Estimating the Dispersion Parameter (r)
Once you've determined that a player's data is overdispersed (VMR > 1.3), the next step is to estimate the dispersion parameter r. This value captures the player's inherent boom-or-bust nature and is essential for calculating Negative Binomial probabilities.
The Method of Moments Formula
Dispersion Parameter (r)
r = μ² / (Variance - μ)=AVERAGE(range)^2 / (VAR.S(range) - AVERAGE(range))This formula derives from rearranging the Negative Binomial variance equation:
Variance = μ + (μ² / r)
Solving for r:
Variance - μ = μ² / r
r = μ² / (Variance - μ)
Warning
Important: This formula only works when Variance > Mean (VMR > 1.0). If variance equals or is less than the mean, the denominator becomes zero or negative, and you should use Poisson instead.
Worked Example: Boom-or-Bust Running Back
Let's calculate r for Running Back B from the previous lesson:
Game Log: 2, 0, 0, 3, 0, 0, 2, 0, 3, 0
| Metric | Calculation | Value |
|---|---|---|
| Mean (μ) | Sum/Count = 10/10 | 1.00 |
| Variance | VAR.S calculation | 1.78 |
| VMR | 1.78/1.00 | 1.78 ✓ Overdispersed |
Now calculate r:
r = μ² / (Variance - μ)
r = 1.00² / (1.78 - 1.00)
r = 1.00 / 0.78
r = 1.28
Note
r = 1.28 indicates a moderately boom-or-bust player. This value will be used with μ to calculate probabilities using NEGBINOM.DIST.
Interpreting r Values
| r Value | Player Profile | Typical Players |
|---|---|---|
| r < 1 | Extremely boom-or-bust | Backup specialists, gadget players |
| r = 1-2 | Highly volatile | TD-dependent WR3s, streaky shooters |
| r = 2-5 | Moderately volatile | Most backup RBs, role players |
| r = 5-10 | Slightly volatile | Consistent starters with occasional big games |
| r > 10 | Near-Poisson | Very consistent performers |
The Critical Insight: r Reflects Personality, Not Matchups
Key Insight
The dispersion parameter r reflects a player's inherent boom-or-bust nature. It doesn't change game-to-game. When adjusting for matchups, you adjust μ (the expected count), not r. Think of r as the player's "personality," and μ as their expected output in a given situation.
Example: Matchup Adjustment
Scenario: Your boom-or-bust RB (r = 1.28) faces a weak run defense.
| Parameter | Base Value | Matchup Adjusted | What Changed |
|---|---|---|---|
| μ | 1.00 | 1.15 (+15%) | ✓ Increased for plus matchup |
| r | 1.28 | 1.28 (unchanged) | ✗ Personality stays constant |
The player is still boom-or-bust (same r), but you expect more TDs on average (higher μ) due to the favorable matchup.
Excel Implementation
Complete Formula Set
// In cells with historical game data in B2:B21 (20 games)
Mean (μ):
=AVERAGE(B2:B21)
Variance:
=VAR.S(B2:B21)
VMR:
=VAR.S(B2:B21)/AVERAGE(B2:B21)
r (Dispersion Parameter):
=AVERAGE(B2:B21)^2 / (VAR.S(B2:B21) - AVERAGE(B2:B21))
// Error handling version (returns "Use Poisson" if not overdispersed):
=IF(VAR.S(B2:B21)>AVERAGE(B2:B21),
AVERAGE(B2:B21)^2/(VAR.S(B2:B21)-AVERAGE(B2:B21)),
"Use Poisson")
Sample Size Requirements
Warning
Minimum Data Requirement: You need at least 15-20 games of data to reliably estimate r. With smaller samples, the variance estimate is too noisy to trust, and you should default to Poisson or use empirical frequencies (just counting historical outcomes directly).
What to Do with Limited Data
| Games Available | Recommended Approach |
|---|---|
| 1-5 games | Use Poisson with qualitative adjustment |
| 6-10 games | Use Poisson, note uncertainty |
| 11-15 games | Calculate VMR; if clearly >1.5, consider NB |
| 16+ games | Full VMR and r calculation |
| 30+ games | High confidence in r estimate |
Practice Calculation
Let's work through another example:
Streaky NBA Shooter (Three-Pointers Made)
Game log (20 games): 1, 4, 0, 2, 6, 1, 0, 3, 0, 5, 2, 0, 4, 1, 0, 3, 0, 5, 2, 1
Step 1: Calculate Mean
Sum = 1+4+0+2+6+1+0+3+0+5+2+0+4+1+0+3+0+5+2+1 = 40
Mean = 40/20 = 2.00
Step 2: Calculate Variance
Using Excel: =VAR.S({1,4,0,2,6,1,0,3,0,5,2,0,4,1,0,3,0,5,2,1})
Variance = 3.68
Step 3: Calculate VMR
VMR = 3.68 / 2.00 = 1.84
→ VMR > 1.3, proceed with Negative Binomial
Step 4: Calculate r
r = μ² / (Variance - μ)
r = 2.00² / (3.68 - 2.00)
r = 4.00 / 1.68
r = 2.38
Interpretation: This shooter has moderate boom-or-bust tendencies (r = 2.38). They're not as volatile as our touchdown RB (r = 1.28), but still overdispersed enough to warrant Negative Binomial modeling.
📝 Exercise
Instructions
A tight end has the following touchdown data over 16 games:
0, 1, 0, 2, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 3, 0
Calculate the mean, variance, VMR, and dispersion parameter r.