Chapter 15: Player Performance Metrics

In Chapter 14 we examined how to evaluate teams as collective units. Now we turn the lens inward and ask a more granular question: How good is this individual player, and how do we measure it?

Player evaluation is one of the oldest pursuits in sport. Scouts, coaches, and journalists have always formed opinions about individual talent. What modern analytics adds is structure, repeatability, and context. A well-designed metric does not replace expert judgment---it sharpens it by stripping away narrative bias and anchoring discussion in evidence.

This chapter builds a complete toolkit for player performance measurement. We begin with the philosophy of comprehensive evaluation, move through position-specific metrics, tackle the statistical pitfalls of playing time, model how performance changes with age, measure form and consistency, assemble composite profiles, and finish with algorithms that answer the question every recruitment department asks: Who else plays like this player? We also address two of the most challenging problems in the field: isolating individual contributions from team effects, and building unified valuation frameworks such as WAR and VAEP.

15.1 Comprehensive Player Evaluation

15.1.1 Why Single Metrics Fail

The most common way fans evaluate players is through goals and assists. While these are important outputs, they suffer from several well-documented problems:

1. Role blindness. A holding midfielder who shields the back four brilliantly will never top a goals chart.
2. Team dependency. A striker's goal tally depends heavily on the quality of service from teammates.
3. Sample size volatility. Goals are rare events; a player who scores 5 in 10 matches may simply have experienced positive variance relative to a player who scored 2 in 10.
4. Context ignorance. Scoring a consolation goal in a 4-1 defeat is not the same as scoring a match-winner.

Intuition: Think of player evaluation like a medical checkup. A doctor does not diagnose your health from blood pressure alone---they measure heart rate, cholesterol, blood oxygen, reflexes, and more. Similarly, a single metric gives a dangerously incomplete picture of a soccer player.

The temptation to rely on a single number is strong. Media narratives thrive on simplicity: "Player X scored 20 goals, so Player X had a better season than Player Y who scored 15." But consider that Player Y may have created 15 assists, pressed with elite intensity, and contributed defensively---contributions that a goals-only analysis completely ignores. The history of soccer analytics is, in many ways, a history of moving beyond single-metric evaluation toward increasingly holistic frameworks.

15.1.2 The Multi-Dimensional Framework

Modern player evaluation operates along multiple dimensions simultaneously. A useful taxonomy divides player contributions into four broad categories:

Within each dimension we can define volume metrics (how much?) and efficiency metrics (how well?). A complete evaluation considers both.

$$ \text{Player Value} \approx f(\text{Scoring}, \text{Creation}, \text{Possession}, \text{Defense}, \text{Context}) $$

The function $f$ is what the rest of this chapter seeks to define---not as a single formula, but as a flexible framework adaptable to the question being asked.

Callout --- The Importance of Context: Two players with identical per-90 metrics may have very different true abilities if one plays for a dominant team that controls 65% of possession and the other plays for a relegation-threatened team that defends deep. Context-aware evaluation adjusts for team quality, league strength, tactical system, and game state. We return to this challenge in Section 15.9.

15.1.3 Raw Counts vs. Rates vs. Percentiles

Every metric can be expressed in three forms:

• Raw count: "Player A made 47 interceptions this season."
• Rate (per 90): "Player A makes 3.1 interceptions per 90 minutes."
• Percentile rank: "Player A is in the 89th percentile for interceptions per 90 among centre-backs in the top five leagues."

Each form has its use. Raw counts reward durability and availability. Rates allow comparison across different playing times. Percentile ranks contextualize a player against a relevant peer group.

import numpy as np
from scipy import stats

def compute_percentile(player_value: float, population: np.ndarray) -> float:
    """Compute the percentile rank of a player within a population.

    Args:
        player_value: The player's metric value.
        population: Array of metric values for the comparison group.

    Returns:
        Percentile rank between 0 and 100.
    """
    return stats.percentileofscore(population, player_value, kind="rank")

Common Pitfall: Percentile ranks are only meaningful when the comparison group is well-defined. Comparing a centre-back's interceptions against all outfield players inflates their ranking because attackers rarely intercept. Always filter by position and league level first.

15.1.4 Choosing a Comparison Group

The choice of comparison group---sometimes called the peer set---is one of the most consequential decisions in player evaluation. Common peer sets include:

• All players in the same league and season
• All players in the same position within that league
• All players in the same position across the top five European leagues
• All players in the same age bracket and position

A midfielder who ranks in the 70th percentile among all Premier League midfielders might rank in the 95th percentile among Championship midfielders. Neither ranking is "wrong," but they answer different questions.

When building peer sets for recruitment purposes, analysts typically apply several filters simultaneously: position, age range, league tier, and minimum minutes played. The result is a carefully curated comparison group that answers the specific question at hand---for example, "Among all central midfielders aged 21-25 in the top two tiers of European football who have played at least 1,500 minutes this season, where does this player rank?"

The granularity of the positional filter also matters. "Midfielder" is a broad category that encompasses deep-lying playmakers, ball-winning midfielders, box-to-box runners, and attacking number 10s. Many analytics departments now use sub-position classifications---often derived from clustering analysis (Section 15.7)---to create more meaningful peer groups.

15.2 Position-Specific Metrics

15.2.1 Goalkeepers

Goalkeepers occupy a unique statistical universe. The metrics that matter for an outfield player are largely irrelevant for a keeper, and vice versa.

Core goalkeeper metrics:

The most informative modern goalkeeper metric is PSxG - GA (Post-Shot Expected Goals minus Goals Allowed). Unlike raw save percentage, PSxG accounts for the difficulty of each shot faced by considering not just shot location but also the trajectory of the ball after it is struck.

$$ \text{PSxG} - \text{GA} = \sum_{i=1}^{n} \text{PSxG}_i - \text{Goals Allowed} $$

A positive value means the goalkeeper has saved more goals than the average keeper would have, given the same shots. A negative value indicates underperformance.

Real-World Application: When Brentford signed goalkeeper Mark Flekken in 2023, analytics departments noted his PSxG-GA ranked in the top 10 among Bundesliga keepers over the prior two seasons, even though his raw save percentage was merely average. The difference was explained by the high volume of difficult shots Flekken faced playing behind a high defensive line.

Modern goalkeeper evaluation beyond shot-stopping:

The evolution of the goalkeeper role---particularly in possession-based systems---has expanded the relevant metric set considerably. Today's elite goalkeepers are expected to function as an additional outfield player during build-up play. Key metrics for this expanded role include:

• Passes attempted per 90 under pressure: How often the goalkeeper is involved in build-up play under pressing situations, and how often they retain possession.
• Progressive passing distance per 90: The total forward distance of successful passes, capturing the goalkeeper's ability to bypass lines of pressure with distribution.
• Sweeper actions per 90: The number of defensive actions taken outside the penalty area, reflecting the goalkeeper's willingness and ability to act as a sweeper-keeper behind a high defensive line.
• Goal kick pass completion rate: The percentage of goal kicks that successfully find a teammate, broken down by short (inside the box) and long distribution.

Callout --- The Goalkeeper Sample Size Problem: Goalkeepers face far fewer measurable events than outfield players. A goalkeeper may face only 3-5 shots on target per match, meaning that over a 38-match season, even a full-time starter faces only 115-190 shots. This makes goalkeeper metrics inherently noisier than outfield metrics. PSxG-GA stabilizes meaningfully only after approximately 1,500-2,000 shots faced, which may require 2-3 full seasons. This is one reason goalkeeper transfer markets are notoriously inefficient---clubs often overreact to single-season performances that are substantially influenced by randomness.

15.2.2 Defenders

Defenders---both centre-backs and full-backs---require metrics that capture their dual responsibilities: preventing opposition attacks and initiating their own team's build-up play.

Centre-back metrics:

• Aerial duel win rate: $\frac{\text{Aerial Duels Won}}{\text{Aerial Duels Contested}}$
• Tackles + Interceptions per 90: Combined ball-winning volume
• Clearances per 90: How often the defender resolves danger
• Progressive passes per 90: Passes that move the ball at least 10 yards toward the opponent's goal
• Errors leading to shots/goals: A critical negative metric

Full-back / wing-back metrics add:

• Crosses per 90 and cross accuracy
• Carries into the final third per 90
• Assists and xA per 90 (especially for attacking full-backs)

def defender_composite_score(
    tackles_per90: float,
    interceptions_per90: float,
    aerial_win_pct: float,
    progressive_passes_per90: float,
    errors_leading_to_goals: float,
    weights: dict[str, float] | None = None
) -> float:
    """Compute a weighted composite score for a centre-back.

    Args:
        tackles_per90: Tackles won per 90 minutes.
        interceptions_per90: Interceptions per 90 minutes.
        aerial_win_pct: Aerial duel win percentage (0-100).
        progressive_passes_per90: Progressive passes per 90 minutes.
        errors_leading_to_goals: Errors leading to goals (negative metric).
        weights: Optional dictionary of weights. Defaults to equal weighting.

    Returns:
        Composite score (higher is better).
    """
    if weights is None:
        weights = {
            "tackles": 0.2,
            "interceptions": 0.2,
            "aerial": 0.2,
            "progressive": 0.25,
            "errors": 0.15,
        }
    score = (
        weights["tackles"] * tackles_per90
        + weights["interceptions"] * interceptions_per90
        + weights["aerial"] * (aerial_win_pct / 100.0)
        + weights["progressive"] * progressive_passes_per90
        - weights["errors"] * errors_leading_to_goals
    )
    return score

Callout --- The Defensive Metrics Paradox: A persistent challenge in defensive evaluation is that the best defenders often record fewer tackles and interceptions because their positioning is so good that they do not need to make recovery actions. Virgil van Dijk in his prime was a classic example: his tackles-per-90 ranked in only the 40th-50th percentile among Premier League centre-backs, yet he was widely regarded as the best defender in the world. This is because excellent positioning means opponents rarely get close enough to require a tackle. Analysts must supplement counting stats with positional and spatial metrics (see Chapter 17) to capture this "invisible" defensive quality.

15.2.3 Midfielders

Midfielders are the most tactically diverse position group. A deep-lying playmaker, a box-to-box midfielder, and an attacking midfielder occupy the same broad position label but perform radically different functions.

Holding / defensive midfielder:

• Tackles and interceptions per 90
• Pass completion % (short and medium)
• Pressure success rate
• Ball recoveries in the defensive and middle thirds

Box-to-box midfielder:

• Progressive carries and progressive passes per 90
• Tackles + interceptions per 90
• Goal-creating actions per 90
• Distance covered and high-intensity sprints

Attacking midfielder / number 10:

• xG + xA per 90 (combined threat)
• Key passes per 90
• Shot-creating actions per 90
• Successful dribbles per 90

Advanced: For midfielders, the concept of ball progression is particularly revealing. A midfielder's progressive value can be measured as the sum of the forward distance (in yards) of all successful passes and carries, normalized per 90 minutes. This captures the essential midfield task of moving the ball from defense to attack.

$$ \text{Progressive Value}_{p90} = \frac{\sum_{i} \Delta y_i^{\text{pass}} + \sum_{j} \Delta y_j^{\text{carry}}}{90 \text{-min units played}} $$

where $\Delta y$ represents the forward displacement of each action toward the opponent's goal.

The distinction between passing progression and carrying progression is itself informative. Some midfielders progress play primarily through incisive passing (the Kevin De Bruyne archetype), while others do so through driving ball carries (the Naby Keita archetype). A scatter plot of progressive passing distance vs. progressive carrying distance per 90 effectively separates these styles.

15.2.4 Forwards

Forwards are evaluated primarily on their output---goals and the actions leading to goals---but a modern evaluation must go deeper.

Striker metrics:

• Non-penalty goals per 90 (npG/90): Removes penalty distortion
• Non-penalty xG per 90 (npxG/90): Shot quality regardless of finishing
• xG outperformance (npG - npxG): Finishing skill or luck (controversial---see Chapter 10)
• Shot volume per 90
• Aerial duels won per 90 (for target-man types)
• Pressing actions per 90 (for high-pressing systems)

Winger metrics add:

• Successful dribbles per 90 and dribble success rate
• Crosses and cross accuracy
• Touches in the penalty area per 90
• xA per 90

Common Pitfall: Penalty goals distort forward evaluation significantly. A striker who takes all of their team's penalties may appear to be a far more prolific scorer than they truly are from open play. Always examine non-penalty figures alongside total figures.

Callout --- The Forward Contribution Beyond Goals: Modern tactical systems increasingly demand that forwards contribute to the team's defensive effort. Pressing metrics---such as pressures per 90, successful pressure rate, and pressures in the attacking third---are now standard components of forward evaluation at elite clubs. A forward who scores 15 non-penalty goals but never presses creates a structural weakness that opponents can exploit. Conversely, a forward who scores 10 goals but leads the league in pressing intensity may contribute more to overall team performance through the chances won from high turnovers.

15.3 Minutes and Playing Time Adjustments

15.3.1 The Per-90 Standard

The soccer analytics community has settled on per 90 minutes as the standard unit of normalization. This is analogous to baseball's "per plate appearance" or basketball's "per 36 minutes."

The per-90 formula is straightforward:

$$ \text{Metric}_{p90} = \frac{\text{Raw Count}}{\text{Minutes Played}} \times 90 $$

For example, if a player has 6 goals in 1,350 minutes:

$$ \text{Goals}_{p90} = \frac{6}{1350} \times 90 = 0.40 $$

def per_90(raw_count: float, minutes_played: float) -> float:
    """Normalize a raw count to a per-90-minute rate.

    Args:
        raw_count: The raw event count.
        minutes_played: Total minutes played.

    Returns:
        The per-90 rate.

    Raises:
        ValueError: If minutes_played is zero or negative.
    """
    if minutes_played <= 0:
        raise ValueError("Minutes played must be positive.")
    return (raw_count / minutes_played) * 90

15.3.2 Minimum Appearance Thresholds

Per-90 rates become unreliable at small sample sizes. A substitute who plays 45 minutes and scores a goal has a per-90 rate of 2.0 goals---which is meaningless.

The standard practice is to impose a minimum minutes threshold before including a player in per-90 rankings. Common thresholds include:

The choice of threshold involves a bias-variance trade-off. A higher threshold reduces variance (noisy per-90 rates from small samples) but introduces selection bias (excluding part-time players, new signings, and injured players who may be extremely good or bad).

Intuition: Imagine flipping a coin. After 3 flips, getting 100% heads is unremarkable. After 300 flips, getting 100% heads would be miraculous. Per-90 rates work the same way---more minutes mean more confidence that the rate reflects true ability rather than chance.

15.3.3 Sample Size Requirements by Metric Type

Different metrics stabilize at different rates. Some metrics require only a few hundred minutes to become reliable indicators of true talent; others need thousands. Research by analysts at StatsBomb and others has produced approximate stabilization points:

The stabilization point is the number of minutes after which the metric's "true talent" signal exceeds the noise. Metrics with high stabilization requirements should be treated with extreme caution in small samples.

Callout --- Why Conversion Rate Is Almost Useless in Small Samples: Conversion rate (goals divided by shots) is one of the most commonly cited statistics in media, yet it is among the slowest to stabilize. A player who converts 20% of shots over 10 matches may have a true conversion rate anywhere between 8% and 32% at a 95% confidence level. Only after approximately 3,000+ minutes (about 33 full matches) does conversion rate begin to reliably distinguish elite finishers from average ones. This is why xG-based evaluation, which considers shot quality rather than outcomes, is far more informative for forward assessment.

15.3.4 Bayesian Shrinkage for Small Samples

An elegant alternative to hard thresholds is Bayesian shrinkage (also called regression to the mean). Instead of excluding low-minute players, we adjust their rates toward the population average, with the adjustment inversely proportional to playing time.

The shrunk estimate is:

$$ \hat{\mu}_{\text{player}} = w \cdot \bar{x}_{\text{player}} + (1 - w) \cdot \bar{x}_{\text{population}} $$

where the weight $w$ depends on the player's sample size:

$$ w = \frac{n}{n + \kappa} $$

Here, $n$ is the player's minutes (or match count) and $\kappa$ is a smoothing parameter representing the "prior strength." A common heuristic sets $\kappa$ equal to the number of minutes at which we would trust the player's observed rate about 50%.

def bayesian_shrinkage(
    player_rate: float,
    population_rate: float,
    minutes_played: float,
    kappa: float = 900.0
) -> float:
    """Apply Bayesian shrinkage to a player's per-90 rate.

    For players with few minutes, the estimate is pulled toward the
    population average. As minutes increase, the player's own data
    dominates.

    Args:
        player_rate: The player's observed per-90 rate.
        population_rate: The population (prior) mean per-90 rate.
        minutes_played: Minutes the player has played.
        kappa: Smoothing parameter (minutes for 50% weight).

    Returns:
        Shrunk per-90 rate estimate.
    """
    weight = minutes_played / (minutes_played + kappa)
    return weight * player_rate + (1 - weight) * population_rate

Real-World Application: StatsBomb and other analytics providers use shrinkage-style adjustments when ranking players with limited minutes. This prevents a substitute who happened to score in a brief cameo from appearing at the top of per-90 leaderboards.

15.3.5 Adjusting for Substitution Patterns

Players who frequently appear as substitutes face a systematic bias: they tend to enter matches when the game state favors certain actions. A forward brought on at 70 minutes when the team is trailing may face a more open defense, inflating their per-90 goal rate. Conversely, a defensive midfielder brought on to protect a lead may face fewer attacking opportunities.

While correcting for this bias is complex, a simple approach is to separate starter and substitute appearances and analyze them independently. More sophisticated methods include game-state adjustments (Chapter 14) applied at the player level.

A related issue is the end-of-match effect: players who enter in the final 10-15 minutes often face tired opponents and chaotic game states, which can inflate attacking metrics. Analysts should be cautious about including very short substitute appearances (under 20 minutes) in rate calculations, or alternatively should apply game-state and match-minute adjustments.

15.4 Age Curves and Development

15.4.1 The General Age-Performance Relationship

One of the most robust findings in sports analytics is that athletic performance follows an inverted-U shape with age. Players improve through their early twenties, peak in their mid-to-late twenties, and decline thereafter.

In soccer, research consistently shows:

• Physical peak (sprints, distance): Ages 24-26
• Technical/creative peak (passing, dribbling): Ages 26-29
• Tactical/positional peak (interceptions, positioning): Ages 28-31
• Goalkeeping peak: Ages 27-32

The general age curve can be modeled as a quadratic (parabolic) function:

$$ \text{Performance}(a) = \beta_0 + \beta_1 a + \beta_2 a^2 + \epsilon $$

where $a$ is the player's age. The peak age is at $a^* = -\frac{\beta_1}{2\beta_2}$ when $\beta_2 < 0$.

A more flexible approach uses a delta method, which measures the average year-over-year change in a metric for players at each age:

$$ \Delta_a = \bar{x}_{a+1} - \bar{x}_{a} $$

Cumulating these deltas from a baseline age produces an aging curve.

import pandas as pd
import numpy as np

def compute_delta_aging_curve(
    df: pd.DataFrame,
    metric: str,
    age_col: str = "age",
    player_col: str = "player_id",
    min_minutes: float = 900.0
) -> pd.Series:
    """Compute an aging curve using the delta method.

    Requires a panel dataset with repeated observations of players
    across multiple seasons.

    Args:
        df: DataFrame with player-season observations.
        metric: Column name of the per-90 metric.
        age_col: Column name for age.
        player_col: Column name for player identifier.
        min_minutes: Minimum minutes for inclusion.

    Returns:
        Series indexed by age with cumulative performance delta.
    """
    # Filter by minimum minutes
    qualified = df[df["minutes"] >= min_minutes].copy()

    # Compute year-over-year changes for each player
    qualified = qualified.sort_values([player_col, age_col])
    qualified["delta"] = qualified.groupby(player_col)[metric].diff()
    qualified["prev_age"] = qualified.groupby(player_col)[age_col].shift(1)

    # Keep only consecutive-age observations
    valid = qualified.dropna(subset=["delta", "prev_age"])
    valid = valid[valid[age_col] == valid["prev_age"] + 1]

    # Average delta at each age
    avg_delta = valid.groupby(age_col)["delta"].mean()

    # Cumulate from median age to get the curve
    curve = avg_delta.cumsum()
    return curve

15.4.2 Position-Specific Age Curves

The peak age varies substantially by position and by the specific metric being examined:

Advanced: The aging curve is not deterministic. Individual players deviate from population averages due to genetics, injury history, training methods, and positional evolution. Cristiano Ronaldo's sustained peak well into his thirties is exceptional but does not invalidate the average curve---it simply illustrates the wide variance around the mean trajectory.

15.4.3 Development Trajectories for Young Players

For recruitment departments, a crucial application of age curves is projecting the future performance of young players. If a 20-year-old centre-back is already performing at the 70th percentile among top-five-league centre-backs, and the average centre-back improves by 15% between ages 20 and 27, the projected peak performance is approximately the 85th percentile.

However, projection is fraught with uncertainty. Key complicating factors include:

• Non-linear development: Some players make sudden leaps in performance (often coinciding with a move to a better team or a coaching change), while others plateau.
• Injury risk: Young players with certain injury profiles (e.g., recurring hamstring problems, early knee injuries) may have truncated development curves.
• Positional evolution: A player who develops from a winger into a striker may follow a completely different trajectory than the winger or striker age curves would predict.
• League quality adjustment: A 21-year-old who dominates the Eredivisie may not experience the same trajectory when moving to the Premier League, because the competition level changes.

Callout --- The Projection Confidence Interval: When projecting a young player's future performance, always attach a confidence interval. For a 20-year-old with 2,000 minutes of top-flight data, a 90% confidence interval for their age-27 performance level might span from the 50th to the 95th percentile. This uncertainty must be communicated to decision-makers, because the difference between the 50th and 95th percentile centre-back is the difference between a squad player and a cornerstone signing.

15.4.4 Survivorship Bias in Age Curves

A subtle but important problem in aging curve analysis is survivorship bias. Players who are still playing at age 34 are, by definition, those good enough to retain professional contracts. The weaker players from their cohort have already retired or dropped to lower leagues. This makes the aging curve appear flatter at older ages than it truly is, because we only observe the survivors.

Corrections for survivorship bias include:

1. Weighted regression that accounts for the number of players at each age
2. Heckman selection models that jointly model performance and the probability of remaining in the sample
3. Within-player analysis using only players observed across a continuous age span

$$ \text{Observed mean at age } a = E[\text{Performance} | \text{Survived to } a] $$

This conditional expectation is higher than the unconditional $E[\text{Performance at age } a]$ for all players who were once professionals, because the conditioning event (survival) is positively correlated with performance.

15.4.5 Applications: Valuation and Recruitment

Age curves are central to player valuation models. The transfer fee for a player implicitly prices their remaining career value---the area under the age curve from the current age to the expected retirement age.

$$ V_{\text{remaining}} = \sum_{a=a_{\text{current}}}^{a_{\text{retire}}} \delta^{a - a_{\text{current}}} \cdot \text{Performance}(a) $$

where $\delta$ is a discount factor (reflecting the time value of performance). This is why a 23-year-old with identical current performance to a 29-year-old commands a much higher transfer fee---more of the curve lies ahead.

Real-World Application: Clubs like Brentford, Brighton, and RB Salzburg systematically exploit age curves by purchasing players at 20-23 (before their peak, when fees are lower), developing them, and selling at 25-28 (at or near peak, when fees are highest). This "buy low, sell high" strategy is built on aging curve analysis.

15.5 Form and Consistency Measurement

15.5.1 Defining Form

"Form" in soccer refers to a player's recent performance level relative to their long-term average. A player "in form" is performing above their baseline; a player "out of form" is performing below it.

Quantifying form requires two components:

1. A measure of recent performance (typically a rolling window)
2. A measure of baseline performance (typically a season-long or career average)

$$ \text{Form Index} = \frac{\bar{x}_{\text{recent}}}{\bar{x}_{\text{baseline}}} $$

A Form Index above 1.0 indicates positive form; below 1.0 indicates negative form.

15.5.2 Rolling Averages and Exponential Weighting

The most straightforward approach to measuring recent performance is a simple rolling average over the last $k$ matches:

$$ \bar{x}_{\text{recent}} = \frac{1}{k} \sum_{i=1}^{k} x_i $$

A more nuanced approach uses exponentially weighted moving averages (EWMA), which assign greater weight to more recent observations:

$$ \text{EWMA}_t = \alpha \cdot x_t + (1 - \alpha) \cdot \text{EWMA}_{t-1} $$

where $\alpha \in (0, 1)$ is the smoothing parameter. A higher $\alpha$ makes the average more responsive to recent results; a lower $\alpha$ produces a smoother, more stable estimate.

import pandas as pd

def compute_form_metrics(
    match_ratings: pd.Series,
    rolling_window: int = 5,
    ewma_span: int = 5
) -> pd.DataFrame:
    """Compute rolling and EWMA form indicators for a player.

    Args:
        match_ratings: Series of per-match performance ratings, ordered
            chronologically.
        rolling_window: Number of matches for the simple rolling average.
        ewma_span: Span parameter for exponential weighting.

    Returns:
        DataFrame with columns: raw, rolling_avg, ewma, form_index.
    """
    form = pd.DataFrame({"raw": match_ratings})
    form["rolling_avg"] = form["raw"].rolling(window=rolling_window, min_periods=1).mean()
    form["ewma"] = form["raw"].ewm(span=ewma_span, adjust=False).mean()
    baseline = form["raw"].expanding().mean()
    form["form_index"] = form["ewma"] / baseline
    return form

The choice of window size matters considerably. A 3-match window is highly reactive but very noisy; a 10-match window is stable but slow to detect genuine changes in performance. In practice, a 5-match window offers a reasonable compromise, and EWMA with a span of 5-7 matches provides a good balance of reactivity and stability.

15.5.3 Consistency: Variance and Coefficient of Variation

Two players with identical per-90 rates may have very different match-to-match profiles. One might deliver a steady 7/10 every week, while the other alternates between 9/10 and 5/10. For many coaches, the consistent player is more valuable because they are more predictable and dependable.

Consistency can be quantified using:

• Standard deviation of match-level ratings: $\sigma = \sqrt{\frac{1}{n-1}\sum(x_i - \bar{x})^2}$
• Coefficient of variation (CV): $\text{CV} = \frac{\sigma}{\bar{x}}$, which normalizes variability relative to the mean
• Interquartile range (IQR): $Q_3 - Q_1$, which is robust to outliers

$$ \text{Consistency Score} = 1 - \text{CV}_{\text{normalized}} $$

where $\text{CV}_{\text{normalized}}$ scales the CV to [0, 1] relative to the peer group.

Intuition: Consider two strikers who each score 15 goals in a 38-match season. Striker A scores exactly 1 goal every 2-3 matches. Striker B scores hat-tricks in three matches and is goalless in the other 32. Their total output is identical, but Striker A's steady contribution is arguably more tactically valuable.

15.5.4 Streak Analysis

Beyond variance, we can analyze the sequential structure of a player's performance to detect hot and cold streaks.

A simple streak detection algorithm:

def detect_streaks(
    performances: list[float],
    threshold: float
) -> list[dict]:
    """Detect consecutive streaks above or below a threshold.

    Args:
        performances: List of per-match metric values.
        threshold: Value above which a match counts as "hot."

    Returns:
        List of streak dictionaries with type, start, end, and length.
    """
    streaks = []
    current_type = "hot" if performances[0] >= threshold else "cold"
    start = 0

    for i in range(1, len(performances)):
        match_type = "hot" if performances[i] >= threshold else "cold"
        if match_type != current_type:
            streaks.append({
                "type": current_type,
                "start": start,
                "end": i - 1,
                "length": i - start,
            })
            current_type = match_type
            start = i

    # Final streak
    streaks.append({
        "type": current_type,
        "start": start,
        "end": len(performances) - 1,
        "length": len(performances) - start,
    })
    return streaks

Common Pitfall: Humans are notoriously bad at distinguishing genuine streaks from random clustering. Before concluding that a player is "streaky," apply a statistical test (such as the runs test or the Wald-Wolfowitz test) to determine whether the observed streak pattern differs significantly from what random chance would produce.

15.5.5 Big-Game Performance

A specialized form of consistency analysis examines whether players perform differently in high-stakes matches. The "big game player" narrative is pervasive in media, but how often does it hold up statistically?

To test this, we partition matches into "big" (against top-6 opponents, knockout stages, derby matches) and "regular" categories, then compare per-90 metrics in each group. A player's Big Game Index can be computed as:

$$ \text{BGI} = \frac{\bar{x}_{\text{big games}}}{\bar{x}_{\text{all games}}} $$

A BGI above 1.0 suggests the player elevates their performance in important matches; below 1.0 suggests they shrink.

Callout --- Statistical Caution on Big-Game Analysis: The sample of "big games" is typically very small---perhaps 8-12 matches per season. This makes the BGI extremely noisy. Most apparent "big game players" or "flat-track bullies" are simply experiencing normal variance. Only with 3+ seasons of data can we begin to distinguish genuine tendencies from randomness. Even then, the effect sizes tend to be small.

15.6 Building Player Profiles

15.6.1 Feature Selection

A player profile is a curated collection of metrics that, taken together, characterize the player's style and quality. The first step is feature selection---choosing which metrics to include.

Good features for a player profile are:

• Relevant to the player's position and role
• Stable enough across samples to reflect true ability (not noise)
• Non-redundant (avoid including both "goals" and "goals per 90" if minutes are constant)
• Diverse in the dimensions they capture (scoring, creation, defense, possession)

A typical profile for an attacking midfielder might include:

15.6.2 Standardization with Z-Scores

To combine metrics measured on different scales, we standardize each metric into z-scores:

$$ z_i = \frac{x_i - \mu}{\sigma} $$

where $\mu$ and $\sigma$ are the mean and standard deviation of the metric across the comparison group (e.g., all attacking midfielders in the top five leagues with 900+ minutes).

A z-score of 0 means the player is average; +1 means one standard deviation above average; -2 means two standard deviations below average.

import pandas as pd
import numpy as np

def standardize_metrics(
    df: pd.DataFrame,
    metrics: list[str],
    group_col: str | None = None
) -> pd.DataFrame:
    """Standardize metrics to z-scores, optionally within groups.

    Args:
        df: DataFrame with player rows and metric columns.
        metrics: List of column names to standardize.
        group_col: Optional column to group by (e.g., position).

    Returns:
        DataFrame with z-score columns named '{metric}_z'.
    """
    result = df.copy()
    for metric in metrics:
        if group_col:
            grouped = result.groupby(group_col)[metric]
            result[f"{metric}_z"] = grouped.transform(
                lambda x: (x - x.mean()) / x.std()
            )
        else:
            result[f"{metric}_z"] = (
                (result[metric] - result[metric].mean()) / result[metric].std()
            )
    return result

15.6.3 Radar Charts (Spider Plots)

The radar chart is the most popular visualization for player profiles. Each axis represents one metric, and the player's values are plotted as a polygon.

To create effective radar charts:

1. Standardize all metrics to the same scale (percentile ranks from 0-100 work well)
2. Orient all axes so that higher values are better (invert metrics like "errors leading to goals")
3. Limit the number of axes to 6-10 for readability
4. Order adjacent axes to represent related dimensions (all defensive metrics next to each other)

import matplotlib.pyplot as plt
import numpy as np

def radar_chart(
    categories: list[str],
    values: list[float],
    player_name: str,
    ax: plt.Axes | None = None,
    color: str = "steelblue",
    alpha: float = 0.25,
) -> plt.Axes:
    """Create a radar chart for a player profile.

    Args:
        categories: List of metric names.
        values: List of percentile values (0-100).
        player_name: Name for the chart title.
        ax: Optional matplotlib axes (must be polar).
        color: Fill color.
        alpha: Fill transparency.

    Returns:
        The matplotlib Axes object.
    """
    n = len(categories)
    angles = np.linspace(0, 2 * np.pi, n, endpoint=False).tolist()

    # Close the polygon
    values_closed = values + [values[0]]
    angles_closed = angles + [angles[0]]

    if ax is None:
        fig, ax = plt.subplots(figsize=(8, 8), subplot_kw={"projection": "polar"})

    ax.plot(angles_closed, values_closed, "o-", color=color, linewidth=2)
    ax.fill(angles_closed, values_closed, color=color, alpha=alpha)

    ax.set_xticks(angles)
    ax.set_xticklabels(categories, size=10)
    ax.set_ylim(0, 100)
    ax.set_title(player_name, size=14, fontweight="bold", pad=20)

    return ax

Callout --- Radar Chart Best Practices: While radar charts are visually appealing and widely used, they have known perceptual limitations. Humans are poor at comparing areas of irregular polygons, so two players whose radar charts look different may actually be very close in overall quality. Always pair radar charts with numerical tables or bar charts. Additionally, the ordering of axes matters: placing two strong metrics adjacent to each other creates a visually larger polygon segment that can mislead the viewer. Randomize or standardize axis ordering when making comparative analyses.

15.6.4 Composite Indices

Sometimes we want to reduce a multi-dimensional profile to a single number---a composite performance index. While this sacrifices nuance, it can be useful for ranking and screening.

Methods for constructing composite indices:

1. Weighted z-score sum: $\text{CPI} = \sum_j w_j \cdot z_j$ with expert-chosen weights $w_j$
2. Principal Component Analysis (PCA): Let the data determine weights via the first principal component
3. Goal-based: Weight each metric by its estimated contribution to goal-scoring or match outcomes (from a regression model)

$$ \text{CPI}_i = \sum_{j=1}^{p} w_j \cdot z_{ij}, \quad \sum_j w_j = 1 $$

Common Pitfall: Composite indices compress rich, multi-dimensional information into a single number. They are useful for initial screening, but final evaluations should always return to the full profile. A single number cannot tell you how a player contributes---only a rough sense of how much.

15.7 Similarity Scores and Comparisons

15.7.1 The Player Comparison Problem

One of the most frequent questions in player recruitment is: "We're losing Player X. Who is the most similar replacement?" Or: "We want someone who plays like Player Y but is younger and cheaper."

Answering these questions requires a formal definition of similarity between players. If each player is represented as a vector of standardized metrics, similarity becomes a problem of measuring the distance (or closeness) between vectors in multi-dimensional space.

15.7.2 Cosine Similarity

Cosine similarity measures the angle between two player vectors, ignoring their magnitudes. Two players with identical profiles but different absolute levels (e.g., a world-class and a good player with the same shape of strengths and weaknesses) would have a cosine similarity of 1.0.

$$ \text{cos}(\mathbf{a}, \mathbf{b}) = \frac{\mathbf{a} \cdot \mathbf{b}}{||\mathbf{a}|| \cdot ||\mathbf{b}||} = \frac{\sum_i a_i b_i}{\sqrt{\sum_i a_i^2} \cdot \sqrt{\sum_i b_i^2}} $$

Values range from -1 (opposite profiles) to +1 (identical profiles).

from sklearn.metrics.pairwise import cosine_similarity
import numpy as np

def find_similar_players_cosine(
    target_vector: np.ndarray,
    player_matrix: np.ndarray,
    player_names: list[str],
    top_n: int = 10
) -> list[tuple[str, float]]:
    """Find the most similar players using cosine similarity.

    Args:
        target_vector: 1D array of the target player's standardized metrics.
        player_matrix: 2D array where each row is a player's metrics.
        player_names: List of player names corresponding to rows.
        top_n: Number of similar players to return.

    Returns:
        List of (player_name, similarity_score) tuples, sorted descending.
    """
    similarities = cosine_similarity(
        target_vector.reshape(1, -1), player_matrix
    ).flatten()

    ranked = sorted(
        zip(player_names, similarities),
        key=lambda x: x[1],
        reverse=True
    )
    return ranked[:top_n]

15.7.3 Euclidean Distance

Euclidean distance measures the straight-line distance between two player vectors. Unlike cosine similarity, it is sensitive to both the shape and the magnitude of the profile.

$$ d(\mathbf{a}, \mathbf{b}) = \sqrt{\sum_i (a_i - b_i)^2} $$

Euclidean distance is appropriate when you want similar players to be close in both style and level. A similarity score can be derived as:

$$ \text{Similarity} = \frac{1}{1 + d(\mathbf{a}, \mathbf{b})} $$

Intuition: Cosine similarity asks "Do these players have the same shape of strengths and weaknesses?" Euclidean distance asks "Do these players have the same values across all metrics?" The right choice depends on whether you are looking for a stylistic match or a quality-and-style match.

15.7.4 Weighted Similarity

Not all metrics are equally important for defining a player's style. A recruitment analyst looking for a replacement centre-back might care more about aerial duel rate and progressive passing than about tackles. Weighted similarity incorporates this:

$$ d_w(\mathbf{a}, \mathbf{b}) = \sqrt{\sum_i w_i (a_i - b_i)^2} $$

where $w_i$ is the weight assigned to the $i$-th metric.

15.7.5 Clustering for Player Archetypes

Rather than comparing individual players, we can use clustering algorithms to discover natural groups or archetypes of players.

K-Means clustering partitions players into $k$ groups by minimizing within-cluster variance:

$$ \min_{\{C_1, \ldots, C_k\}} \sum_{j=1}^{k} \sum_{\mathbf{x} \in C_j} ||\mathbf{x} - \boldsymbol{\mu}_j||^2 $$

Each cluster center $\boldsymbol{\mu}_j$ defines an archetype---a prototypical player profile.

from sklearn.cluster import KMeans
from sklearn.preprocessing import StandardScaler
import pandas as pd
import numpy as np

def discover_archetypes(
    df: pd.DataFrame,
    metrics: list[str],
    n_clusters: int = 6,
    random_state: int = 42
) -> pd.DataFrame:
    """Cluster players into archetypes using K-Means.

    Args:
        df: DataFrame with player rows and metric columns.
        metrics: List of metric column names.
        n_clusters: Number of archetypes to discover.
        random_state: Random seed for reproducibility.

    Returns:
        Original DataFrame with added 'archetype' column.
    """
    scaler = StandardScaler()
    X = scaler.fit_transform(df[metrics].values)

    kmeans = KMeans(n_clusters=n_clusters, random_state=random_state, n_init=10)
    df = df.copy()
    df["archetype"] = kmeans.fit_predict(X)

    return df

Common archetypes that emerge from clustering forwards in top European leagues include:

1. Poacher: High npxG, low creative output, few dribbles
2. Complete forward: High across scoring, creation, and dribbling
3. Target man: High aerial duels, hold-up play, moderate scoring
4. Pressing forward: High pressures, moderate scoring, high work rate
5. Playmaking forward: High xA, progressive passes, drops deep

15.7.6 Dimensionality Reduction for Visualization

When working with many metrics, dimensionality reduction techniques help visualize player similarity in two or three dimensions.

Principal Component Analysis (PCA) projects the high-dimensional metric space onto a lower-dimensional space that preserves the maximum variance:

from sklearn.decomposition import PCA
import matplotlib.pyplot as plt
import numpy as np

def plot_player_map(
    player_matrix: np.ndarray,
    player_names: list[str],
    highlight: str | None = None
) -> plt.Figure:
    """Create a 2D player map using PCA.

    Args:
        player_matrix: Standardized metric matrix (players x features).
        player_names: List of player names.
        highlight: Optional player name to highlight.

    Returns:
        Matplotlib figure.
    """
    pca = PCA(n_components=2)
    coords = pca.fit_transform(player_matrix)

    fig, ax = plt.subplots(figsize=(12, 8))
    ax.scatter(coords[:, 0], coords[:, 1], alpha=0.5, s=30)

    if highlight and highlight in player_names:
        idx = player_names.index(highlight)
        ax.scatter(
            coords[idx, 0], coords[idx, 1],
            color="red", s=100, zorder=5, label=highlight
        )
        ax.legend()

    ax.set_xlabel(f"PC1 ({pca.explained_variance_ratio_[0]:.1%} variance)")
    ax.set_ylabel(f"PC2 ({pca.explained_variance_ratio_[1]:.1%} variance)")
    ax.set_title("Player Similarity Map (PCA)")

    return fig

t-SNE and UMAP are nonlinear alternatives that often produce better visual separation of clusters, at the cost of losing interpretable axis meanings.

Advanced: When building similarity models for recruitment, consider adding contextual features beyond on-ball statistics. These might include the league level of the player's competition, the team's average possession, the tactical system (back three vs. back four), and the player's age. A player's raw numbers in the Eredivisie may not translate directly to the Premier League, and contextual features help account for this.

15.7.7 Putting It All Together: A Similarity Pipeline

A complete player similarity pipeline involves the following steps:

1. Define the target player and the metric profile that characterizes them
2. Select the candidate pool (age range, position, league level, contract situation)
3. Extract per-90 metrics for all candidates, applying minimum minute thresholds
4. Standardize metrics to z-scores within the candidate pool
5. Compute similarity scores (cosine, Euclidean, or weighted) between the target and all candidates
6. Rank and filter the results
7. Visualize top matches with radar charts and PCA maps
8. Validate with video analysis and scouting reports

Real-World Application: Liverpool's data science team reportedly used a similarity-based approach when identifying Diogo Jota as a stylistic match for their front three. The algorithm considered metrics like pressing intensity, movement patterns, and shot quality to find players whose profiles complemented the existing squad.

15.8 Comparing Players Across Leagues and Eras

15.8.1 The Cross-League Challenge

One of the most frequent demands placed on analytics departments is comparing players across different leagues. A Brazilian midfielder in the Serie A, a French winger in Ligue 1, and a Nigerian striker in the Eredivisie may all be transfer targets---but how do their metrics compare when the competitive environments are so different?

Raw per-90 statistics are not directly comparable across leagues because leagues differ in:

• Pace of play: The number of possessions, passes, and shots per match varies by league. The Premier League averages more shots per match than La Liga, which in turn averages more than Serie A.
• Defensive intensity: Leagues with aggressive pressing cultures (like the Bundesliga) produce more turnovers, tackles, and interceptions per match.
• Overall quality: A player recording 0.5 xG per 90 in the Eredivisie faces weaker opposition than one recording the same figure in the Premier League.
• Tactical norms: Serie A's tradition of low-block defending produces different metric baselines than the Bundesliga's high-line approach.

15.8.2 League-Adjustment Methods

Several approaches exist for cross-league comparison:

1. Within-league percentile ranks: Compute percentile ranks within each league separately, then compare percentile ranks across leagues. This implicitly assumes that the 90th percentile in the Eredivisie is comparable to the 90th percentile in the Premier League, which is a strong assumption but a useful starting point.

2. League-quality scaling factors: Estimate scaling factors from players who transfer between leagues. If players moving from League A to League B experience, on average, a 15% decline in npxG per 90, we can adjust League A metrics by 0.85 for comparison purposes.

3. Regression-based adjustment: Fit a model that predicts a metric as a function of player ability, league, and team context:

$$ Y_{i,l} = \mu + \alpha_i + \beta_l + \gamma \cdot \text{TeamQuality}_i + \epsilon_{i,l} $$

The player effect $\alpha_i$ is the league-adjusted estimate of the player's true ability.

Callout --- The Transfer Validation Approach: The most rigorous method for calibrating cross-league comparisons is to study players who actually moved between leagues. By comparing their performance before and after the transfer (controlling for age effects), we can estimate league difficulty coefficients. Research by several analytics groups has produced rough conversion factors: for example, moving from the Eredivisie to the Premier League is associated with approximately a 20-30% decline in most offensive metrics, while moving from the Bundesliga to the Premier League shows a smaller 5-15% decline.

15.8.3 Comparing Across Eras

Historical comparisons are even more challenging than cross-league comparisons. How does a 2010 Lionel Messi compare to a 1990 Marco van Basten? The game itself has changed:

• Tactical evolution: Modern pressing systems, positional play, and high defensive lines create different opportunities and challenges than the more individualistic systems of earlier decades.
• Physical conditioning: Today's players cover 10-13 km per match with significantly more high-intensity sprints than players in the 1990s. The physical demands have changed the types of players who succeed.
• Data availability: Pre-2010 data is sparse, making direct metric comparison impossible for most metrics beyond goals and assists.
• Rule changes: The back-pass rule (1992), changes to offside interpretation, and increases in added time have all altered the statistical landscape.

The most defensible approach to cross-era comparison is to measure a player's dominance relative to their contemporaries. A player in the 99th percentile of their era can reasonably be compared to a player in the 99th percentile of another era, even if the absolute numbers differ.

15.9 Isolating Individual Contribution from Team Effects

15.9.1 The Fundamental Attribution Problem

Perhaps the most challenging problem in player evaluation is separating a player's individual quality from their team context. A midfielder who completes 92% of their passes might be elite---or might simply play for a dominant possession team where most passes are short and uncontested. A striker scoring 0.6 npxG per 90 might be a brilliant movement player---or might benefit from an elite creative midfield that generates world-class chances for anyone playing the number 9 role.

This is the fundamental attribution problem of player analytics: how do we attribute outcomes to individuals when those outcomes are the product of collective interaction?

15.9.2 Approaches to Disentangling Player and Team Effects

Several methodological approaches attempt to separate individual from team contributions:

1. Plus-minus models: Borrowed from basketball and hockey, plus-minus measures how a team's performance changes when a specific player is on the field versus off. The adjusted plus-minus (APM) or regularized APM (RAPM) uses ridge regression to estimate each player's marginal contribution while controlling for teammate and opponent effects:

$$ Y_{\text{stint}} = \sum_{i \in \text{on}} \beta_i - \sum_{j \in \text{opp}} \beta_j + \epsilon $$

However, plus-minus models are extremely data-hungry in soccer because substitutions are rare (only 3-5 per match), creating severe collinearity problems.

2. Transfer-based natural experiments: When a player moves to a new team, the change in that player's metrics (and the change in the old team's metrics) provides causal evidence about the player's individual contribution. If a team's xG drops by 0.3 per match after losing their star playmaker, that provides evidence that the playmaker was individually responsible for roughly that amount.

3. Within-player variation across team contexts: By tracking players across multiple seasons and teams, we can estimate the portion of a metric's variance that is attributable to the player (persistent across contexts) versus the team (changes when the player changes teams).

Callout --- The LeBron James Problem in Soccer: In basketball analytics, the "LeBron James problem" refers to the difficulty of evaluating a transformational player who elevates every teammate's performance. Soccer has analogous cases: when Virgil van Dijk joined Liverpool in January 2018, not only did his individual defensive metrics improve, but every other defender's metrics also improved. His organizing presence made the entire defensive unit better. Standard player metrics will underestimate such a player's total contribution because they miss the "raising all boats" effect.

15.9.3 Possesion-Adjusted and Context-Adjusted Metrics

A practical partial solution is to adjust metrics for contextual factors. Common adjustments include:

• Possession adjustment: Normalize defensive metrics by opponent possession time and attacking metrics by own-team possession time. A centre-back making 3 tackles per 90 when the team has 40% possession is working harder than one making 3 tackles per 90 with 60% possession.
• Opponent quality adjustment: Weight each match's metrics by the strength of the opponent faced (using Elo ratings from Chapter 16).
• Game-state adjustment: Account for the score at the time of each action. Players perform differently when leading, drawing, or trailing.

15.10 WAR/VAEP-Based Player Valuation

15.10.1 The Quest for a Unified Metric

The holy grail of player evaluation is a single number that captures a player's total contribution to winning---analogous to baseball's WAR (Wins Above Replacement). While no soccer metric has achieved the acceptance of baseball's WAR, several frameworks represent significant progress.

15.10.2 VAEP: Valuing Actions by Estimating Probabilities

VAEP (Valuing Actions by Estimating Probabilities), developed by Tom Decroos, Lotte Bransen, and colleagues at KU Leuven, assigns a value to every on-ball action based on how it changes the probability of scoring and conceding.

For each action $a_i$, VAEP computes:

$$ \text{VAEP}(a_i) = \Delta P_{\text{score}}(a_i) - \Delta P_{\text{concede}}(a_i) $$

where: - $\Delta P_{\text{score}}(a_i) = P(\text{score} | a_i, a_{i-1}, a_{i-2}) - P(\text{score} | a_{i-1}, a_{i-2})$ is the change in scoring probability - $\Delta P_{\text{concede}}(a_i) = P(\text{concede} | a_i, a_{i-1}, a_{i-2}) - P(\text{concede} | a_{i-1}, a_{i-2})$ is the change in conceding probability

The scoring and conceding probabilities are estimated using gradient-boosted tree models trained on historical event data, with features including action type, location, body part, and the context of the preceding actions.

A player's total VAEP over a season is the sum of all their individual action values. This can be normalized to a per-90 rate or expressed as total value.

Callout --- VAEP's Strengths and Limitations: VAEP's main strength is its generality: it values every type of on-ball action (passes, shots, dribbles, tackles, interceptions, clearances) on a common scale, making it possible to compare a midfielder's passing contribution directly with a striker's shooting contribution. Its main limitation is that it cannot value off-ball actions---movement, pressing runs, decoy runs, and spatial positioning---because these are not captured in event data. Players whose primary contribution is off-ball (think Thomas Muller's "Raumdeuter" role) will be systematically undervalued by VAEP.

15.10.3 WAR: Wins Above Replacement

The WAR framework in soccer follows the same logic as in baseball: estimate how many additional wins a player contributes relative to a replacement-level player (defined as a freely available player from the reserve team or lower division).

$$ \text{WAR}_i = \frac{\text{VAEP}_i - \text{VAEP}_{\text{replacement}}}{C} $$

where $C$ is the conversion factor from VAEP units to wins. Estimating $C$ requires calibrating total VAEP against actual match outcomes, and defining "replacement level" requires assumptions about the talent available outside the squad.

In practice, a top Premier League player might accumulate 3-5 WAR per season, meaning they contribute 3-5 additional wins over a full season compared to a freely available replacement. A league average starter typically has a WAR of 1-2, and a replacement-level player has a WAR near zero by definition.

15.10.4 Alternative Unified Frameworks

Beyond VAEP, several other frameworks attempt unified player valuation:

• xT (Expected Threat): Assigns a value to each zone on the pitch based on the probability of scoring from that zone. Player actions that move the ball to higher-value zones earn positive xT. Simple but limited to ball-moving actions.
• OBSO (Off-Ball Scoring Opportunity): Developed by Karun Singh, OBSO attempts to value off-ball positioning by computing the probability that a player in a given position could receive a pass and score.
• g+ (Goals Added): Developed by American Soccer Analysis for MLS, g+ values each action by its contribution to expected goals, using a possession-value framework.

Each framework makes different modeling choices and captures different aspects of player value. No single framework is definitively superior; the best practice is to triangulate across multiple methods.

Real-World Application: Several leading clubs---including Liverpool, Manchester City, and FC Bayern Munich---employ custom versions of action-value models as part of their recruitment process. These models do not replace scouting; rather, they serve as a first-pass filter to identify candidates from the vast global player pool, narrowing thousands of potential targets down to a shortlist of 20-50 that can be assessed through video analysis and live scouting.

Summary

This chapter has built a comprehensive toolkit for individual player evaluation:

The methods in this chapter form the backbone of modern player recruitment analytics. They provide the quantitative foundation upon which scouting departments build shortlists, negotiation teams set price targets, and coaching staffs monitor player development.

However, it is essential to close with a note of humility. Reducing the infinite complexity of human athletic performance to a set of numbers is an inherently reductive exercise. The best analytics departments treat metrics as one input---alongside video analysis, medical assessments, psychological profiling, and expert judgment---in a holistic evaluation process. Numbers can reveal what the eye misses, but the eye can perceive what numbers cannot capture: leadership, adaptability, mentality under pressure, and the intangible quality of making teammates better.

In Chapter 16, we will extend these individual metrics to understand how players interact within teams, moving from individual profiles to the chemistry of collective performance.

Chapter Notation Reference