The VMR Decision Framework

The Variance-to-Mean Ratio (VMR) is your first diagnostic tool for choosing the right distribution. It tells you how "spread out" a player's outcomes are relative to their average—and that single number often determines which model you should use.

The VMR Formula

Variance-to-Mean Ratio (VMR)

VMR = σ² / μ

Excel: =VAR.S(A1:A20)/AVERAGE(A1:A20)

Where:

σ² = sample variance of historical outcomes
μ = mean of historical outcomes

Tip

In Excel, use VAR.S() (sample variance) rather than VAR.P() when working with historical game data. You're using past games as a sample to estimate the underlying process.

Interpreting VMR: The Decision Rules

VMR Range	Interpretation	Recommended Model
VMR < 0.8	Underdispersed (rare in sports)	Poisson may overestimate tails
VMR ≈ 1.0	Perfect Poisson fit	Use Poisson
VMR = 1.0–1.3	Acceptable Poisson fit	Use Poisson
VMR = 1.3–1.8	Moderate overdispersion	Consider Negative Binomial
VMR > 1.8	Strong overdispersion	Use Negative Binomial

Key Insight

The decision tree is simple: Calculate VMR first, then choose your distribution. This prevents systematic errors in your probability estimates.

VMR in Action: Three Player Profiles

Let's examine three players with different consistency profiles to see how VMR guides our choice:

Player A: The Consistent Goal-Line Back

Last 10 games (TDs): 1, 1, 0, 1, 1, 0, 1, 1, 0, 1

Mean: 0.70
Variance: 0.23
VMR: 0.33

Interpretation: VMR well below 1.0 indicates this player is more consistent than Poisson predicts. He's a steady producer—use Poisson (it will be conservative on tails).

Player B: The Boom-or-Bust Backup

Last 10 games (TDs): 2, 0, 0, 3, 0, 0, 2, 0, 3, 0

Mean: 1.00
Variance: 1.78
VMR: 1.78

Interpretation: VMR significantly above 1.0 indicates this player is boom-or-bust. He either has a big game or disappears. Use Negative Binomial to capture the fatter tails.

Player C: The Consistent Red Zone Target

Last 10 games (TDs): 1, 0, 1, 1, 0, 1, 0, 1, 1, 0

Mean: 0.60
Variance: 0.27
VMR: 0.45

Interpretation: VMR well below 1.0—another consistent player. Use Poisson.

Warning

Notice the pattern: Consistent players have VMR well below 1.0, while boom-or-bust players have VMR well above 1.0. The market often prices all players as if they follow Poisson (VMR = 1.0), creating opportunities when you can identify the boom-or-bust types.

Beyond VMR: The Zero-Inflation Check

VMR tells you about overdispersion, but it doesn't catch everything. You also need to check for excess zeros—situations where you see more zeros than the distribution predicts.

The Zero-Inflation Diagnostic

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

Poisson Zero Probability

P(X=0) = e^(-λ)

Excel: =EXP(-A1)

Rule of thumb: If observed zeros are 10-20% higher than Poisson predicts, consider Zero-Inflated Poisson (ZIP) or Hurdle models.

Example: Detecting Zero-Inflation

Backup RB Receptions (last 17 games):

Games with 0 receptions: 11 out of 17 (64.7%)
Mean receptions: 1.41
Poisson-predicted P(X=0): e^(-1.41) = 24.4%

Gap: 64.7% - 24.4% = +40.3 percentage points

This massive gap tells us the zeros aren't coming from the count process alone—there's a structural component (games where this RB never runs routes, only blocks).

Key Insight

If VMR > 1 AND your data shows excess zeros (more than Poisson predicts), this could indicate zero-inflation rather than (or in addition to) overdispersion. ZIP or Hurdle models may provide a better fit.

The Complete Decision Framework

Here's your step-by-step workflow for selecting the right distribution:

Step 1: Is this a count-based prop or continuous?
         ├── Continuous (yards, points with high volume) → Use Normal
         └── Count-based (TDs, receptions, strikeouts) → Continue...

Step 2: Calculate VMR from historical data
         ├── VMR ≤ 1.3 → Poisson candidate
         └── VMR > 1.3 → Negative Binomial candidate

Step 3: Check for excess zeros
         ├── Observed zeros ≤ Poisson prediction → Use distribution from Step 2
         └── Observed zeros >> Poisson prediction → Continue...

Step 4: Identify zero source
         ├── Structural (no opportunity) → Use ZIP
         └── Qualitative (any vs. none matters) → Use Hurdle

VMR Calculator

Use this tool to quickly assess whether a player's data fits Poisson or requires a more complex model:

Negative Binomial Calculator

Try the interactive calculator for this concept

Open Tool

Tip

When using the calculator, input the player's game-by-game counts. The tool will compute VMR and recommend the appropriate distribution automatically.

📝 Exercise

Instructions

Exercise: VMR Analysis Practice

Calculate VMR for the following player data and determine which distribution to use.

Player Data (Touchdowns over 20 games):

Player A (Consistent): 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1
Player B (Moderate): 2, 0, 1, 0, 2, 1, 0, 1, 2, 0, 1, 0, 2, 1, 0, 1, 2, 0, 1, 0
Player C (Extreme): 3, 0, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0, 3, 0, 0, 0, 2, 0, 3

Player A has Mean ≈ 0.65 and Variance ≈ 0.24. What is the VMR and recommended model?

Player B has Mean ≈ 0.85 and Variance ≈ 0.66. What is the VMR and recommended model?

Player C has Mean ≈ 0.90 and Variance ≈ 1.67. What is the VMR and recommended model?

For Player C, Poisson predicts P(X=0) = e^(-0.90) ≈ 40.7%. If the observed zero rate is 65%, what does this suggest?