Key Takeaways: Descriptive Statistics in Football

Core Concepts at a Glance

Central Tendency

Football Rule of Thumb: Use median for per-play metrics (yards per carry, per target), mean for totals and rates calculated from aggregates.

Variability Measures

Consistency Insight: Lower SD = more consistent performance. A QB with 250 yards/game (SD=30) is more reliable than one with 250 yards/game (SD=75).

Distribution Shapes

Right-Skewed (Positive)     Normal              Left-Skewed (Negative)
        ▲                    ▲                        ▲
       ██                   ███                      ██
      ███                  █████                    ███
     ████                 ███████                  ████
    █████ → tail         █████████               ████████
   Mode < Median < Mean  Mode = Median = Mean   Mean < Median < Mode

   Example: Rushing yards  Example: Pass %      Example: Veteran experience

Correlation Quick Reference

Critical Reminder: Correlation ≠ Causation. Third variables, reverse causation, and coincidence can create misleading correlations.

Z-Score Interpretation

z = (Value - Mean) / Standard Deviation

z = +2.0  →  Top ~2.5% (excellent)
z = +1.0  →  Top ~16% (above average)
z = 0.0   →  Average (50th percentile)
z = -1.0  →  Bottom ~16% (below average)
z = -2.0  →  Bottom ~2.5% (poor)

Application: Compare players across different statistical categories by converting to z-scores.

Essential Code Patterns

Calculate All Central Tendency Measures

def central_tendency_summary(data: pd.Series) -> dict:
    """Complete central tendency analysis."""
    return {
        'mean': data.mean(),
        'median': data.median(),
        'mode': data.mode().iloc[0] if len(data.mode()) > 0 else None,
        'trimmed_mean': stats.trim_mean(data.dropna(), 0.1)
    }

Robust Variability Analysis

def variability_summary(data: pd.Series) -> dict:
    """Complete variability analysis."""
    q1, q3 = data.quantile([0.25, 0.75])
    return {
        'std': data.std(),
        'variance': data.var(),
        'range': data.max() - data.min(),
        'iqr': q3 - q1,
        'cv': (data.std() / data.mean()) * 100
    }

Outlier Detection

def find_outliers_iqr(data: pd.Series) -> pd.Series:
    """Identify outliers using IQR method."""
    q1, q3 = data.quantile([0.25, 0.75])
    iqr = q3 - q1
    lower = q1 - 1.5 * iqr
    upper = q3 + 1.5 * iqr
    return data[(data < lower) | (data > upper)]

Correlation Matrix with Significance

def correlation_analysis(df: pd.DataFrame, cols: list) -> pd.DataFrame:
    """Calculate correlation matrix with interpretation."""
    corr_matrix = df[cols].corr()

    # Add interpretation
    def interpret(r):
        if abs(r) >= 0.8: return "Very Strong"
        if abs(r) >= 0.6: return "Strong"
        if abs(r) >= 0.4: return "Moderate"
        if abs(r) >= 0.2: return "Weak"
        return "Very Weak"

    return corr_matrix.applymap(interpret)

Decision Framework

Choosing the Right Statistic

START
  │
  ▼
Is your data categorical?
  │
  ├─ YES → Use MODE
  │
  └─ NO → Is there significant skew or outliers?
           │
           ├─ YES → Use MEDIAN and IQR
           │
           └─ NO → Use MEAN and STANDARD DEVIATION

Comparing Players or Teams

1. Same metric, same context: Direct comparison okay
2. Same metric, different contexts: Use z-scores or percentiles
3. Different metrics: Normalize to common scale (z-scores)
4. Over time: Consider trends, not just snapshots

Common Pitfalls to Avoid

Football Analytics Cheat Sheet

Quick Formulas

Mean = Σx / n
Median = Middle value (or average of two middle values)
SD = √(Σ(x - mean)² / (n-1))
IQR = Q3 - Q1
CV = (SD / Mean) × 100%
z-score = (x - mean) / SD
Outlier bounds = [Q1 - 1.5×IQR, Q3 + 1.5×IQR]

Interpretation Benchmarks

• Completion %: Mean ~62%, SD ~8% in FBS
• Yards per Carry: Mean ~4.0, SD ~1.5 (right-skewed)
• Points per Game: Mean ~28, SD ~10 in FBS
• Turnover Margin: Mean ~0, range typically -2 to +2 per game

Chapter Summary

1. Central tendency describes typical values; choose based on data distribution
2. Variability measures spread and consistency; essential for player evaluation
3. Distributions reveal data shape; skewness affects which statistics to use
4. Correlation quantifies relationships; strength and direction both matter
5. Z-scores enable cross-metric comparisons; standard deviations are the universal unit
6. Always consider context: sample size, competition level, era, and position

The best analysts don't just calculate statistics—they interpret them within the context of the game.