The Two Numbers That Define Everything
Every normal distribution is defined by just two numbers. Get these right, and you can calculate exact probabilities for any over/under line. Get them wrong, and your entire analysis falls apart.
1. Mean (μ) — The Average
The mean is the center of the bell curve. It's where outcomes are most likely to cluster.
Mean (Average)
μ = Sum of all values ÷ Number of observations=AVERAGE(A1:A50)For LeBron James in 2023-24:
- μ = 25.7 points (his scoring average over 71 games)
This tells you: If you had to guess LeBron's point total for any random game with no other information, 25.7 is your best single estimate.
Note
For Prop Betting
The mean is your starting point for any projection. But raw season averages often aren't good enough—you need to adjust for context (matchup, pace, rest, injuries). We'll cover this in Lesson 6.
2. Standard Deviation (σ) — The Spread
Standard deviation measures how much outcomes vary around the average. This is what you learned in Chapter 6.
Standard Deviation
σ = √[Σ(xᵢ - μ)² / n]=STDEV.S(A1:A50)For LeBron James in 2023-24:
- σ = 6.7 points
This tells you: On a typical night, LeBron's scoring varies by about 6.7 points from his average.
What Standard Deviation Means in Practice
| σ Value | Interpretation | Betting Implication |
|---|---|---|
| Small σ | Consistent performer | More predictable outcomes, easier to bet |
| Large σ | Volatile performer | Wider range of outcomes, harder to predict |
Comparing Two Players: Same Average, Different Variance
Key Insight
This is crucial for prop betting. Two players can have the same average but completely different risk profiles.
Player A (Consistent)
- Average: 25 points
- Standard Deviation: 5 points
- Very consistent. Scores 20-30 points in 68% of games.
Player B (Volatile)
- Average: 25 points
- Standard Deviation: 10 points
- Very volatile. Scores 15-35 points in 68% of games.
Same average, but Player A is far more predictable.
If the line is 24.5 points:
- Player A goes over about 55% of the time
- Player B goes over about 52% of the time
The difference? Player B's wider distribution means more outcomes fall on both extremes, so the probability of landing just above the line is lower.
Tip
Lower variance = More predictable = Better for betting
When two players have similar averages, prefer betting on the more consistent player if the line is close to their mean.
How μ and σ Shape the Bell Curve
The mean shifts the curve left or right:
- Higher μ → Curve shifts right (higher outcomes more likely)
- Lower μ → Curve shifts left (lower outcomes more likely)
The standard deviation changes the width:
- Smaller σ → Taller, narrower curve (outcomes cluster tightly)
- Larger σ → Shorter, wider curve (outcomes spread out)
Visual Comparison
| Player | μ (Average) | σ (Std Dev) | Curve Shape |
|---|---|---|---|
| Player A | 25 | 5 | Tall and narrow |
| Player B | 25 | 10 | Short and wide |
| Player C | 30 | 5 | Tall, narrow, shifted right |
Sample Size: How Many Games Do You Need?
Your estimates of μ and σ are only as good as your data.
For Estimating Mean (μ)
| Sample Size | Reliability |
|---|---|
| 10 games | Minimum viable |
| 20 games | Recommended |
| 30+ games | Ideal (full season) |
For Estimating Standard Deviation (σ)
| Sample Size | Reliability |
|---|---|
| 15 games | Minimum viable |
| 25 games | Recommended |
| 40+ games | Ideal |
Warning
Standard deviation is harder to estimate than mean. It requires more data to stabilize.
For new situations (role change, injury return, new team):
- Use 10-15 games in the new situation
- Widen your σ by 15-25% to account for uncertainty
- Be more conservative with bet sizing until you have 20+ games
Calculating μ and σ in Excel
Here's how to calculate both parameters from a player's game log:
Step 1: Enter the data
Put each game's stat in a column (e.g., A1:A30 for 30 games of points data)
Step 2: Calculate the mean
=AVERAGE(A1:A30)
Step 3: Calculate the standard deviation
=STDEV.S(A1:A30)
Use STDEV.S for sample data (a subset of games), which is almost always what you have. Use STDEV.P only if you have the complete population of data.
📝 Exercise
Instructions
A player has scored the following points in their last 5 games: 22, 28, 25, 30, 20. Calculate the mean.
What is the mean (average) of these 5 games?
📝 Exercise
Instructions
Two players both average 20 points per game. Player X has a standard deviation of 3 points. Player Y has a standard deviation of 8 points.
If you're betting Over 19.5 points, which player gives you the more reliable bet?
📝 Exercise
Instructions
A player just joined a new team 8 games ago. You've calculated their standard deviation to be 6.5 points in this new role.
What adjustment should you make when using this σ for betting?