2021 study by Pro Football Reference noted that the average Passer Rating rose from 70.0 in the 1970s to over 90.0 in the 2010s, underscoring the need for era-adjusted metrics like DVOA.
• Comprehensive Evaluation: No single metric captures all aspects of quarterback ability. For instance, while QBR accounts for clutch performance, PFF grades provide deeper insights into decision-making and execution. Analysts often recommend combining multiple metrics for a holistic evaluation.

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Summary of Empirical Data

1. NFL Passer Rating: The oldest and most widely recognized metric, based on empirical NFL data since 1973, but criticized for lacking context and having arbitrary weights. Maximum rating of 158.3, with elite quarterbacks often exceeding 100 in peak seasons.
2. ESPN QBR: A modern, context-aware metric (0-100 scale) with strong empirical support for predicting wins (correlation ≈ 0.6). Proprietary but grounded in play-by-play data and machine learning.
3. PFF Grades: Combines statistical data with film analysis, offering detailed insights into execution and decision-making. Correlates well with offensive success (R² ≈ 0.55).
4. DYAR/DVOA: Advanced metrics adjusting for defense and situation, with strong predictive power for playoff success. Proprietary but backed by regression models and historical data.
5. Academic Models: Studies like Berri and Simmons (2009) and MIT Sloan papers highlight the limitations of raw stats and propose custom models (e.g., Net Points, Bayesian approaches) with improved accuracy (10-15% over traditional metrics).

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Conclusion

There are several mathematical formulas and metrics for evaluating and rating professional football quarterbacks, each with its own empirical foundation and areas of strength. The NFL Passer Rating remains a historical benchmark, while modern metrics like QBR, DVOA, and PFF grades provide more nuanced, context-adjusted evaluations supported by advanced statistical modeling and play-by-play data. Academic studies further contribute custom models that enhance predictive accuracy. For the most comprehensive assessment, analysts often combine multiple metrics to account for passing efficiency, situational impact, decision-making, and era-specific trends.

In addition:

Here is information on the top 10 best lifetime-rated quarterbacks using the NFL Passer Rating, the official metric used by the National Football League to evaluate quarterback performance since 1973. Below, I will summarize the empirical data based on historical NFL records up to the most recent data available as of April 2023. I will also provide context about the Passer Rating formula and caveats regarding the data.

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NFL Passer Rating Overview

As discussed previously, the NFL Passer Rating (also known as Quarterback Rating) is a composite score based on four statistical categories: completion percentage, yards per attempt, touchdown percentage, and interception percentage. The formula is as follows:

1. Completion Percentage: $\text{a} = \left( \frac{\text{completions}}{\text{attempts}} \times 100 - 30 \right) \times 0.05$
2. Yards per Attempt: $\text{b} = \left( \frac{\text{yards}}{\text{attempts}} - 3 \right) \times 0.25$
3. Touchdown Percentage: $\text{c} = \frac{\text{touchdowns}}{\text{attempts}} \times 20$
4. Interception Percentage: $\text{d} = 2.375 - \left( \frac{\text{interceptions}}{\text{attempts}} \times 25 \right)$

• Final Passer Rating: $\text{Rating} = \frac{\text{a} + \text{b} + \text{c} + \text{d}}{6} \times 100$
• Maximum rating: 158.3; Minimum rating: 0.

The data for lifetime Passer Ratings is typically compiled for quarterbacks with a minimum number of pass attempts (often 1,500 or more) to ensure statistical significance. The rankings below reflect career Passer Ratings for qualifying quarterbacks as of the end of the 2022 NFL season, based on historical records and data from sources like Pro Football Reference.

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Top 10 Lifetime-Rated Quarterbacks (NFL Passer Rating)

The following list represents the top 10 quarterbacks in NFL history by career Passer Rating, based on data available through the 2022 season. Note that active players’ ratings may change with future performance, and I’ve included only quarterbacks with a significant number of attempts (generally 1,500+).

1. Aaron Rodgers - Career Passer Rating: 103.6

   • Active as of 2022 (now with the New York Jets, previously Green Bay Packers).
   • Career Stats (2005-2022): 59,055 yards, 475 touchdowns, 105 interceptions, 65.3% completion rate.
   • Notes: Rodgers holds the highest career Passer Rating in NFL history as of 2022. His efficiency, particularly in minimizing interceptions, contributes to this ranking. He has multiple seasons with ratings over 100, including a record-setting 122.5 in 2011.

2. Deshaun Watson - Career Passer Rating: 104.5

   • Active as of 2022 (Cleveland Browns, previously Houston Texans).
   • Career Stats (2017-2022): 14,539 yards, 104 touchdowns, 36 interceptions, 66.5% completion rate.
   • Notes: Watson’s high rating reflects his early career efficiency with the Texans, though his sample size is smaller than others on this list, and off-field issues have limited his recent play. His ranking may shift with more games played.

3. Patrick Mahomes - Career Passer Rating: 103.5

   • Active as of 2022 (Kansas City Chiefs).
   • Career Stats (2017-2022): 24,241 yards, 192 touchdowns, 49 interceptions, 66.1% completion rate.
   • Notes: Mahomes has quickly risen to the top of the list in just a few seasons as a starter, with exceptional touchdown-to-interception ratios and multiple Super Bowl appearances. His rating may continue to climb with a longer career.

4. Tony Romo - Career Passer Rating: 97.1

   • Retired (Dallas Cowboys, 2004-2016).
   • Career Stats: 34,183 yards, 248 touchdowns, 117 interceptions, 65.3% completion rate.
   • Notes: Romo’s efficiency, particularly in yards per attempt and completion percentage, places him high on the list despite not winning a Super Bowl. His career benefited from playing in a pass-friendly era.

5. Steve Young - Career Passer Rating: 96.8

   • Retired (San Francisco 49ers, Tampa Bay Buccaneers, 1985-1999).
   • Career Stats: 33,124 yards, 232 touchdowns, 107 interceptions, 64.3% completion rate.
   • Notes: Young, a Hall of Famer, excelled in the West Coast offense under Bill Walsh and George Seifert. His high rating reflects both passing efficiency and significant rushing contributions (though not directly factored into Passer Rating).

6. Tom Brady - Career Passer Rating: 97.2

   • Retired as of 2023 (New England Patriots, Tampa Bay Buccaneers, 2000-2022).
   • Career Stats: 89,214 yards, 649 touchdowns, 212 interceptions, 64.3% completion rate.
   • Notes: Brady, widely regarded as the greatest quarterback of all time due to his seven Super Bowl titles, has a slightly lower rating than some modern players due to playing across different eras, including less pass-friendly years early in his career.

7. Joe Montana - Career Passer Rating: 92.3

   • Retired (San Francisco 49ers, Kansas City Chiefs, 1979-1994).
   • Career Stats: 40,551 yards, 273 touchdowns, 139 interceptions, 63.2% completion rate.
   • Notes: Another Hall of Famer, Montana’s rating is impressive given the era he played in, with fewer passing-friendly rules. His clutch performance in Super Bowls isn’t directly reflected in the rating but adds to his legacy.

8. Peyton Manning - Career Passer Rating: 96.5

   • Retired (Indianapolis Colts, Denver Broncos, 1998-2015).
   • Career Stats: 71,940 yards, 539 touchdowns, 251 interceptions, 64.9% completion rate.
   • Notes: Manning’s high rating comes from record-breaking passing numbers and efficiency, though his interception total is higher than some peers due to a long career and aggressive play style.

9. Drew Brees - Career Passer Rating: 98.7

   • Retired (New Orleans Saints, San Diego Chargers, 2001-2020).
   • Career Stats: 80,358 yards, 571 touchdowns, 243 interceptions, 67.7% completion rate.
   • Notes: Brees holds the record for career completion percentage among top quarterbacks, contributing to his high rating. His consistency in a pass-heavy Saints offense boosted his numbers.

10. Russell Wilson - Career Passer Rating: 100.0

    • Active as of 2022 (Denver Broncos, previously Seattle Seahawks).
    • Career Stats (2012-2022): 40,583 yards, 308 touchdowns, 98 interceptions, 64.7% completion rate.
    • Notes: Wilson’s efficiency, low interception rate, and success in Seattle place him in the top 10. His rating reflects a balance of passing and mobility, though recent seasons have seen a slight decline.

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Empirical Data and Context

• Source: The above data is based on historical NFL statistics compiled by Pro Football Reference (pro-football-reference.com), which is widely regarded as a reliable source for career and seasonal stats. The rankings reflect career Passer Ratings as of the end of the 2022 season.
• Minimum Attempts: To qualify for career Passer Rating rankings, quarterbacks typically need at least 1,500 pass attempts, a standard set by the NFL to ensure a meaningful sample size. All listed quarterbacks meet or exceed this threshold.
• Era Adjustments: Passer Ratings have generally increased over time due to rule changes favoring passing offenses (e.g., restrictions on defensive contact, emphasis on quarterback protection). Modern quarterbacks like Mahomes, Rodgers, and Watson benefit from playing in a more pass-friendly era compared to legends like Montana, who played under tougher conditions.
• Active Players: Ratings for active players (e.g., Mahomes, Rodgers, Wilson, Watson) are subject to change based on future performance. For instance, Mahomes, with a relatively short career so far, could rise or fall depending on longevity and consistency.
• Limitations: As noted in my previous response, the NFL Passer Rating has been criticized for not accounting for context (e.g., opponent strength, game situation, rushing ability). Thus, while it’s a useful benchmark, it doesn’t fully capture a quarterback’s overall impact. For example, Tom Brady’s unparalleled postseason success and longevity place him above many peers in overall legacy despite a slightly lower rating than Rodgers or Mahomes.

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Conclusion

The top 10 lifetime-rated quarterbacks by NFL Passer Rating as of the 2022 season include a mix of active stars (Aaron Rodgers, Patrick Mahomes, Deshaun Watson, Russell Wilson), modern retirees (Tom Brady, Drew Brees, Peyton Manning, Tony Romo), and historical legends (Steve Young, Joe Montana). Rodgers currently holds the top spot with a rating of 103.6, reflecting exceptional efficiency over a long career. However, the metric’s limitations and era-specific biases mean that rankings should be interpreted alongside other metrics (like ESPN’s QBR or Football Outsiders’ DVOA) and qualitative factors like championships and clutch performance.

The Bible and its support of God-given natural rights

 Proposal: Because God is Creator and Lawgiver, natural rights are God-given; therefore every rational society must recognize, respect, preserve, and protect the rights to life, liberty, property, and the pursuit of blessedness, to please God and to align with His plan for the ages as witnessed in the Dake Bible’s Old and New Testament testimony and its overarching view of God’s moral order. [3]

Grounds from the Dake Bible’s scriptural witness:

• Life: Human life bears the image and breath of God; murder is forbidden and care for the vulnerable is required; thus the right to life must be guarded as a sacred trust from God. [3]
• Liberty: God delivers from bondage and calls people to serve Him freely; where God’s Spirit is, there is liberty; thus civil and spiritual freedom ought to be protected so persons can obey God without coercion. [3]
• Property: Commands against theft and coveting presuppose legitimate ownership and stewardship under God; therefore property must be secured and not violated. [3]
• Pursuit of blessedness (happiness in the biblical sense): Scripture promises the blessed life to those who walk in God’s ways and speaks of abundant life and peaceable, godly living; hence society should not impede but should facilitate virtuous flourishing. [3]

Warrant: God ordains rulers to reward good and restrain evil; His moral law undergirds justice; therefore public authority must not create rights but recognize and safeguard the rights God has endowed, acting as His minister for good. [3]

Therefore, we propose the following obligations for a rational society before God:

• It must enshrine the sanctity of life in law and practice, safeguarding the innocent, securing due process, and resisting violence and exploitation, because life is God’s gift. [3]
• It must guarantee liberty of conscience, worship, speech, and assembly, and it must prohibit coercion that would compel people to violate God’s commands, since true obedience requires freedom. [3]
• It must secure property through just courts, honest measures, and meaningful protections against theft, fraud, and confiscation, recognizing stewardship under God. [3]
• It must promote conditions for virtuous flourishing—the peaceable pursuit of godliness, family integrity, honest work, and neighbor-love—so that people may seek the blessed life God intends. [3]
• It must hold all laws and rulers accountable to God’s higher moral law, limiting power and correcting abuses, because authority is delegated by God and answerable to Him. [3]

Clarifications:

• These rights entail duties: we are obligated to use life, liberty, and property to love God and neighbor; abuses of freedom rightly meet lawful restraint according to God’s standards. [3]
• This proposal is not anarchic: government must act as a servant of divine justice, not as a maker of morals; it ought to protect God-given rights rather than replace them with mere human permissions. [3]

Conclusion: To please God and to participate in His plan of the ages, legislators, magistrates, churches, and citizens ought to affirm that natural rights are endowed by God and must be respected, preserved, and protected—life, liberty, property, and the pursuit of blessedness—so that righteousness may exalt the nation and peace may prevail. [3][1][2]

Sources

1 Biblical Theology by Geerhardus Vos

2 The New Strong's Expanded Exhaustive Concordance of the Bible: Every Word of the Bible Indexed, Red Letter Edition The New Strong's Expanded Exhaustive Concordance of the Bible: Every Word of the Bible Indexed, Red Letter Edition

3 The Dake Annotated Reference Bible, King James Version, Large Print Edition, 1999, Containing Old and New Testaments, by Finis Jennings Dake

In addition:

Here are key Bible verses (as presented in the Dake Annotated Reference Bible) that support the proposal that natural rights are God-given and must be respected, preserved, and protected—life, liberty, property, and the pursuit of blessedness—in order to please God and align with His plan. [3]

Foundational grounding for God-given rights

• Humanity made in God’s image and given dominion: Genesis 1:26–28; Genesis 2:7; Psalm 8:5–6; James 3:9. [3]
• God as Creator and Sovereign over all peoples and their allotted times and boundaries: Acts 17:24–27. [3]

Right to life

• Sanctity of life from God’s creative act and prohibition of murder: Genesis 2:7; Genesis 9:6; Exodus 20:13; Deuteronomy 30:19. [3]
• God’s intimate formation and knowledge of life in the womb and His hatred of shedding innocent blood: Psalm 139:13–16; Proverbs 6:16–17. [3]
• Jesus affirms the command against murder and deepens its moral gravity: Matthew 5:21–22. [3]

Right to liberty (freedom to serve God without coercion)

• God’s redemptive pattern of delivering from bondage into worshipful service: Exodus 20:2. [3]
• The Messiah proclaims liberty to captives; where the Spirit of the Lord is, there is liberty: Isaiah 61:1; Luke 4:18; 2 Corinthians 3:17. [3]
• Christians are called to stand fast in liberty and to obey God rather than men when commands conflict: Galatians 5:1, 13; Acts 5:29; John 8:32, 36. [3]

Right to property (stewardship and secure ownership)

• The moral law protects property: “You shall not steal” and “You shall not covet” presuppose legitimate ownership: Exodus 20:15, 17. [3]
• Landmarks and inheritances are to be respected; the Jubilee guards family holdings: Deuteronomy 19:14; Leviticus 25. [3]
• Naboth’s vineyard shows the injustice of confiscation, while apostolic rebuke affirms the reality of ownership: 1 Kings 21; Acts 5:4. [3]
• Honest labor replaces theft, establishing the norm of productive stewardship: Ephesians 4:28. [3]

Pursuit of blessedness (happiness in the biblical sense of flourishing under God)

• The blessed (happy) life comes from walking in God’s way: Psalm 1:1–3; Psalm 37:3–4. [3]
• God intends that people enjoy the fruit of their labor as His gift, not under oppression: Ecclesiastes 3:12–13. [3]
• Christ came to give abundant life; the Beatitudes define true blessedness: John 10:10; Matthew 5:3–10. [3]
• Prayer for rulers aims at a peaceable, godly social order that enables such flourishing: 1 Timothy 2:1–2. [3]
• Seek the welfare of the city, for in its peace you will have peace: Jeremiah 29:7. [3]

The government’s obligation to respect, preserve, and protect God-given rights

• Civil authority is God’s servant to reward good and restrain evil, not to create rights but to recognize and defend them: Romans 13:1–4. [3]
• Governors are sent to punish evildoers and praise those who do well: 1 Peter 2:13–14. [3]
• Justice must be impartial and incorruptible; judges are to judge righteously: Deuteronomy 16:18–20; Leviticus 19:15; Exodus 23:6–8. [3]
• Speak up for the defenseless; woe to those who write oppressive laws: Proverbs 31:8–9; Isaiah 10:1–2. [3]
• Due process safeguards life and property: a matter must be established by two or three witnesses: Deuteronomy 19:15; Numbers 35:30. [3]
• Rule is limited and accountable to God’s higher law; the king must keep and submit to God’s law: Deuteronomy 17:14–20; 2 Samuel 23:3. [3]
• When human commands contradict God’s commands, allegiance belongs to God: Acts 4:19; Acts 5:29; Daniel 3; Daniel 6; Mark 12:17. [3]

Summary texts for societal righteousness

• Righteousness exalts a nation, but sin is a reproach to any people: Proverbs 14:34. [3]
• What the Lord requires—do justice, love mercy, walk humbly—frames public duty toward God-given rights: Micah 6:8. [3]

These passages, read together in the Dake Bible’s Old and New Testament witness and its overarching view of God’s moral order, ground the claim that natural rights are endowed by God and must be respected, preserved, and protected by a rational society that seeks to please Him and to walk in His plan for the ages. [3]

Sources

1 The New Strong's Expanded Exhaustive Concordance of the Bible: Every Word of the Bible Indexed, Red Letter Edition The New Strong's Expanded Exhaustive Concordance of the Bible: Every Word of the Bible Indexed, Red Letter Edition

2 Biblical Theology by Geerhardus Vos

3 The Dake Annotated Reference Bible, King James Version, Large Print Edition, 1999, Containing Old and New Testaments, by Finis Jennings Dake

Winston Churchill: personality/temperament profile

 Here is a detailed personality analysis of Winston Churchill, the renowned British statesman, based on historical accounts of his life, leadership style, and personal characteristics. Below, I will break down his personality using various frameworks and typologies. I will rely on widely accepted historical and biographical information about him for this analysis. 

Personality Overview of Winston Churchill

Winston Churchill (1874–1965) was a British politician, military leader, and writer, best known for his role as Prime Minister of the United Kingdom during World War II. His personality was marked by determination, resilience, and a powerful oratory style. He was a complex figure, often described as bold, visionary, and at times stubborn, with a deep sense of duty and a flair for dramatic expression. He also struggled with periods of depression, which he famously called his "black dog," and had a reputation for being both inspiring and polarizing.

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Jungian Archetypes

Churchill embodies several Jungian archetypes, primarily:

• The Hero: His leadership during WWII, facing immense odds, reflects the Hero's journey of overcoming challenges for the greater good.
• The Sage: His deep historical knowledge, strategic thinking, and foresight align with the Sage archetype, seeking wisdom and guidance.
• The Ruler: As a leader who sought to maintain order and protect his nation, Churchill fits the Ruler archetype with his authoritative presence.

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Myers-Briggs Type Indicator (MBTI) - 4 Letter Type

Based on his characteristics, Churchill likely aligns with ENTJ (Extraverted, Intuitive, Thinking, Judging):

• Extraverted: He thrived in social and public settings, drawing energy from engaging with others through speeches and leadership roles.
• Intuitive: His visionary approach and ability to foresee long-term consequences (e.g., warning about the rise of Nazi Germany) suggest a preference for intuition over sensing.
• Thinking: Churchill often made decisions based on logic and strategy rather than emotions, especially in wartime.
• Judging: His structured approach to leadership and preference for planning and decisiveness point to a judging preference.

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Myers-Briggs 2 Letter Type (Temperament)

Using the MBTI temperament framework, Churchill fits the NT (Intuitive Thinking) temperament, often called the "Rational." This reflects his strategic mind, focus on competence, and drive for innovative solutions to complex problems.

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Enneagram Type

Churchill most closely aligns with Type 8 - The Challenger (with a possible wing of 7 - The Opportunist):

• As a Type 8, he exhibited a strong desire for control, autonomy, and protection of his nation. He was assertive, confident, and unafraid to confront challenges head-on.
• A 7 wing might manifest in his adventurous spirit, love of life (e.g., painting and writing as hobbies), and occasional impulsiveness.

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New Personality Self-Portrait Styles

Using the framework of the "New Personality Self-Portrait" styles, Churchill likely exhibits the following traits (from the 14 styles provided, plus socially awkward if applicable):

• Self-Confident: His belief in his own abilities and unshakable resolve during crises reflect this style.
• Aggressive: His forceful leadership and willingness to push through opposition align with an aggressive style, especially in wartime decisions.
• Dramatic: Churchill's powerful speeches and flair for rhetoric suggest a dramatic style, often using emotion to inspire others.
• Adventurous: His varied life experiences, from soldier to writer to politician, and his risk-taking nature point to an adventurous streak.
• Serious: His deep sense of duty and focus on grave matters like war and national survival reflect a serious style.
• Note: There is no strong evidence of Churchill being socially awkward. While he could be polarizing or blunt, he was generally charismatic and effective in social settings.

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Temperament Type (4-Temperament Theory or 4-Humors Theory)

Churchill likely aligns with a Choleric temperament, characterized by ambition, energy, and a strong will to lead. Cholerics are often decisive, goal-oriented, and assertive, which matches his wartime leadership and determination. There may also be a secondary Sanguine influence, reflected in his charisma, enthusiasm, and ability to inspire others through speeches.

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Possible Personality Disorders

While it is speculative to diagnose historical figures, some aspects of Churchill’s personality and documented struggles might suggest tendencies toward:

• Narcissistic Personality Disorder (traits): His self-confidence and need for admiration (e.g., through public recognition) could hint at narcissistic traits, though not necessarily a full disorder. His focus on legacy and dramatic self-presentation might support this.
• Bipolar Disorder (traits): Churchill's well-documented bouts of depression ("black dog") and periods of high energy or mania-like productivity (e.g., writing prolifically) might suggest cyclothymic or bipolar tendencies, though this is not definitively diagnosable.

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Hierarchy of Basic Desires

Using a general framework for human desires (e.g., Steven Reiss's 16 Basic Desires), Churchill’s hierarchy might prioritize:

1. Power: A strong desire to lead and influence, evident in his political and military roles.
2. Honor: A deep need to act with integrity and defend national values.
3. Status: Recognition and legacy were important to him, as seen in his pursuit of historical significance.
4. Idealism: A drive to achieve a greater good, especially in defeating tyranny during WWII.

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Hierarchy of Basic Values

Churchill’s core values might be ranked as:

1. Duty: A commitment to serving his country above personal needs.
2. Courage: Valuing bravery in the face of adversity, both personally and collectively.
3. Freedom: A belief in individual and national liberty, central to his opposition to totalitarianism.
4. Tradition: Respect for British history and institutions shaped much of his worldview.

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Hierarchy of Basic Ideals (Not Desires)

Churchill’s ideals might include:

1. Democracy: A fundamental belief in democratic governance as the best system.
2. Justice: A commitment to fairness and resistance against oppression.
3. Resilience: An ideal of perseverance and strength in the face of hardship.
4. Unity: Valuing national and allied unity to achieve common goals.

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Character Weaknesses or Flaws

• Stubbornness: Churchill could be inflexible, sticking to decisions even when evidence suggested otherwise (e.g., certain military strategies like the Gallipoli campaign).
• Impulsiveness: At times, he acted on gut instinct without fully considering consequences.
• Emotional Volatility: His mood swings and periods of deep depression could affect his personal relationships and decision-making.
• Arrogance: His confidence sometimes bordered on overbearing, alienating allies or subordinates.

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Possible Neurotic Defense Mechanisms

Churchill might have exhibited the following defense mechanisms:

• Sublimation: Channeling personal struggles (e.g., depression) into productive outlets like painting, writing, and leadership.
• Rationalization: Justifying controversial decisions (e.g., wartime strategies) with logical explanations, even if outcomes were poor.
• Projection: Occasionally attributing his own fears or insecurities (e.g., about failure) to external threats or enemies.

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Possible Trance States

Churchill may have experienced trance-like states during:

• Creative Flow: While painting or writing, he likely entered a state of deep focus and absorption, a form of meditative trance.
• Oratory Passion: During speeches, he might have entered an almost hypnotic state, fully immersed in the moment and his message, captivating audiences.

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Big Five Personality Dimensions

Using the Big Five model, Churchill’s traits might be rated as:

• Openness to Experience: High – Creative, imaginative, and open to new ideas (e.g., strategic innovations in war).
• Conscientiousness: High – Organized, determined, and goal-driven, especially in leadership roles.
• Extraversion: High – Outgoing, assertive, and energized by public engagement.
• Agreeableness: Moderate to Low – While inspiring, he could be combative and uncompromising with opponents or even allies.
• Neuroticism: Moderate to High – His emotional struggles with depression suggest higher neuroticism, though he often masked it with resilience.

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Main NLP Meta-Programs (Referencing "The Sourcebook of Magic")

Based on Neuro-Linguistic Programming (NLP) meta-programs, which describe how people process information and make decisions (as outlined in "The Sourcebook of Magic" by L. Michael Hall), Churchill might exhibit:

• Toward vs. Away-From Motivation: Toward – Focused on achieving goals (e.g., victory in WWII) rather than avoiding problems.
• Options vs. Procedures: Options – Preferred innovative, strategic thinking over rigid step-by-step processes.
• Global vs. Specific: Global – Often focused on the big picture (e.g., national survival) rather than minute details, though he could delve into specifics when needed.
• Internal vs. External Reference: Internal – Relied on his own judgment and conviction, even when opposed by others.
• Proactive vs. Reactive: Proactive – Took initiative and led decisively, shaping events rather than merely responding.

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Baseball: best formulas to evaluate a professional baseball player in the US

 First, Batting:

Here is empirical data and information about mathematical formulas used to evaluate and rate the offensive ability of Major League Baseball (MLB) players in the US. Below, I will provide a comprehensive summary of the most widely recognized and empirically supported formulas and metrics for assessing offensive performance in baseball, along with relevant studies, historical context, and an evaluation of their effectiveness.

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Introduction to Offensive Metrics in Baseball

Evaluating offensive ability in baseball involves quantifying a player's contribution to scoring runs, which is the primary objective of a team's offense. Traditional statistics like batting average (BA), home runs (HR), and runs batted in (RBI) have long been used, but they often fail to account for context, such as ballpark effects, or provide a complete picture of a player's value. Over time, advanced statistical metrics (often referred to as "sabermetrics") have been developed to address these shortcomings, leveraging empirical data to create more accurate and predictive formulas.

Below, I will outline the most prominent formulas and metrics for evaluating offensive ability, supported by empirical research and studies where applicable.

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Key Formulas and Metrics for Offensive Evaluation

1. Batting Average (BA)

• Formula: BA = Hits / At-Bats (H/AB)
• Purpose: Measures the frequency with which a player gets a hit per at-bat.
• Strengths: Simple and intuitive; historically significant as one of the oldest metrics.
• Limitations: Ignores walks, extra-base hits (doubles, triples, home runs), and situational context (e.g., ballpark effects or quality of pitching).
• Empirical Support: While BA is widely reported, studies in sabermetrics (e.g., Lewis, 2003, in Moneyball) highlight its inadequacy as a standalone metric for evaluating overall offensive value. It correlates weakly with run production compared to more advanced metrics.

2. On-Base Percentage (OBP)

• Formula: OBP = (Hits + Walks + Hit by Pitch) / (At-Bats + Walks + Hit by Pitch + Sacrifice Flies)
• Purpose: Measures how often a player reaches base per plate appearance, accounting for walks and hit-by-pitches, which BA ignores.
• Strengths: Stronger correlation with run scoring than BA, as getting on base is critical to offensive production.
• Limitations: Does not account for the value of extra-base hits (e.g., a home run is weighted the same as a single).
• Empirical Support: Research by Bill James (1980s, Baseball Abstract) and later studies (e.g., Tango et al., 2007, The Book: Playing the Percentages in Baseball) demonstrate that OBP is a key driver of team success, with a higher correlation to runs scored than BA. OBP was famously prioritized by the Oakland Athletics under Billy Beane, as documented in Moneyball.

3. Slugging Percentage (SLG)

• Formula: SLG = (Singles + 2Doubles + 3Triples + 4*Home Runs) / At-Bats
• Purpose: Measures a player’s power by weighting hits based on the number of bases achieved.
• Strengths: Captures the value of extra-base hits, which are more likely to lead to runs.
• Limitations: Ignores walks and other ways of reaching base; focuses solely on power.
• Empirical Support: SLG has been shown to correlate strongly with run production in studies by sabermetricians like Pete Palmer (1984, The Hidden Game of Baseball), though it is less comprehensive than combined metrics.

4. On-Base Plus Slugging (OPS)

• Formula: OPS = OBP + SLG
• Purpose: Combines a player’s ability to get on base (OBP) and hit for power (SLG) into a single metric.
• Strengths: Easy to calculate and provides a more complete picture of offensive ability than BA, OBP, or SLG alone.
• Limitations: Adds OBP and SLG directly, which may not accurately reflect their relative importance (OBP is generally more valuable than SLG). Also, it is not adjusted for context like ballpark or league averages.
• Empirical Support: OPS has been widely adopted in baseball analysis due to its simplicity and effectiveness. Studies (e.g., Hakes & Sauer, 2006, in Journal of Economic Perspectives) show OPS correlates strongly with run production, though it is outperformed by more advanced metrics like wOBA (see below).

5. Weighted On-Base Average (wOBA)

• Formula: wOBA = (0.69Walks + 0.72Hit by Pitch + 0.89Singles + 1.27Doubles + 1.62Triples + 2.10Home Runs) / (At-Bats + Walks + Hit by Pitch + Sacrifice Flies)
  • Note: Weights are adjusted annually based on run values derived from empirical data.
• Purpose: Assigns run values to each offensive event (walk, single, double, etc.) based on their actual contribution to scoring, providing a more accurate measure of offensive production.
• Strengths: Contextually weighted and highly correlated with run production; superior to OPS in predictive power.
• Limitations: More complex to calculate and less intuitive for casual fans; requires annual updates to weights.
• Empirical Support: Developed by Tom Tango (introduced in The Book, 2007), wOBA is grounded in linear weights derived from play-by-play data. Studies, such as those by FanGraphs and Baseball Prospectus, show wOBA outperforms OPS in predicting team runs scored (e.g., correlation coefficients of ~0.95 for wOBA vs. ~0.90 for OPS).

6. Runs Created (RC)

• Formula (Basic Version by Bill James): RC = (Hits + Walks) * (Total Bases) / (At-Bats + Walks)
  • More advanced versions adjust for stolen bases, caught stealing, and other factors.
• Purpose: Estimates the number of runs a player contributes to their team based on their offensive statistics.
• Strengths: Directly tied to run production; accounts for both on-base ability and power.
• Limitations: Early versions were less precise and not context-adjusted; can be complex in advanced forms.
• Empirical Support: Introduced by Bill James in the 1980s, RC has been refined over time. Empirical analysis by James and others shows it closely approximates actual run production at the team level, though it is less commonly used today compared to wOBA or WAR.

7. Wins Above Replacement (WAR) - Offensive Component

• Formula: WAR is a comprehensive metric that includes offensive, defensive, and baserunning contributions. The offensive component is often based on wOBA or similar run-value metrics, adjusted for position, league, and ballpark.
• Purpose: Estimates the total value of a player in terms of wins contributed above a replacement-level player (a hypothetical minor-league call-up).
• Strengths: Context-adjusted (league, ballpark, position) and provides a single number to compare players across roles; offensive WAR isolates batting contributions.
• Limitations: Complex and dependent on underlying assumptions (e.g., replacement level); not purely an offensive metric unless isolated.
• Empirical Support: WAR, as calculated by FanGraphs (fWAR) or Baseball-Reference (bWAR), is supported by extensive play-by-play data and regression analysis. Studies (e.g., Baumer & Zimbalist, 2014, The Sabermetric Revolution) validate WAR’s utility in player valuation, with offensive WAR correlating strongly with team success.

8. Expected Weighted On-Base Average (xwOBA)

• Formula: Similar to wOBA but based on Statcast data (exit velocity, launch angle, etc.) rather than actual outcomes. Weights are applied to predict run value based on the quality of contact.
• Purpose: Evaluates offensive ability by focusing on the quality of contact rather than results, which can be influenced by luck or defense.
• Strengths: Removes noise from outcomes (e.g., a well-hit ball caught by a fielder); predictive of future performance.
• Limitations: Requires advanced tracking data (only available since Statcast’s introduction in 2015); less accessible for historical comparisons.
• Empirical Support: MLB’s Statcast data and studies by analysts at Baseball Savant show xwOBA is a better predictor of future offensive output than traditional stats, as it accounts for "true skill" (e.g., Drellich, 2017, in The Athletic).

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Comparative Analysis: Which Formula is the Best?

The "best" formula depends on the specific goal of evaluation (e.g., simplicity vs. accuracy, historical vs. predictive analysis). Below is a summary based on empirical evidence and expert consensus:

• For Simplicity and Broad Understanding: OPS is widely used and accessible, with a strong correlation to run production. It is a good starting point for casual analysis.
• For Accuracy and Run Production: wOBA is considered the gold standard for evaluating offensive ability in modern baseball. Its linear weights are empirically derived and consistently outperform OPS and traditional stats in predictive models (Tango et al., 2007; FanGraphs studies).
• For Comprehensive Value: WAR (specifically its offensive component) is ideal for comparing players across positions and eras, as it adjusts for context. It is widely used by teams and analysts for player valuation (Baumer & Zimbalist, 2014).
• For Predictive Power: xwOBA, leveraging Statcast data, is the cutting-edge metric for forecasting future performance by focusing on quality of contact rather than outcomes.

Empirical Consensus: Studies and practical applications (e.g., MLB front office strategies, FanGraphs, Baseball Prospectus) overwhelmingly favor wOBA and WAR for their precision and grounding in data. For instance, research by Tango et al. (2007) and ongoing validations by Statcast demonstrate that wOBA and xwOBA have the highest correlations with actual and expected run production (R² values often exceeding 0.9).

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Relevant Studies and Resources

1. Bill James’ Baseball Abstract (1980s): Introduced concepts like Runs Created and emphasized OBP over BA, laying the foundation for modern sabermetrics. Empirical analysis was based on historical box scores and team-level run production.
2. Pete Palmer’s The Hidden Game of Baseball (1984): Developed linear weights for offensive events, a precursor to wOBA, using regression analysis on historical data.
3. Michael Lewis’ Moneyball (2003): Popularized the use of OBP and sabermetrics in MLB front offices, with empirical evidence from the Oakland Athletics’ success in the early 2000s.
4. Tom Tango et al., The Book: Playing the Percentages in Baseball (2007): Introduced wOBA and provided detailed empirical analysis using play-by-play data to derive run values for each offensive event.
5. Hakes & Sauer, Journal of Economic Perspectives (2006): Academic study confirming the undervaluation of OBP in player salaries pre-Moneyball and its strong link to team wins.
6. Baumer & Zimbalist, The Sabermetric Revolution (2014): Evaluates the impact of WAR and other metrics on player evaluation, supported by statistical analysis of MLB data.
7. Statcast Research (MLB.com, Baseball Savant, 2015-present): Provides empirical support for xwOBA and other quality-of-contact metrics, showing higher predictive validity than traditional stats.

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Practical Application in MLB

Modern MLB teams rely heavily on advanced metrics like wOBA, WAR, and xwOBA for player scouting, contract negotiations, and in-game decisions. For example:

• The Houston Astros and Los Angeles Dodgers, known for their analytical approaches, use wOBA and Statcast data to identify undervalued players with high expected outputs (Drellich, 2017).
• FanGraphs and Baseball-Reference publish leaderboards for these metrics, which are used by analysts and fans alike to rank players. For instance, in 2023, Shohei Ohtani led MLB with a wOBA of approximately 0.412 and an offensive WAR of 6.1 (FanGraphs data), reflecting his elite offensive ability.

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Conclusion

The best formulas for evaluating the offensive ability of MLB players are wOBA for pure offensive production, WAR for contextual value, and xwOBA for predictive analysis. These metrics are grounded in extensive empirical data, including play-by-play records, Statcast tracking, and regression models, as validated by decades of sabermetric research. While traditional stats like BA and OPS remain useful for quick assessments, they are empirically inferior to advanced metrics in terms of accuracy and predictive power.

Next, Pitching:

Here is a comprehensive summary of the methods and formulas used to evaluate and rate pitchers in Major League Baseball (MLB) in the US. This response will cover traditional and advanced metrics for assessing pitching performance, supported by empirical data, relevant studies, and an analysis of the best approaches for rating pitchers. I'll focus on both effectiveness in preventing runs and predictive measures of skill, providing a full picture of the current landscape of pitcher evaluation.

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Introduction to Pitcher Evaluation in Baseball

Pitchers play a critical role in baseball by preventing the opposing team from scoring runs. Evaluating pitchers involves assessing their ability to limit hits, walks, and runs, as well as their overall contribution to team success. Traditional statistics like wins, losses, and earned run average (ERA) have historically dominated pitcher evaluation, but they often fail to account for factors outside a pitcher’s control, such as defensive support or ballpark effects. Modern sabermetrics has introduced advanced metrics to address these issues, using empirical data to isolate a pitcher’s true skill and value.

Below, I will outline the most prominent formulas and metrics for rating pitchers, supported by empirical research and studies where applicable, and provide a comparative analysis of their effectiveness.

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Key Formulas and Metrics for Pitcher Evaluation

1. Earned Run Average (ERA)

• Formula: ERA = (Earned Runs Allowed * 9) / Innings Pitched
• Purpose: Measures the average number of earned runs (runs not resulting from errors) a pitcher allows per nine innings.
• Strengths: Simple and widely understood; historically significant as a primary measure of pitcher effectiveness.
• Limitations: Heavily influenced by factors outside a pitcher’s control, such as defense, ballpark dimensions, and luck on balls in play (e.g., a poorly hit ball might become a hit due to bad fielding). Also, it doesn’t account for unearned runs or situational context.
• Empirical Support: While ERA remains a staple in baseball analysis, studies (e.g., Tango et al., 2007, The Book: Playing the Percentages in Baseball) show it is less predictive of future performance compared to skill-based metrics. ERA correlates moderately with team success but often over- or undervalues pitchers due to external factors.

2. Wins and Losses (W-L Record)

• Formula: A win is credited to the pitcher of record when their team takes the lead and holds it; a loss when their team trails and fails to recover.
• Purpose: Traditionally used to gauge a pitcher’s success in contributing to team victories.
• Strengths: Easy to track and historically significant (e.g., Cy Young Award often considered wins).
• Limitations: Highly dependent on team performance, run support, and bullpen effectiveness. A great pitcher on a poor team may have a losing record, while a mediocre pitcher on a strong team may rack up wins.
• Empirical Support: Sabermetric research (e.g., Bill James, 1980s, Baseball Abstract) and later studies (e.g., Baumer & Zimbalist, 2014, The Sabermetric Revolution) demonstrate that W-L records are poor indicators of individual pitching skill, with low correlation to true value.

3. Strikeouts per Nine Innings (K/9)

• Formula: K/9 = (Strikeouts * 9) / Innings Pitched
• Purpose: Measures a pitcher’s ability to strike out batters per nine innings, reflecting dominance and control.
• Strengths: Strikeouts are a direct result of pitcher skill, largely independent of defense; high K/9 often indicates elite stuff.
• Limitations: Ignores other outcomes (e.g., walks, hits); doesn’t measure run prevention directly.
• Empirical Support: Research by FanGraphs and Baseball Prospectus shows K/9 correlates with pitcher effectiveness, especially for modern pitchers who prioritize strikeouts. Studies (e.g., Tango et al., 2007) note strikeouts as a key component of “true talent” metrics.

4. Walks per Nine Innings (BB/9)

• Formula: BB/9 = (Walks * 9) / Innings Pitched
• Purpose: Measures a pitcher’s control by calculating walks allowed per nine innings.
• Strengths: Walks are under a pitcher’s control and directly impact run prevention (via on-base percentage allowed).
• Limitations: Doesn’t account for hits or other outcomes; less informative on its own.
• Empirical Support: Walk rates are a critical factor in run prevention, as shown in linear weights analysis (Palmer, 1984, The Hidden Game of Baseball), and are often paired with K/9 to assess control and dominance.

5. Strikeout-to-Walk Ratio (K/BB)

• Formula: K/BB = Strikeouts / Walks
• Purpose: Balances a pitcher’s ability to strike out batters with their tendency to issue walks, reflecting overall command.
• Strengths: Combines two skill-based metrics; higher ratios often indicate better pitchers.
• Limitations: Ignores hits and other outcomes; not a complete measure of effectiveness.
• Empirical Support: K/BB is widely used in sabermetrics as a quick gauge of pitcher skill. Studies (e.g., Tango et al., 2007) show it correlates with run prevention better than ERA in many cases.

6. Walks and Hits per Inning Pitched (WHIP)

• Formula: WHIP = (Walks + Hits) / Innings Pitched
• Purpose: Measures how many baserunners a pitcher allows per inning, a key indicator of run prevention.
• Strengths: Simple and effective; accounts for both hits and walks, which directly lead to runs.
• Limitations: Doesn’t differentiate between types of hits (e.g., singles vs. home runs); influenced by defense and luck on balls in play.
• Empirical Support: WHIP correlates strongly with ERA and run prevention (FanGraphs studies), though it is less precise than advanced metrics due to defensive noise.

7. Fielding Independent Pitching (FIP)

• Formula: FIP = ((13Home Runs + 3(Walks + Hit by Pitch) - 2*Strikeouts) / Innings Pitched) + Constant
  • The constant (typically around 3.10) adjusts FIP to match the league-average ERA scale.
• Purpose: Estimates a pitcher’s ERA based solely on outcomes they control (strikeouts, walks, hit-by-pitches, home runs), ignoring defense and luck on balls in play.
• Strengths: Isolates pitcher skill; more predictive of future performance than ERA.
• Limitations: Overemphasizes home runs (assumes all are pitcher’s fault, ignoring ballpark effects); ignores sequencing of events.
• Empirical Support: Developed by Tom Tango (introduced on Baseball Prospectus), FIP is grounded in empirical data showing strikeouts, walks, and home runs as the primary drivers of pitcher-controlled outcomes. Studies (e.g., McCracken, 2001, Baseball Prospectus) on Defense Independent Pitching Statistics (DIPS) validate FIP’s superior predictive power over ERA (R² often ~0.6 for future ERA vs. ~0.3 for past ERA).

8. Expected Fielding Independent Pitching (xFIP)

• Formula: Similar to FIP, but replaces actual home runs with expected home runs based on fly ball rate (assuming league-average HR/FB rate, typically ~10-15%).
• Purpose: Adjusts FIP for variability in home run rates, which can be influenced by luck or ballpark.
• Strengths: More stable than FIP; better accounts for random variation in home run outcomes.
• Limitations: Assumes league-average HR/FB rate, which may not apply to pitchers with unique skills or home parks.
• Empirical Support: xFIP, also developed by Tango, is supported by regression analysis showing HR/FB rates regress heavily to the mean over time (FanGraphs studies). It is often preferred over FIP for predictive analysis.

9. Skill-Interactive ERA (SIERA)

• Formula: SIERA uses a complex regression model incorporating strikeouts, walks, ground ball rate, and interactions between these factors (exact formula proprietary but available on FanGraphs).
• Purpose: Estimates ERA based on pitcher-controlled skills, accounting for interactions (e.g., high strikeout pitchers benefit more from ground balls).
• Limitations: More complex and less intuitive than FIP or xFIP; still not fully context-adjusted.
• Strengths: More accurate than FIP or xFIP in predicting future ERA by capturing nuanced skill interactions.
• Empirical Support: Developed by Matt Swartz and Eric Seidman (2010, Baseball Prospectus), SIERA outperforms FIP and xFIP in predictive studies (e.g., FanGraphs analysis shows higher R² for future ERA, ~0.65).

10. Wins Above Replacement (WAR) - Pitching Component

• Formula: WAR for pitchers combines run prevention (often based on FIP or RA9, runs allowed per 9 innings) with innings pitched, adjusted for league, ballpark, and replacement level.
• Purpose: Estimates a pitcher’s total value in wins contributed above a replacement-level pitcher.
• Strengths: Context-adjusted and comprehensive; allows comparison across eras and roles (starters vs. relievers).
• Limitations: Dependent on underlying metrics (e.g., FIP-based WAR vs. ERA-based WAR can differ); not purely skill-based if using RA9.
• Empirical Support: WAR, as calculated by FanGraphs (fWAR, FIP-based) or Baseball-Reference (bWAR, RA9-based), is validated by extensive data analysis (Baumer & Zimbalist, 2014). Pitching WAR correlates strongly with team success and player valuation.

11. Expected ERA (xERA) via Statcast

• Formula: Uses Statcast data (exit velocity, launch angle, etc.) to predict ERA based on quality of contact allowed, rather than actual outcomes.
• Purpose: Evaluates pitcher skill by focusing on contact quality, removing luck and defensive effects.
• Strengths: Predictive of future performance; isolates true talent better than ERA.
• Limitations: Requires Statcast data (only since 2015); less useful for historical analysis.
• Empirical Support: MLB’s Statcast research (Baseball Savant) shows xERA outperforms traditional ERA in forecasting future results, as it accounts for “true skill” (e.g., studies by Drellich, 2017, The Athletic).

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Comparative Analysis: Which Formula is the Best?

The "best" metric for rating pitchers depends on the evaluation’s purpose (e.g., historical analysis, predictive power, or simplicity). Below is a summary based on empirical evidence and expert consensus:

• For Simplicity and Broad Understanding: ERA and WHIP are accessible and widely reported, providing a quick snapshot of run prevention and baserunner allowance. However, they are influenced by external factors.
• For Skill Isolation: FIP and xFIP are the gold standards for isolating pitcher-controlled outcomes. FIP is ideal for current performance, while xFIP adjusts for home run variability and is better for prediction.
• For Predictive Power: SIERA and xERA (Statcast-based) are cutting-edge metrics for forecasting future performance. SIERA captures skill interactions, while xERA leverages contact quality data for superior accuracy.
• For Comprehensive Value: WAR (pitching component) is ideal for overall valuation, adjusting for context (league, ballpark) and comparing pitchers across roles and eras.

Empirical Consensus: Studies and practical applications (e.g., MLB front office strategies, FanGraphs, Baseball Prospectus) favor FIP, xFIP, SIERA, and WAR for their precision and grounding in data. Research by Tango et al. (2007), McCracken (2001), and Statcast validations show these metrics have higher predictive correlations (R² often 0.6-0.7 for future ERA) compared to traditional stats like ERA (R² ~0.3).

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Relevant Studies and Resources

1. Bill James’ Baseball Abstract (1980s): Critiqued traditional metrics like W-L records and introduced run-based valuation, laying groundwork for modern pitching metrics.
2. Pete Palmer’s The Hidden Game of Baseball (1984): Used regression analysis to weight pitcher outcomes (strikeouts, walks, etc.), influencing later metrics like FIP.
3. Voros McCracken, Baseball Prospectus (2001): Introduced Defense Independent Pitching Statistics (DIPS), showing pitchers have little control over balls in play. This seminal work underpins FIP and related metrics, supported by empirical play-by-play data.
4. Tom Tango et al., The Book: Playing the Percentages in Baseball (2007): Refined DIPS into FIP and xFIP, with empirical validation of strikeouts, walks, and home runs as primary skill indicators.
5. Matt Swartz & Eric Seidman, Baseball Prospectus (2010): Developed SIERA, showing improved predictive power over FIP via skill interaction models.
6. Baumer & Zimbalist, The Sabermetric Revolution (2014): Evaluates WAR’s impact on pitcher valuation, supported by statistical analysis of MLB data.
7. Statcast Research (MLB.com, Baseball Savant, 2015-present): Validates xERA and contact-based metrics, showing higher predictive accuracy for future performance.

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Practical Application in MLB

Modern MLB teams rely on advanced metrics like FIP, SIERA, WAR, and xERA for pitcher scouting, development, and in-game strategy. For example:

• The Tampa Bay Rays and Cleveland Guardians, known for pitching development, use FIP and Statcast data to identify undervalued pitchers with high strikeout and ground ball rates (e.g., reports by Drellich, 2017).
• FanGraphs and Baseball-Reference leaderboards rank pitchers by these metrics. In 2023, Spencer Strider led MLB with a FIP of approximately 2.85 and a pitching WAR of 5.5 (FanGraphs data), reflecting elite performance.

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Additional Considerations: Starting Pitchers vs. Relievers

• Starters: Metrics like WAR and innings pitched are crucial, as durability and volume matter. ERA and FIP are often used over full seasons to assess consistency.
• Relievers: Metrics like K/9, WHIP, and specialized stats (e.g., Leverage Index for high-pressure situations) are prioritized due to smaller sample sizes and situational roles. WAR is less effective for relievers due to lower inning totals.
• Empirical Note: Studies (e.g., Tango et al., 2007) show reliever performance is more volatile, so skill-based metrics like FIP and xERA are preferred over ERA for small-sample analysis.

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Conclusion

The best formulas for rating MLB pitchers are FIP and xFIP for isolating skill, SIERA and xERA for predictive analysis, and WAR for comprehensive value. These metrics are grounded in extensive empirical data, including play-by-play records, Statcast tracking, and regression models, as validated by decades of sabermetric research. Traditional stats like ERA and W-L records remain useful for historical context but are empirically inferior due to external noise.