Common Mistakes and Chapter Exercises

Even sharp bettors make mistakes with normal distributions. Here are the seven most costly errors—and how to avoid them.

Mistake 1: Using Normal Distributions for Low-Volume Stats

The Problem: You try to use NORM.DIST for a prop on anytime touchdowns or home runs.

Why It's Wrong: These are discrete, low-frequency events. Normal distributions assume continuous data with high volume. Touchdowns and home runs follow Poisson distributions (Chapter 8), not normal.

Warning

Solution: Use normal distributions only for high-volume, continuous stats: yards, points, total bases. For counting stats (TDs, HRs, 3-pointers, goals), use Poisson.

Quick Reference: Which Distribution?

Stat	Distribution	Why
NFL Passing Yards	Normal	High volume, continuous
NBA Points	Normal	Many scoring opportunities
Anytime Touchdown	Poisson	Discrete, low-count events
Home Runs	Poisson	Rare events (0, 1, 2...)
3-Pointers Made	Poisson	Discrete counts
Completion %	Neither	Bounded (0-100%)

Mistake 2: Using Season Averages Without Context

The Problem: You use a player's season average (μ = 28 points) without adjusting for matchup, pace, or rest.

Why It's Wrong: The normal distribution is only as good as your inputs. Books adjust for context—you must too.

Tip

Solution: Always adjust μ for:

Opponent strength
Pace of play
Rest/fatigue
Home/away
Teammate availability
Game script (spread implications)

Your edge comes from better context modeling than the market.

Mistake 3: Ignoring Player Variance Changes

The Problem: You calculate σ from a player's full season, but their variance has changed due to a role change or injury.

Why It's Wrong: A player's variance isn't constant. Role changes, injuries, and aging all affect consistency. Using outdated σ leads to wrong probabilities.

Solution:

Weight recent games more heavily
Use last 15-20 games for σ, not full season
If a player's role changed, recalculate σ from scratch

Mistake 4: Assuming Perfect Symmetry

The Problem: You assume the normal distribution is perfectly symmetric for all stats.

Why It's Wrong: Some stats are slightly skewed. Rushing yards can have a long right tail (big runs). Points can have a left boundary (can't score negative).

Solution: For lines within ±2σ of the mean, normal distributions work great. For extreme lines (±3σ or more), be more conservative with your edge estimates.

Mistake 5: Overconfidence in Small Edges

The Problem: Your model shows 53% probability, market shows 52.4%. You bet big because "it's +EV."

Why It's Wrong: A 0.6% edge is technically +EV, but it's within your model's margin of error. Small edges require huge sample sizes to validate (Chapter 6).

Key Insight

Solution: Only bet when your edge is:

2%+ for standard props
5%+ for high-variance props

Smaller edges might be model noise, not real edges.

Mistake 6: Forgetting About Vig

The Problem: You calculate that a prop is 50/50, so you think it's fair.

Why It's Wrong: At -110 odds, a 50/50 prop is -EV on both sides. You need 52.4% probability to break even.

Solution: Always compare your probability to the implied probability (including vig), not to 50%. A 51% probability at -110 is still -EV.

Odds	Break-Even %
-110	52.4%
-115	53.5%
-120	54.5%
-130	56.5%

Mistake 7: Using Normal for Bounded Stats

The Problem: You use normal distribution for completion percentage or field goal percentage.

Why It's Wrong: These stats are bounded (0-100%). Normal distributions assume unbounded data and can predict impossible values (like 110% completion rate).

Solution: For bounded percentages, use beta distributions or logistic models. Or stick to volume stats (yards, points) where normal distributions excel.

Key Takeaways

Key Insight

Eight Things to Remember:

Normal distributions model high-volume, continuous stats — yards, points, total bases
Two numbers define everything: μ and σ — Get these right and you can calculate exact probabilities
The 68-95-99.7 rule is your mental model — Quick assessments without Excel
Excel does all the math: NORM.DIST — For under: =NORM.DIST(line, μ, σ, TRUE). For over: =1-NORM.DIST(line, μ, σ, TRUE)
Context adjustments are where edges come from — Don't use raw season averages
Alternate lines often have value — Books may misprice lines away from the main number
Don't use normal for discrete counts — TDs, HRs, 3-pointers use Poisson (Chapter 8)
Small edges need large sample sizes — A 1-2% edge requires 500+ bets to validate

Quick Reference: Normal Distribution Formulas

Calculation	Excel Formula
Probability of UNDER	`=NORM.DIST(line, μ, σ, TRUE)`
Probability of OVER	`=1-NORM.DIST(line, μ, σ, TRUE)`
Z-score	`=(line - μ) / σ`
Value at percentile	`=NORM.INV(percentile, μ, σ)`
Random outcome	`=NORM.INV(RAND(), μ, σ)`
68% confidence interval	μ ± σ
95% confidence interval	μ ± 1.96×σ

For each stat, identify the correct distribution.

Which of these stats should NOT use a normal distribution?

📝 Exercise

Instructions

Exercise 7: Z-Score Interpretation

You project μ = 78.0 yards, σ = 22.0 yards. The line is 101.5 yards.

Calculate the z-score and assess the difficulty.

What is the z-score and what does it tell you?

📝 Exercise

Instructions

Exercise 8: Small-Edge Discipline

Your model says P(Over) = 53.2% at -110.

Market break-even at -110 is 52.4%.

Your edge is 0.8%. What's the right move?

Moving Forward

You now have a powerful tool for analyzing continuous prop stats. The normal distribution converts your estimates (average and variance) into precise probabilities.

But not all props are continuous. Many are discrete counts:

Touchdowns (0, 1, 2, 3...)
Strikeouts (0, 1, 2, 3...)
Three-pointers made (0, 1, 2, 3...)
Hits (0, 1, 2, 3...)
Goals (0, 1, 2, 3...)

For these stats, we need a different distribution: the Poisson distribution.

In Chapter 8, you'll learn how Poisson distributions model counting stats—events that happen a certain number of times per game. This is essential for touchdown props, strikeout props, and any other discrete counting stat.

After that, Chapter 9 covers the Negative Binomial distribution for overdispersed counts (when variance is higher than Poisson predicts), and Chapter 10 ties everything together with correlation and covariance.

You're building a complete statistical toolkit. Keep going.