When Poisson Breaks Down: Overdispersion

Poisson assumes Var(X) = λ—the variance equals the mean. In real sports data, you often see overdispersion: the variance is larger than the mean. This usually happens when:

• Outcomes cluster because the rate isn't truly constant (game script)
• Opportunities are not independent
• The player is genuinely boom-or-bust

The Variance-to-Mean Ratio (VMR)

A simple diagnostic is the variance-to-mean ratio (VMR):

VMR Interpretation Guide

Calculating VMR in Excel

For game-by-game counts in a range (e.g., A1:A20):

Mean (λ̂):     =AVERAGE(A1:A20)
Sample Var:   =VAR.S(A1:A20)
VMR:          =VAR.S(A1:A20)/AVERAGE(A1:A20)

Why Overdispersion Matters for Betting

When Poisson underestimates tail probabilities, you systematically misprice extreme outcomes:

Translation: Overdispersed players are more boom-or-bust than Poisson suggests. They have more 0-event games AND more 3+ event games.

Example: Detecting Overdispersion

Player A: Consistent Receiver Last 15 games receiving TDs: 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1

Mean = 0.67 TDs/game
Sample Variance = 0.24
VMR = 0.24 / 0.67 = 0.36

VMR = 0.36 < 1.0 → Actually underdispersed (very consistent). Poisson works well here.

Player B: Boom-or-Bust Back Last 15 games rushing TDs: 0, 0, 3, 0, 2, 0, 0, 0, 3, 1, 0, 2, 0, 0, 3

Mean = 0.93 TDs/game  
Sample Variance = 1.64
VMR = 1.64 / 0.93 = 1.76

VMR = 1.76 > 1.5 → Overdispersed. This player is boom-or-bust. Consider Negative Binomial (Chapter 9).

Zero-Inflation: A Related Problem

Overdispersion isn't the only common issue with count data in sports. For props with especially low λ (e.g., home runs or anytime touchdowns), you might see even more zeros than Poisson expects—not just from variance, but from structural factors like a player being scripted out of the game entirely.

This is called zero-inflation, and it's addressed by specialized models:

• Zero-Inflated Poisson (ZIP)
• Hurdle distributions

These models handle zero-inflation by separately modeling the probability of a zero versus positive counts.

A Simple Check for Zero-Inflation

Compare your observed P(X=0) to the Poisson-predicted e^(-λ):

Decision Tree: Which Model to Use?

                    Calculate VMR
                         │
            ┌────────────┼────────────┐
            │            │            │
       VMR < 0.8    VMR ≈ 1.0    VMR > 1.3
            │            │            │
       Investigate   Poisson is    Consider
       for patterns    fine     Negative Binomial
                         │
                         ↓
              Check for excess zeros
                         │
              ┌──────────┴──────────┐
              │                     │
        Observed P(0)          Observed P(0)
        matches e^(-λ)         >> e^(-λ)
              │                     │
         Use Poisson          Consider ZIP
                              (Chapter 11)

Practical Implications for Betting

When to Stick with Poisson

• VMR is close to 1.0
• Player has consistent role and usage
• Game script doesn't dramatically affect opportunity
• No structural reasons for excess zeros

When to Consider Alternatives

• VMR > 1.3 → Negative Binomial (Chapter 9)
• Excess zeros from structural factors → ZIP/Hurdle (Chapter 11)
• Very high λ (> 20) → Normal approximation