Introduction to the Normal Distribution
It's January 2023, and you're looking at a prop bet on Buffalo Bills quarterback Josh Allen's passing yards against the Kansas City Chiefs in the playoffs. The line is 275.5 yards at -110 on both sides.
You've done your homework:
- Allen averages 268 yards per game this season
- In high-stakes games, he averages 285 yards
- Against top defenses like KC, he averages 275 yards
- His standard deviation is 85 yards
So... should you bet the over or under?
Your gut says, "He averages 275, the line is 275.5, so it's basically a coin flip." And you'd be right—but only if you understand why it's a coin flip.
This is where the normal distribution comes in. It's the mathematical tool that converts your estimates (average and standard deviation) into precise probabilities.
Key Insight
Without the normal distribution, you're guessing. With it, you can calculate that Allen has exactly a 49.8% chance of going over 275.5 yards—which means the under at -110 (implied 52.4%) is slightly -EV.
But what if the line was 265.5 instead? Or 285.5?
The normal distribution tells you exactly how the probability changes as the line moves. And that's the difference between profitable betting and expensive guessing.
What Is the Normal Distribution?
The normal distribution is a pattern that shows up everywhere in nature and sports: heights, test scores, shooting percentages, passing yards.
It's called the "bell curve" because of its shape:
- Most outcomes cluster around the average (the peak of the bell)
- Outcomes get less likely as you move away from the average
- The curve is symmetric—equally likely to be above or below average
- Extreme outcomes are rare but possible
A Real Example: LeBron James's Points (2023-24 Season)
Over the 2023-24 regular season, LeBron averaged 25.7 points per game (71 games) with a standard deviation of 6.7 points.
His scoring pattern looked like this:
| Range | Description | Frequency |
|---|---|---|
| 19-32 points | Around his average (within ±1 SD) | Most games |
| 12-19 or 32-39 points | 1-2 SD away | Some games |
| Under 12 or over 39 points | 2+ SD away | Rare games |
This is a normal distribution. The bell curve describes exactly how likely each outcome is.
Why This Matters for Props
When you bet a prop, you're betting on where the outcome will fall on the bell curve.
- Over bets: You're betting the outcome lands in the right half (or right tail) of the curve
- Under bets: You're betting the outcome lands in the left half (or left tail) of the curve
The normal distribution tells you exactly how much of the curve is on each side of the line.
Note
Example: LeBron Points at 24.5
Your model: μ = 25.7, σ = 6.7 (his 2023-24 actual stats)
The line (24.5) is slightly below your average (25.7). That means more of the bell curve is above the line than below it. The over should be more likely than the under.
How much more likely? That's what the math tells us in upcoming lessons.
What You'll Learn in This Chapter
By the end of Chapter 7, you'll understand:
- The two numbers that define everything: Mean (μ) and standard deviation (σ)
- The 68-95-99.7 rule: Your mental model for quick assessments
- How to calculate exact probabilities: Using Excel's NORM.DIST function
- Z-scores: Standardizing the distribution for comparison
- When to use normal distributions: And when NOT to
- How to analyze alternate lines: Finding value where books misprice
- Common mistakes to avoid: Pitfalls that cost bettors money
The Volume Rule: When to Use Normal Distributions
Warning
Critical Concept
Normal distributions work best for high-volume, continuous stats—think of passing yards or points as a bucket being filled; the water level can rise to any height.
They don't work well for low-volume "event" stats like touchdowns or home runs, which are discrete. You either have 1 or you have 2; there's no such thing as 1.5 touchdowns.
For those rare, chunky events, use Poisson (Chapter 8).
Stats That Fit Normal Distributions Well
| Stat | Sport | Why It Works |
|---|---|---|
| Passing Yards | NFL | High volume, continuous flow |
| Points | NBA | Many scoring opportunities |
| Total Bases | MLB | Aggregated hitting outcomes |
| Points + Rebounds + Assists (PRA) | NBA | Combined high-volume stats |
| Rushing Yards | NFL | Many carries, yards accumulate |
Stats That DON'T Fit Normal Distributions
| Stat | Sport | Use Instead |
|---|---|---|
| Touchdowns | NFL | Poisson |
| Home Runs | MLB | Poisson |
| Goals | NHL/Soccer | Poisson |
| 3-Pointers Made | NBA | Poisson |
| Steals | NBA | Negative Binomial |
📝 Exercise
Instructions
Test your understanding of when normal distributions apply.
A quarterback's passing yards prop is set at Over/Under 265.5. Why is the normal distribution appropriate for this stat?
📝 Exercise
Instructions
Identify the correct distribution for each prop type.
Which of these props should NOT be modeled with a normal distribution?